MASARYKOVA UNIVERZITA
PŘÍRODOVĚDECKÁ FAKULTA
Habilitation Thesis
BRNO 2018 ZDENĚK SVOBODA
MASARYKOVA UNIVERZITA
PŘÍRODOVĚDECKÁ FAKULTA
ASYMPTOTIC
PROPERTIES OF
FUNCTIONAL-DIFFERENTIAL
EQUATIONS
WITH DELAY
Habilitation Thesis
Zdeněk Svoboda
Brno 2018
Bibliografický záznam
Autor: RNDr. Zdeněk Svoboda CSc.
Přírodovědecká fakulta, Masarykova univerzita
Ústav matematiky a statistiky
Název práce: ASYMPTOTIC PROPERTIES FUNCTIONAL-DIFFERENTIAL
EQUATIONS WITH DELAY
Akademický rok: 2017/2018
Počet stran: 6 + 233
Klíčová slova: zpožděné funkcionálně diferenciálná rovnice; kladná řešení;
zpožděné maticové funkce; toplogická metoda; princip retraktu;
exponenciální stabilita; asymptotický rozklad
Bibliographic Entry
Author: RNDr. Zdeněk Svoboda CSc.
Faculty of Science, Masaryk University
Department of Mathematics and Statistics
Title of Thesis: ASYMPTOTIC PROPERTIES FUNCTIONAL-DIFFERENTIAL
EQUATIONS WITH DELAY
Academic Year: 2017/2018
Number of Pages: 6 + 233
Keywords: delayed functional differential equation; positive solutions; delayed
matrix functions; topological mathod; retract principle;
exponential stability; asymptotic expansion;
Abstrakt
V této habilitační práci jsou shrnuty výsledky mých 13 vybraných vědeckých prací,
které jsou věnovány problematice funkcionálních diferenciálních rovnic a jejich asymptotickým
vlastnostem. Výber těchtopublikací může být rozdělen do 3 částí. První se
zabývá lineárními systémy s konstantními koeficienty a konstantním zpožděním. Druhá
část je věnována toplogické metodě a jejímu užití při studiu asymptotických vlastností
zpožděných funkcionálně diferenciálních rovnic. Třetí oblast výzkumu je věnována exponenciální
stabiltě zpožděných funkcionálně diferenciálních rovnic.
Abstract
In this habilitation thesis are summarized the results of my 13 selected scientific papers
which are devoted to the problems of functional differential equations and their asymptotic
properties. The selection of this papers can be divided into three parts. The first deals with
linear systems with constant coefficients and constant delays. The second part is devoted
to the topological method and its use in the study of asymptotic properties of delayed
functional differential equations. The third area of research is devoted to the exponential
stability of delayed functional differential equations.
Contents
1 Introduction 2
2 Fundamental matrix for linear systems 5
2.1 Linear first-order systems . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Linear second-order systems . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Asymptotic properties of the delayed matrix functions and Lambert function 10
3 Topological method 16
3.1 The system of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 The retract method and the Lyapunov method for p-RFDE’s . . . . . . . 17
3.3 Retract principle for neutral functional differential equations . . . . . . . 19
4 Applications to nonlinear systems 24
4.1 Asymptotic expansion of solution . . . . . . . . . . . . . . . . . . . . . 25
4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3 Positive solutions of nonlinear system . . . . . . . . . . . . . . . . . . . 29
5 Positive solutions of a linear system 32
5.1 A scalar equation with discrete delays . . . . . . . . . . . . . . . . . . . 34
5.2 Positive solutions to a scalar equation in the critical case . . . . . . . . . 36
5.3 A scalar equation with distributed delay . . . . . . . . . . . . . . . . . . 38
5.4 Existence of decreasing positive solutions to linear differential equations
of neutral type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6 Exponential stability 41
6.1 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.3 Statement of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
List of commented research papers . . . . . . . . . . . . . . . . . . . . . . . . 50
1
Chapter 1
Introduction
This thesis is based on a selection of the author’s papers since 1998 dealing with the
asymptotic properties of delayed differential equations. The selection can be divided into
three parts. The first deals with linear systems with constant coefficients and constant
delays. In this section, definitions of delayed matrix functions are introduced as a new
formalization of the well-known step-by-step method. It is a concretization of the previously
used term of the fundamental matrix, see [34]. One of the many motivations is the fact that
these systems are canonical equations based on which the entire class of delayed equations
can be transformed. These results are shown by F. Neuman [49][50], and others [35], [63].
Papers [38], [39] bring an integral representation of the solutions to a first-order system
defining a delayed exponential of matrix and bringing the basic results. This formalization
was widely applied, e.g., in boundary-value problems, control problems, and stability
problems, modification to discrete equations was performed, generalizations to the case
of several delays were developed, etc. (see papers [6], [7], [36], [47], [48]). In the
papers [46], [51] the systems with more constant delays and commutative matrices coefficients
are investigated and the generalization of delayed exponential of matrix is given.
For a second-order system, analogously, two functions are defined - delayed matrix sine
and cosine [16]. In [a12] the relationships are given between these functions, which can
be understood as a generalization of the well-known Euler’s formula, because, for the zero
delay, the relationship for delayed matrix functions is reduced to this identity.
First, it is possible to use this identity as a motivation for studying more general systems.
The representation of the solutions to the second-order system of n equations is obtained
from the representation of the first n components of the solution to the related initial
problem of a first-order linear system of 2n equations, see [a12].
Second it is possible to obtain the asymptotic properties of the delayed matrix functions
sine and cosine from the asymptotic properties of the delayed exponential of matrix. The
results given in the papers [59], [a6] describes the asymptotic properties of a delayed matrix
exponential for k → ∞ proving that the sequence of values of the delayed matrix exponential
at the nodes is approximately represented by a geometric progression. A constant matrix is
found such that its matrix exponential is the “quotient” factor that depends on the principal
branch of Lambert function, see [a6]. In the paper [a13] it was proved that the spectral norm
of the delayed matrix sin and cosine are unbounded for t → ∞ by asymptotic properties of
delayed matrix exponential.
2
CHAPTER 1. INTRODUCTION 3
And finally, a possible application is shown of delayed matrix functions to formalizing
the solution of partial differential equations with constant delay, see [15].
The second part is devoted to the topological method and its use in the study of asymptotic
properties of delayed functional differential equations. We describe a modification of
the Wazewski’s topological method for ordinary differential equations [33] with new use of
the topological method applied to delayed functional-differential equations with bounded
delay. Introduced by Rybakowski [53], this modification uses a topological method for
a system of curves. This idea is further adapted to functional differential systems with
unbounded delay and finite memory [a3]. This type of functional-differential equations
was given in [43] by using of the p function. Later, the topological principle was used to
study the asymptotic properties of neutral differential equations. This was made possible
by introducing a system of subsidiary inequalities in the definition of a polyfacial set.
These modifications of the topological method were also used to study the asymptotic
properties of delayed systems. First, it was used for the asymptotic integration of a
functional-differential equation in which the solution is represented by an asymptotic
series. Using a concrete example of one equation, the existence is shown of different
asymptotic properties to the solution to this equation as depending on the magnitude of the
delay [a11]. Another application of this method made it possible to obtain criteria for the
existence of positive solutions to functional differential equations with unbounded delay.
The criteria given in [a3], [a4], [a5], [58] generalize the criteria for delayed differential
equations with bonded delay.
The topological principle was used to study the asymptotic properties of neutral differential
equations as a tool for the verification of a new existence criterion for the positive
solutions of one neutral equation, see [a9]. In order to achieve a continuous dependence
of solutions, which are moreover continuously differentiable, the definition of a system of
initial functions fulfilling the sewing condition was introduced.
The third area of research is devoted to the exponential stability of delayed functional
differential equations. The problem has been studied by a number of authors and new
results are presented in this section, that are generalizations of previous criteria.
Some existing models with delayed equations.
Delay differential equation arise in many applications in different fields being described in
books such as [41] or [62] with many examples. In [41] part of models is introduced by
the equation of a showering person
˙Tm(t) = −κ(Tm(t −h)−Td), (1.1)
Tm(t) denotes the water temperature at the mixer output, h is the positive constant time that
the water takes to go from the mixer output to the top of the person’s head. Td is the desired
water temperature on the head of the showering person and the coefficient κ depends the
person’s temperament. The paper also studies the modification of the well-known equation
with delayed terms because the new equations are more accurate models of the studied
processes. Hutchinson [62] proposed a logistic delay population model of the form
˙x(t) = γx(t) 1−
x(t −τ)
K
(1.2)
CHAPTER 1. INTRODUCTION 4
where constant γ is the coefficient of linear growth, the constant K is the average population
size related to the ability of the environment to sustain the population, x(t) is the population
size at time t. The delay τ > 0 means that the food resources at time t are determined by
the population size at time T −h. For more details, see [25].
Putting x(t) = K(1+y(t/h), we obtain a new equation for y(t);
˙y(t) = −γhy(t −1)(1+y(t)),
which is encountered in the number theory [65]. Gurney et al. [27] proposed a DDE to
describe the Nicolson blowflies model.
˙N(t) = aN(t −τ)ebN(t−τ)
−dN(t) (1.3)
where N(t) denotes the size of the population at time t, a is the maximum per capita
rate of producing eggs per day, d is the death rate in the adult population, and τ is the time
taken from the birth of a member until it becomes mature.
One of the classical equations of non-linear dynamics was formulated by a Dutch
physicist Van der Pol. Originally, it was a model for an electrical circuit with a triode valve,
and was later extensively studied as a prototype of a rich class of dynamical behavior. This
model is described by the equation
d2x
dt2
− µ(1−x2
)
dx
dt
+x = 0
which may be studied as a system of two first-order equations
˙x = y
˙y = µ(1−x2
)y−x.
Van der Pol found stable oscillations, which he called relaxation-oscillations and which
are now known as limit cycles. [2] also studies a Van der Pol’s oscillator with delay ( feed
back ). This is the equation
¨x(t)+x(t)−ε(1−x2
(t))˙x(t)+kx(t −τ) = 0,
which is also studied by a parameter-expanding method in [40].
Chapter 2
Fundamental matrix for linear systems
with constant coefficients and constant
delay
This chapter is devoted to linear systems with constant coefficients and delay. Our results
are stated in theorems of this chapter are proved in papers [a6], [a12], [a13].
The application of the well-known “step by step” method to solving ordinary differential
equations has recently been in the case of linear first-order systems with single constant
delay and with constant matrix, formalized using special types of delayed matrices (delayed
matrix exponential, delayed matrix sine and delayed matrix cosine). These matrix functions
are defined on intervals (k −1)τ ≤ t < kτ, k = 0,1,... (where τ > 0 is a delay) as matrix
polynomials, and are continuous at the nodes t = kτ, see [39], [16]. The papers [59], [a6]
studies the asymptotic properties of a delayed matrix exponential for k → ∞ proving that
the sequence of values of the delayed matrix exponential at the nodes is approximately
represented by a geometric progression. A constant matrix is found such that its matrix
exponential is the “quotient” factor that depends on the principal branch of Lambert
function. The formulas derived can be applied to the study of the asymptotic properties of
the solutions to linear differential systems with constant matrices and with a single delay.
The well-known “step by step” method is one of the basic concepts for the investigation
of linear differential equations and systems with delay. The application of this method
to linear first-order systems with single constant delay and with constant matrix of linear
terms has recently been formalized using the concept of a delayed matrix exponential eBt
τ
in [38, 39]. For linear second-order “oscillating” systems with constant matrix and with
a single constant delay, analogous results have recently been derived using the so-called
delayed matrix cosine Cosτ Bt and delayed matrix sine Sinτ Bt in [36]. The above special
delayed matrix functions are defined on every interval (k − 1)τ ≤ t < kτ, k = 0,1,...
(where τ > 0 is a delay) as matrix polynomials, and are continuous at nodes t = kτ. Such
“step by step” definitions complicate their asymptotic analysis.
5
CHAPTER 2. FUNDAMENTAL MATRIX FOR LINEAR SYSTEMS 6
2.1 Linear first-order systems
Let B be an s ×s constant matrix, Θ the s ×s null matrix, I the s ×s unit matrix, and let
τ > 0 be a constant. The delayed matrix exponential eBt
τ of the matrix B is an s×s matrix
function mapping R to Rs×s, continuous on R\{−τ}, and defined as follows:
eBt
τ :=
k
∑
j=0
Bj (t −(j −1)τ)j
j!
(2.1)
where k = t/τ is the ceiling function, i.e., the smallest integer greater than or equal
to t/τ.
The main property of the delayed matrix exponential eBt
τ is the following:
deBt
τ
dt
= Be
B(t−τ)
τ , t ∈ R\{0}
and the matrix Y(t) = eBt
τ solves the initial problem for a matrix differential system with a
single delay
˙Y(t) = BY(t −τ), (2.2)
Y(t) = I, t ∈ [−τ,0].
If ϕ : [−τ,0] → Rn is a continuously differentiable vector-function, then the solution of
the initial-value problem
˙y(t) = By(t −τ), t ∈ [−τ,∞), (2.3)
y(t) = ϕ(t), t ∈ [−τ,0] (2.4)
can be represented in the form
y(t) = eBt
τ ϕ(−τ)+
0
−τ
e
B(t−τ−s)
τ ˙ϕ(s)]ds. (2.5)
This definition illustrates general definition of a fundamental matrix to linear functional
differential systems of delayed type given in [34]. For system (2.3), this definition reduces
to (details are omitted)
X(t) =



B
t
−τ
X(u−τ)du+I, for almost all t ≥ −τ,
Θ,−2τ ≤ t < −τ
. (2.6)
Let A be a regular s×s constant matrix satisfying AB = BA and let f(t) be a continuous
function. Then, the solution of the initial-value problem
˙y(t) = Ay(t)+By(t −τ)+ f(t), t ∈ [−τ,∞),
y(t) = ϕ(t), t ∈ [−τ,0]
CHAPTER 2. FUNDAMENTAL MATRIX FOR LINEAR SYSTEMS 7
is given by the formula
y(t) = eA(t+τ)
eB1t
τ ϕ(−τ)+
0
−τ
eA(t−τ−s)
e
B1(t−τ−s)
τ eAτ
[ ˙ϕ(s)−Aϕ(s)]ds
+
t
0
eA(t−τ−s)
e
B1(t−τ−s)
τ eAτ
f(s)ds (2.7)
where B1 = e−AτB. These results, together with the results for a non homogenous system,
are proved in [38, 39].
2.2 Linear second-order systems
The above-mentioned usefullness of the delayed matrix exponential served as a stimulation
to look for another delayed matrix functions capable of simply expressing solutions to some
linear differential systems with constant coefficients. In [36], delayed matrix functions are
defined called the delayed matrix sine SinτAt and delayed matrix cosine CosτAt for t ∈ R
as
SinτAt =
t/τ +1
∑
s=0
(−1)s
A2s+1 (t −(s−1)τ)2s+1
(2s+1)!
(2.8)
and
CosτAt =
t/τ +1
∑
s=0
(−1)s
A2s (t −(s−1)τ)2s
(2s)!
, (2.9)
where · is the floor function. Both the delayed matrix sine and cosine are the fundamental
matrices of a homogeneous second-order linear system with a single delay
¨x(t) = −A2
x(t −τ). (2.10)
In [36] the Cauchy initial value problem is solved for equation (2.10) and the initial
condition
x(t) = ϕ(t), for −τ ≤ t ≤ 0 (2.11)
where ϕ ∈ C2([−τ,0],Rn). Assuming that the matrix A is regular, a representation of the
solution to Cauchy initial problem (2.10), (2.11) is given in the integral form
x(t) = CosτAt ϕ(−τ)+A−1
SinτAt ˙ϕ(−τ)
+A−1
0
−τ
SinτA(t −τ −ξ) ¨ϕ(ξ)dξ. (2.12)
The motivation for the study of properties of solutions to second-order linear differential
systems is the applicability of this fact to the study of solutions to linear partial differential
second-order equations.
CHAPTER 2. FUNDAMENTAL MATRIX FOR LINEAR SYSTEMS 8
In [a12] the relations are studied between the first-order and second-order systems.
The solutions to second-order linear differential systems can be regarded as the first n
components of the solutions to first-order linear differential systems of 2n equations.
In [a12] the following useful identities
Cos2τA(t −τ) = Ree
(iA)t
τ and Sin2τA(t −2τ) = Ime
(iA)t
τ (2.13)
are proved. Equivalently,
e
(iA)t
τ = Cos2τA(t −τ)+iSin2τA(t −2τ),
which can be understood as a generalization of the well-known Euler formula, since we
obtain this formula if we put A = 1, τ = 0 in the above identity. For the delayed matrix
functions, we have
˙y(t) = A y(t −τ/2),
where
A :=
Θ A
−A Θ
, y :=
y1
y2
,
is equivalent with (2.10) through the substitution x(t) = y1(t). In much the same way as
above, we can derive (for details we refer to [a12])
X (t) = eA t
τ/2 =
CosτA(t −τ/2) SinτA(t −τ)
−SinτA(t −τ) CosτA(t −τ/2)
.
These facts may serve as motivation for the study of a more general Cauchy initial problem
¨x(t)+P˙x(t −τ)+Qx(t −2τ) = θ, (2.14)
x(t) = ξ(t), t ∈ [−τ,τ] (2.15)
where P, Q are n×n constant matrices provided that there exists an n×n matrix Λ satisfying
the equation
Λ2
+PΛexp(−τΛ)+Qexp(−2τΛ) = Θ. (2.16)
We assume that a solution of (2.14) can be found in the form
x(t) = exp(Λt) (2.17)
where Λ is a suitable n×n constant matrix. By substituting (2.17) into (2.14), we get
Λ2
exp(2Λt)+PΛexp(Λ(t −τ))+Qexp(Λ(t −2τ)) = Θ
and further simplification gives equation (2.16). Let Y = Λexp(Λτ) be a new unknown
matrix. Then, equation (2.16) can be written as
Y2
+PY +Q = Θ. (2.18)
As this is a quadratic equation with respect the matrix Y, its solution has three forms:
The first one is
¨x(t)−2A˙x(t −τ)+(A2
+B2
)x(t −2τ) = θ, t ≥ τ, (2.19)
x(i)
(t) = ξ(i)
(t), i = 0,1, t ∈ [−τ,τ] (2.20)
where the n × n matrices A, B commute, i.e., AB = BA, the matrix B is regular, and the
function ξ : [−τ,τ] → Rn is assumed to be twice continuously differentiable.
CHAPTER 2. FUNDAMENTAL MATRIX FOR LINEAR SYSTEMS 9
Theorem 1. Let AB = BA and let the matrix B be invertible. Then, the solution of the
initial problem (2.19), (2.20) can be expressed as
x(t) = Ree
(A+iB)t
τ −Ime
(A+iB)t
τ B−1
A ξ(−τ)
+ Ime
(A+iB)t
τ B−1 ˙ξ(0)+
0
−τ
Ree
(A+iB)(t−τ−s)
τ
˙ξ(s)
+ Ime
(A+iB)(t−τ−s)
τ B−1
( ¨ξ(s+τ)−A ˙ξ(s)) ds (2.21)
where t ≥ τ.
The second one is the problem (2.22), (2.20) where
¨x(t)−(A+B)˙x(t −τ)+ABx(t −2τ) = θ, t ≥ τ, (2.22)
with matrices A and B commuting but the regularity of B not assumed.
Theorem 2. Let AB = BA. Then, the solution to the Cauchy initial problem (2.22), (2.20)
has the form
x(t) = eAt
τ ξ(−τ)+e
(A,B)t
τ ( ˙ξ(0)−Aξ(−τ))
+
0
−τ
e
A(t−τ−s)
τ
˙ξ(s)+e
(A,B)(t−τ−s)
τ ( ¨ξ(s+τ)−A ˙ξ(s)) ds (2.23)
where t ≥ τ and the matrix function e
(A,B)t
τ is defined as
e
(A,B)t
τ =
t/τ
∑
s=0
(t −(s−1)τ)s
s!
s
∑
i=0
As−i
Bi
.
The third initial problem has the form of a solution to the initial problem given by the
initial condition (2.20) and by the equation:
¨x(t)−2A˙x(t −τ)+A2
x(t −2τ) = θ, t ≥ τ, (2.24)
where ξ : [−τ,τ] → Rn
Theorem 3. A solution to initial problem (2.24), (2.20) has the form
x(t) = eAt
τ ξ(−τ)+DAeAt
τ ( ˙ξ(0)−Aξ(−τ))
+
0
−τ
e
A(t−τ−s)
τ
˙ξ(s)+DAe
A(t−τ−s)
τ ( ¨ξ(s+τ)−A ˙ξ(s)) ds (2.25)
where t ≥ τ and the function DAeAt
τ is defined as
DAeAt
τ =
t/τ
∑
s=0
(t −(s−1)τ)s
s!
sAs−1
=
∂
∂ A
eAt
τ .
CHAPTER 2. FUNDAMENTAL MATRIX FOR LINEAR SYSTEMS 10
The proofs of the theorems that describe the formalization of a solution to the initial
value problem consisting of second-order systems with constant delay, n × n constant
matrices and an initial condition are based on the study of solutions to the initial value
problem of a first-order system with constant 2n×2n matrices and one constant delay. For
more details, see [a12].
2.3 Asymptotic properties of the delayed matrix functions
and Lambert function
The delayed matrix functions are defined on the intervals (k−1)τ ≤ t < kτ, k = 0,1,... as
matrix polynomials and are continuous at the nodes t = kτ. The asymptotic properties of a
delayed matrix exponential are studied for k → ∞ and the sequence of values of the delayed
matrix exponential at the nodes is approximately represented by a geometric progression.
There is a constant matrix C such that the exponential eCτ is a “quotient”, i.e.
lim
k→∞
eBkτ
τ e
B(k+1)τ
τ
−1
= e−Cτ
, (2.26)
where (·)−1 denotes the inverse matrix whose existence is assumed.
In the scalar case, the constant C can be expressed by the principal branch of the
Lambert function, named after Johann Heinrich Lambert. He sent his paper [44] to
Leonhard Euler, who in [20] introduced the Lambert function as the inverse function to
the function
f(w) = wew
.
Thus, the Lambert function, usually denoted by W = W(z), is defined implicitly by the
equation
z = W(z)eW(z)
. (2.27)
Such a function is multi-valued (except for the point z = 0). For real arguments z = x,
W(x) satisfying
x > −1/e W(x) > −1,
the equation (2.27) defines a single-valued function W = W0(x) called the principal branch
of the Lambert function W(z), which may be extended to an analytic function in the
complex plane except for the real numbers x < −1/e since the point −1/e is a branch point
of Lambert function. Of all Lambert function branches, the principal branch assumes the
greatest real part values. We refer to [11] for a survey of the basic properties of Lambert
function.
The Maclaurin expansion of W0(x) about the point x = 0 can be found easily and is
given by the series
W0(x) =
∞
∑
n=1
(−n)n−1
n!
xn
, (2.28)
having the radius of convergence r = 1/e.
In [59] the following Theorem is proved.
CHAPTER 2. FUNDAMENTAL MATRIX FOR LINEAR SYSTEMS 11
Theorem 4. Let λj, j = 1,...,n be the eigenvalues of a matrix A and let its Jordan
canonical form be
diag(λ1,...,λn) = D−1
BD, (2.29)
where D is a regular matrix. If |λj|τe < 1, j = 1,...,n, then the sequence
e
B(k+1)τ
τ (eBkτ
τ )−1
, for k → ∞
is convergent and (2.26) holds where
eCτ
= Dexp(diag(W0(λ1τ),...,W0(λn,τ))D−1
. (2.30)
This theorem was proved using Abel’s extension (see [1]) of the well-known binomial
theorem by comparing the Maclaurin expansions for both terms.
In [a6] the theorem is generalized, since the diagonal shape of the Jordan canonical form
is not required. Moreover, the asymptotic equation for the sequence {eBkτ
τ } is described in
the following theorem.
Theorem 5. Let τ > 0 and let an n×n constant matrix B ≡ Θ be given. If the eigenvalues
λi, i = 1,...,n of the matrix B satisfy the inequality |λi|τe < 1, then
lim
k→∞
eBkτ
τ exp(−kW0(Bτ)) = Bτ (W0(Bτ)(I +W0(Bτ)))−1
. (2.31)
For the asymptotic properties of the exponential exp(λx), the real part of the complex
number λ is fundamental. The set of the complex numbers z = x + iy for which the real
part of the Lambert function equals zero is defined in the parametric form
x = −vsinv, y = vcosv.
This parametric specification follows from fact that ℜW(x + iy) = u = 0. Analyzing
the part of this curve corresponding to the principal branch W0(x+iy), we conclude that it
is a simple closed curve for the admissible range v ∈ [−π/2,π/2]. This curve is depicted
in Figure 2.1. The real part of the principal branch of the Lambert function is negative for
|z| < −arctan
Rez
|Imz|
. (2.32)
This domain is bounded by the above curve (see Figure 2.1). Note that a Lambert function
W cannot be expressed in terms of elementary functions. For more details, see [11].
Let F(k) = { fij(k)}n
i,j=1 and G = {gij(k)}n
i,j=1
be matrices defined for all sufficiently large
k. We say that
F(k) G(k), k → ∞ (2.33)
if
fij(k) = gij(k)(1+o(1)), k → ∞, (2.34)
where o(1) is the Landau order symbol “small” o.
Remark 1. Let all assumptions of Theorem 5 be valid. From formula (2.31), we get the
asymptotic relation
eBkτ
τ Bτ exp(kW0(Bτ))(W0(Bτ)(I +W0(Bτ)))−1
, k → ∞. (2.35)
This formula can be useful, e.g., in the investigation of the asymptotic behaviour of the
solutions to the problem at nodes t = kτ.
CHAPTER 2. FUNDAMENTAL MATRIX FOR LINEAR SYSTEMS 12
Re z−
π
2
Im zRe W0(z) > 0
Re W0(z) < 0
1
Figure 2.1: The curve ReW0(z) = 0
Some consequences
Recall that the spectral radius ρ(·) is the maximal absolute value of the spectrum of a given
matrix and the spectral norm A ρ = ρ(A A T )
1/2
) is defined for a matrix A . The
following theorem describes the behaviour of the sequence of values of delayed exponential
eBkτ
τ for (discrete) k → ∞ and of delayed exponential eBt
τ for (continuous) t → ∞, see [a6].
Theorem 6. Let τ > 0 and let an n×n constant matrix B ≡ Θ be given. Assume that the
eigenvalues λi, i = 1,...,n of the matrix B satisfy the inequality τ|λi| < 1/e, i = 1,...,n.
The following three statements hold:
(i) If all the eigenvalues λi, i = 1,...,n satisfy
τ|λi| < −arctan
Reλi
|Imλi|
, (2.36)
then
lim
k→∞
ρ eBkτ
τ = 0. (2.37)
(ii) If there exists, an index i0 ∈ {1,...,n} such that
τ|λi0| > −arctan
Reλi0
|Imλi0|
, (2.38)
then
limsup
k→∞
eBkτ
τ
ρ
= ∞. (2.39)
(iii) If all the eigenvalues λi, i = 1,...,n are real and satisfy
τ|λi| > −arctan
Reλi
|Imλi|
, (2.40)
then
lim
t→∞
eBt
τ ρ
= ∞. (2.41)
Figure 2.2 details the eigenvalue domain for each case considered.
CHAPTER 2. FUNDAMENTAL MATRIX FOR LINEAR SYSTEMS 13
Re z−
π
2τ
1
e
Im z
Re W0(z) > 0
case i case ii case iii
Re W0(z) < 0
1
Figure 2.2: Detailed eigenvalue domains
Equation of a showering person System (2.3) often describe mathematical models
of real-world phenomena. The solution of the initial problem (2.3), (2.4) is given by
formula (2.5). We investigate the long-time behaviour of the solutions generated by
constant initial functions, i.e., assume ϕ(t) ≡ Cϕ for every fixed t ∈ [−τ,0] and Cϕ ∈ Rn.
Then,
˙ϕ(t) ≡ θ, t ∈ [−τ,0],
where θ is the null vector. Formula (2.5) becomes
y(t) = eBt
τ ϕ(−τ) = eBt
τ Cϕ. (2.42)
If all assumptions of Theorem 5 hold, by formula (2.35), we get the asymptotic expression
for (2.42) at nodes t = kτ as k → ∞
y(kτ) = eBkτ
τ Cϕ Bτ exp(kW0(Bτ))(W0(Bτ)(I +W0(Bτ)))−1
Cϕ. (2.43)
The above sentence will be used for a model that generalizes the description of the
water temperature controlled by the person in the shower, i.e., the generalization of the
equation (1.1). Setting y(t) = T(t)−Td in (1.1), we get
˙y(t) = −γy(t −τ), t ∈ [0,∞). (2.44)
Assuming that the water temperature before regulation is constant, i.e., the initial condition
is given by the equation
y(t) = y0, t ∈ [−τ,0], (2.45)
the solution of (2.44), (2.45) is
y(t) = e
−γt
τ y0, t ∈ [−τ,∞)
and, if γτe < 1, then, by (2.33)–(2.35) and (2.43),
y(kτ) = e
−γkτ
τ y0 = −γτ exp(kW0(−γτ))
y0(1+o(1))
W0(−γτ)(1+W0(−γτ))
CHAPTER 2. FUNDAMENTAL MATRIX FOR LINEAR SYSTEMS 14
as k → ∞. By (2.27), the last formula can be simplified to
y(kτ) =
y0(1+o(1))
1+W0(−γτ)
e(1+k)W0(−γτ) , k → ∞.
Since, by (2.28),
W0(−γτ) = −γτ −(γτ)2
−
3
2
(γτ)3
+··· ,
we have y(kτ) > 0 and limk→∞ y(kτ) = 0. This means that the regulated temperature T(kτ)
will tend to the desired value Td as k → ∞.
The above example can be generalized, e.g., for the two showering persons. Suppose
that hot and cold water is supplied in two separate pipes to a bathroom with two showers.
Inside the bathroom, each pipe branches into two pipes leading to the shower mixers. A
person taking a shower regulates the water temperature flowing from the mixer to the
sprinkler. Due to the changes in the water pressure caused by water being regulated by two
persons simultaneously, there is a mutual dependence between the temperatures T1 and T2
of the water flowing from mixer one to sprinkler one and from mixer two to sprinkler two,
respectively. Then, a simple model modeling the behaviour of two showering persons is
˙T1(t) =−γ11[T1(t −τ)−Td1]+γ12[T2(t −τ)−Td2], (2.46)
˙T2(t) = γ21[T1(t −τ)−Td1]−γ22[T2(t −τ)−Td2] (2.47)
where γij > 0, i, j = 1,2 and Tdi, i = 1,2 are the desired water temperatures agreeable for
two showering person, respectively. Substituting yi(t) = Ti(t) − Tdi in (2.46), (2.47), we
get
˙y1(t) =−γ11y1(t −τ)+γ12y2(t −τ), (2.48)
˙y2(t) = γ21y1(t −τ)−γ22y2(t −τ). (2.49)
Assuming that the water temperature before regulation is constant, i.e., the initial condition
is given by the equation
y1(t) = y2(t) = y0, t ∈ [−τ,0], (2.50)
the solution of (2.48)–(2.50) is
y(t) = (y1(t),y2(t))T
= e−Γt
τ y0
, t ∈ [−τ,∞) (2.51)
where y0 = (y0,y0)T and
Γ =
−γ11 γ12
γ21 −γ22
.
Let the eigenvalues
λi =
1
2
−(γ11 +γ22)+(−1)i
(γ11 −γ22)2 +4γ12γ21 , i = 1,2
of the matrix Γ satisfy |λi|τe < 1, i = 1,2. Then, by formula (2.43), at nodes t = kτ, the
solution (2.51) has the asymptotic behavior
y(kτ) Γτ exp(kW0(Γτ))(W0(Γτ)(I +W0(Γτ)))−1
y0
as k → ∞.
CHAPTER 2. FUNDAMENTAL MATRIX FOR LINEAR SYSTEMS 15
Delayed matrix sine and cosine
To describe the asymptotic properties of other delayed matrix functions, we use the equations
(2.13). Our recent result has been proved in [a13].
Theorem 7. Let λj, j = 1,...,n be the eigenvalues of a matrix A and let its Jordan
canonical form be given by (2.29). If |λj| < 1/(eτ), j = 1,...,n and there exists at least
one j = j∗ ∈ {1,...,n} such that λj∗ = 0, then
limsup
t→∞
Cosτ At ρ = ∞
and
limsup
t→∞
Sinτ At ρ = ∞.
Another direction of research is to show that the condition about eigenvalues of matrix
|λj| < 1/(eτ), j = 1,...,n is not necessary and can be weakened.
Chapter 3
Topological method for functional
differential equations
The oscillation of solutions and existence of positive solutions are the essential problems
encountered when studying the asymptotic properties of differential equations. Many
criteria for the existence of positive solutions may be derived applying the retract or the
Lyapunoff method to a system of differential equations with unbounded delay but with
finite memory in the sense given in [43].
The results for the existence of a solution in a predetermined set are given in [a3].
To arrive at these results, two principles were used. First the retract principle which is
often used in the theory of ordinary differential equations (see e.g. [33]) and goes back to
Wa˙zewski [64]. For RFDE’s with bounded retardation, this principle was modified, e.g.,
by Rybakowski [53]. Here, Rybakowski’s modified result was used, which concerns the
existence of at least one curve in a given family of curves, with the graph lying in an open
set. Second, an inverse principle was used, which originates in the theory of Lyapunov
stability, and for retarded functional differential equations, it was developed by Razumikhin
(e.g. [52]).
In both principles the concept system of curves is used.
3.1 The system of curves
If a set A ⊂ R×Rn is given, then intA, A and ∂A denote, as usual, the interior, the closure,
and the boundary of A, respectively.
Definition 1. Let Λ be a topological space, let a subset ˜Ω ⊂ R×Λ be open in R×Λ, and
let x be a mapping associating with every (δ,λ) ∈ ˜Ω a function x(δ,λ) : Dδ,λ → Rn where
Dδ,λ is an interval in R. Assume 1 through 3:
1) δ ∈ Dδ,λ .
2) If t ∈ intDδ,λ , then there is an open neighborhood O(δ,λ) of (δ,λ) in ˜Ω such that
t ∈ Dδ ,λ holds for all (δ ,λ ) ∈ O(δ,λ).
16
CHAPTER 3. TOPOLOGICAL METHOD 17
3) If (δ ,λ ), (δ,λ) ∈ ˜Ω, and t ∈ Dδ ,λ , t ∈ Dδ,λ , then
lim
(δ ,λ ,t )→(δ,λ,t)
x(δ ,λ )(t ) = x(δ,λ)(t).
If all these conditions are satisfied, then (Λ, ˜Ω,x) is called a system of curves in Rn.
Studying the proof of Theorem 2.1, in [53, p.119], we formulated in [a3] two results
Lemma 1 (Retract Principle) and Lemma 2 (Lyapunov Principle) suitable for our
applications to retarded functional-differential equations with unbounded delay.
3.2 The retract method and the Lyapunov method for p-
RFDE’s
Let us recall the notion of a p function.
Definition 2. ([43, p. 8]) The function p ∈ C[R×[−1,0],R] is called a p-function if it has
the following properties:
(i) p(t,0) = t;
(ii) p(t,−1) is a nondecreasing function of t;
(iii) there exists a σ ≥ −∞ such that p(t,ϑ) is an increasing function for ϑ for each
t ∈ (σ,∞).
Definition 3. ([43, p. 8]) Let t0 ∈ R, A > 0 and y ∈ C([p(t0,−1),t0 + A),Rn). For any
t ∈ [t0,t0 + A), we define function yt by yt(ϑ) = y(p(t,ϑ)), −1 ≤ ϑ ≤ 0 and we write
yt ∈ C ≡ C[[−1,0],Rn].
We investigate the system
˙y(t) = f(t,yt), (3.1)
with a functional f ∈ C[[t0,t0 +A)×C ,Rn], called a system of p-type retarded functional
differential equations (p-RFDE’s). The function y ∈C([p(t0,−1),t0 +A),Rn)∩C1([t0,t0 +
A),Rn) satisfying (3.1) on [t0,t0 + A) is called a solution of this system of p-RFDE’s on
[[ p(t0,−1),t0 +A).
Remark 2. System (3.1) with yt defined in accordance with Definition 3 is called a system
with unbounded delay and with finite memory. Note that the frequently used symbol “ yt”
(e.g., in accordance with [34, p. 38], yt(s) = y(t +s), where −τ ≤ s ≤ 0, τ > 0, τ = const)
for an equation with bounded delay is a partial case of the above definition of yt. Indeed,
in this case, we can put p(t,ϑ) ≡ t +τϑ.
Let Ω be an open subset of R×C and the function f ∈ C(Ω,Rn). If (t0,φ) ∈ Ω, there
exists a solution y = y(t0,φ) of the system of p -RFDE’s (3.1) through (t0,φ) (see [43,
p. 25]). Moreover, this solution is unique if f(t,φ) is locally Lipschitzian with respect
to φ ([43, p. 30]) and is continuable in the usual sense of extended existence if f is
quasibounded ([43, p. 41]). Suppose that the solution y = y(t0,φ) of p -RFDE’s (3.1)
CHAPTER 3. TOPOLOGICAL METHOD 18
through (t0,φ) ∈ Ω, defined on [t0,A], is unique. Then, the property of the continuous
dependence holds, too (see [43, p. 33]), i.e., for every ε > 0, there exists a δ(ε) > 0 such
that (s,ψ) ∈ Ω, |s−t0| < δ and ψ −φ < δ implies
yt(s,ψ)−yt(t0,φ) < ε, for all t ∈ [ζ,A]
where y(s,ψ) is the solution of the system of p - RFDE’s (3.1) through (s,ψ), ζ =
max{s,t0}, and · is the supremum norm in Rn. Note that the system of solutions to (3.1)
under the above assumptions is a system of curves in the sense of definition 1 and this fact
can be adapted easily for the case of Ω having the form Ω = [ p∗,∞) × C where p∗ ∈ R
and the cross-section {(˜t,ϕ) ∈ Ω} being an open set for every ˜t ∈ [p∗,∞).
Let li, mj, i = 1,... p, j = 1,...s, p + s > 0 be real-valued C1-functions defined on
R×Rn. The set
˜ω = {(t,y) ∈ [p∗
,∞)×Rn
, li(t,y) < 0, mj(t,y) < 0, for all i, j }
will be called a polyfacial set.
Definition 4. A polyfacial set ˜ω is called regular with respect to equation (3.1) if α), β), γ)
below hold:
α) If (t,φt) ∈ R×C and if (p(t,ϑ),φt(ϑ)) ∈ ˜ω for all ϑ ∈ [−1,0), then (t,φt) ∈ ˜Ω.
β) For all i = 1,..., p, all (t,y) ∈ ∂ ˜ω for which li(t,y) = 0 and for all φt ∈ C for which
φt(0) = y and (p(t,ϑ),φt(ϑ)) ∈ ˜ω for all ϑ ∈ [−1,0), it follows that
Dli(t,y) ≡
n
∑
r=1
∂li
∂yr
(t,y)· fr(t,φt)+
∂li
∂t
(t,y) > 0.
γ) For all j = 1,...,s, all (t,y) ∈ ∂ ˜ω for which mj(t,y) = 0 and for all φt ∈ C for which
φt(0) = y and (p(t,ϑ),φt(ϑ)) ∈ ˜ω for all ϑ ∈ [−1,0), it follows that
Dmj(t,y) ≡
n
∑
r=1
∂mj
∂yr
(t,y)· fr(t,φt)+
∂mj
∂t
(t,y) < 0.
Lemma 1 (Retract Method). Let p > 0. Let ˜ω be a nonempty polyfacial set, regular with
respect to equation (3.1), let the function f ∈C( ˜Ω,Rn) be locally Lipschitzian with respect
to the second argument, and
W = {(t,y) ∈ ∂ ˜ω : mj(t,y) < 0, j = 1,...,s}. (3.2)
Let Z be a subset of ˜ω ∪W and let the mapping q : B = Z ∩(Z ∪W) → C be continuous
and such that if z = (δ,y) ∈ B, then (δ,q(z)) ∈ ˜Ω, and :
1) If z ∈ Z ∩ ˜ω, then (p(δ,ϑ),q(z)(p(δ,ϑ))) ∈ ˜ω for ϑ ∈ [−1,0],
2) If z ∈W ∩B, then (δ,q(z)(δ)) = z and (p(δ,ϑ),q(z)(p(δ,ϑ))) ∈ ˜ω for ϑ ∈ [−1,0).
Let, moreover, Z ∩W be a retract of W, but not a retract of Z. Then, there exists a z0 =
(δ0,y0) ∈ Z ∩ ˜ω such that (t,y(δ0,q(z0))(t)) ∈ ˜ω for every t ∈ Dδ0,q(z0).
CHAPTER 3. TOPOLOGICAL METHOD 19
Lemma 2 (Lyapunov Method). Let p = 0. Let ˜ω be a nonempty polyfacial set, regular with
respect to equation (3.1) and let the function f ∈ C( ˜Ω,Rn) be locally Lipschitzian with
respect to the second argument. Let a mapping q : B → C , B = ˜ω ∩{(t∗,y),t∗ ∈ R,t∗ =
const,y ∈ Rn} be continuous and such that if z = (t∗,y) ∈ B, then (t∗,q(z)) ∈ ˜Ω, and:
1) If z ∈ ˜ω, then (p(t∗,ϑ),q(z)(p(t∗,ϑ))) ∈ ˜ω for ϑ ∈ [−1,0].
2) If z ∈ ∂ ˜ω, then (t∗,q(z)(t∗)) = z and (p(t∗,ϑ),q(z)(p(t∗,ϑ))) ∈ ˜ω for ϑ ∈ [−1,0).
Then, for every z0 = (t∗,y0) ∈ B∩ ˜ω and every t ∈ Dt∗,q(z0):
(t,y(t∗
,q(z0))(t)) ∈ ˜ω. (3.3)
3.3 Retract principle for neutral functional differential
equations
This part discusses a problem of extending the retract principle to neutral differential
equations. A common basis with the previous results is the reuse of the retract principle
for a system of curves. The problem given by the fact that the value of the derivative of
a solution depends on the values of the derivative of this solution in the past is solved by
modifying the notion of a regular polyfacial set to the notion of a regular polyfacial set with
respect to an equation and subsidiary inequalities. Particular problems are solved in [a7]
and [a9].
Neutral functional differential equations We consider a neutral functional differential
system of the form
˙y(t) = f(t,yt, ˙yt) (3.4)
where the symbol ˙y (sometimes we use y ) stands for the derivative (considered, if necessary,
as one-sided). First, we give the necessary auxiliary background for this equation.
Let C be the set of all continuous functions ϕ : [−h,0] → Rn and C 1 be the set of all
continuously differentiable functions ϕ : [−h,0] → Rn.
We assume t ≥ t0, yt(θ) = y(t +θ), θ ∈ [−h,0] where h > 0 is a constant and f : Eh →
Rn with Eh := [t0,∞)×C ×C . We pose an initial problem for (3.4):
yt0 = ϕ, ˙yt0 = ˙ϕ (3.5)
where ϕ ∈ C 1. The norm of ϕ ∈ C is defined as ϕ h := max
θ∈[−h,0]
ϕ(θ) and, if ϕ ∈ C 1,
then
ϕ h := max
θ∈[−h,0]
ϕ(θ) + max
θ∈[−h,0]
˙ϕ(θ) .
A function y: [t0 −h,tϕ) → Rn, tϕ ∈ (t0,∞], is a solution of (3.4), (3.5) if yt0 = ϕ, ˙yt0 = ˙ϕ
and (3.4) is satisfied for any t ∈ [t0,tϕ). The following result is taken from a book [41, p.
107] by Kolmanovskii and Myshkis.
CHAPTER 3. TOPOLOGICAL METHOD 20
Theorem 8. Let f : Eh → Rn be a continuous functional satisfying, in some neighborhood
of any point of Eh, the condition
f(t,ψ1,χ1)− f(t,ψ2,χ2) ≤ L ψ1 −ψ2 h + χ1 − χ2 h (3.6)
with constants L ∈ [0,∞), ∈ [0,1). In addition, assume that ϕ ∈ C 1 and that the sewing
condition
˙ϕ(0) = f(t0,ϕ, ˙ϕ) (3.7)
is fulfilled. Then, there exists a tϕ ∈ (t0,∞] such that:
a) There exists a solution y of (3.4), (3.5) on [t0 −h,tϕ).
b) On any interval [t0 −h,t1] ⊂ [t0 −h,tϕ), t1 > t0, this solution is unique.
c) If tϕ < ∞, then ˙y(t) has not a finite limit as t → t−
ϕ .
d) The solution y and its derivative ˙y depend continuously on f, ϕ.
For a particular case of system (3.4) given by
f(t,yt, ˙yt) :=
f(t,y(t −h1(t)),...,y(t −ho(t)), ˙y(t −g1(t)),..., ˙y(t −g (t)))
where the indices o and are non-negative, i.e.,
˙y(t) = f(t,y(t −h1(t)),...,y(t −ho(t)), ˙y(t −g1(t)),..., ˙y(t −g (t))), (3.8)
a more general result can be proved easily by the method of steps (compare [41, pp. 111,
96] and [32]).
Theorem 9. Let f : [t0,∞)×Ro+ → Rn, hi : [t0,∞) → (0,h], i = 1,...,o and gj : [t0,∞) →
(0,h], j = 1,..., be continuous functions. In addition, assume that ϕ ∈ C 1 and that the
sewing condition (3.7), in the case considered, having the form
˙ϕ(0) = f(t0,ϕ(−h1(t0)),...,ϕ(−ho(t0)), ˙ϕ(−g1(t0)),..., ˙ϕ(−g (t0))) (3.9)
is fulfilled. Then:
a) There exists a solution y of (3.4), (3.5) on [t0 −h,∞).
b) On any interval [t0 −h,t1] ⊂ [t0 −h,∞), t1 > t0, this solution is unique.
c) The solution y and its derivative ˙y depend continuously on f, ϕ.
CHAPTER 3. TOPOLOGICAL METHOD 21
Polyfacial set Let Λ = C 1, ˜Ω ⊂ {(t,λ) ∈ [t0,∞)×C 1 such that ˙λ(0) = f(t0,λ, ˙λ)} and
function f satisfy all the assumptions of Theorem 8. In this case, through each (t0,λ) ∈ ˜Ω,
there passes a unique solution y(t0,λ) of (3.4) defined on the maximal interval [t0 −h,aλ ).
Let Dt0,λ = [t0 −h,aλ ) where aλ > t0. Then, (Λ, ˜Ω,y) is a system of curves in Rn. In [a7]
we define the polyfacial set as:
Definition 5. Let p and s be nonnegative integers, p+s > 0, t∗ > t0, and let
li : [t0 −r,t∗) → R×Rn
, i = 1,..., p,
mj : [t0 −r,t∗) → R×Rn
, j = 1,...,s
be continuously differentiable functions. The set
ω := {(t,y) ∈ [t0 −r,t∗)×Rn
, li(t,y) < 0, mj(t,y) < 0, for all i, j }
is called a polyfacial set provided that the cross-section
ω ∩{(t,y): t = t∗
,y ∈ Rn
}
is an open and simply connected set for every fixed t∗ ∈ [t0 −r,t∗).
In order to prove the existence of a solution of (3.4) lying in a polyfacial set, ω should
meet some additional requirements. Because of the neutrality of the equations, we need to
be able to foresee the properties of the derivatives of solutions as described by the auxiliary
inequalities.
Definition 6. Let q be a nonnegative integer, t∗ > t0, and let
ck : [t0 −r,t∗)×Rn
×Rn
→ R, k = 1,...,q,
be continuous functions. A polyfacial set ω is called regular with respect to equation (3.4)
and auxiliary inequalities
ck(t,y,x) ≤ 0, k = 1,...,q (3.10)
if α) – δ) below hold:
α) If (t,φ) ∈ R×C 1 and (t +θ,φ(θ)) ∈ ω for θ ∈ [−r,0), then (t,φ, ˙φ) ∈ Er.
β) If (t,φ) ∈ R×C 1, (t +θ,φ(θ)) ∈ ω for θ ∈ [−r,0) and, moreover,
ck(t +θ,φ(θ), ˙φ(θ)) ≤ 0, θ ∈ [−r,0), k = 1,...,q, (3.11)
then also
ck(t +θ,φ(θ), f(t,φ, ˙φ)) ≤ 0, k = 1,...,q. (3.12)
γ) For all i = 1,..., p, all (t,y) ∈ ∂ω for which li(t,y) = 0 and for all φ ∈ C 1 for which
φ(0) = y, (t +θ,φ(θ)) ∈ ω, θ ∈ [−r,0) and
ck(t +θ,φ(θ), ˙φ(θ)) ≤ 0, θ ∈ [−r,0), k = 1,...,q, (3.13)
it follows that:
Dli(t,y) ≡
∂li
∂t
(t,y)+
n
∑
r=1
∂li
∂yr
(t,y)· fr(t,φ, ˙φ) > 0.
CHAPTER 3. TOPOLOGICAL METHOD 22
δ) For all j = 1,...,s, all (t,y) ∈ ∂ω for which mj(t,y) = 0 and for all φ ∈ C 1 for
which φ(0) = y, (t +θ,φ(θ)) ∈ ω, θ ∈ [−r,0) and
ck(t +θ,φ(θ), ˙φ(θ)) ≤ 0, θ ∈ [−r,0), k = 1,...,q
for all θ ∈ [−1,0) , it follows that:
Dmj(t,y) ≡
∂mj
∂t
(t,y)+
n
∑
r=1
∂mj
∂yr
(t,y)· fr(t,φ, ˙φ) < 0.
If ω is a polyfacial set, then define the setW used in Lemma 1 (Retract Principle) (see [a3])
as
W := {(t,y) ∈ ∂ω : mj(t,y) < 0, j = 1,...,s}. (3.14)
Moreover, we need to specify the properties of the mapping q in Lemma 1 (Retract Principle)
(see [a3]). The following definition describes the admissible behavior of functions
with respect to ω. A fixed set of functions generated by this mapping and satisfying the
properties listed in the following definition is called a set of initial functions.
Definition 7 (Set of initial functions). Let Z be a subset of ω ∪W and let the mapping
q: B → C 1
, B := Z ∩(Z ∪W)
be continuous. We assume that, if z = (δ,y) ∈ B, then (δ,q(z)) ∈ ˜Ω. If moreover,:
1) For z ∈ Z ∩ω, we have (δ +θ,q(z)(θ)) ∈ ω for θ ∈ [−r,0].
2) For z ∈ W ∩B, we have (δ,q(z)(δ)) = z, and
either
2a) (δ +θ,q(z)(θ)) ∈ ω for θ ∈ [−r,0)
or
2b) (δ +θ,q(z)(θ)) ∈ ω for θ ∈ [−r,0) and, for all σ > 0, there is a t = t(σ,z),
δ < t ≤ δ +σ such that t is within the domain of definition of solution y(δ,q(z))
of (3.4) and (t,y(δ,q(z))(t)) ∈ ω,
then such a set of functions is called a set of initial functions for (3.4) with respect to ω
and Z.
Finally, we will formulate the below theorem as an application of Lemma 1 (Retract
Principle) (see [a3]) to a system of neutral equations (3.4). Therefore, its proof is omitted.
Theorem 10. Let ω be a nonempty polyfacial set, regular with respect to (3.4) and inequalities
(3.10). Assume that φ ∈ C 1 and that the sewing condition (3.7) is fulfilled. Let
a fixed t∗ ∈ (t0,∞] exist such that:
a) There exists a solution y of (3.4), (3.5) on [t0 −r,t∗).
b) On any interval [t0 −r,t1] ⊂ [t0 −h,t∗), t1 > t0, this solution is unique.
CHAPTER 3. TOPOLOGICAL METHOD 23
c) If t∗ < ∞, then ˙y(t) has not a finite limit as t → t−
∗ .
d) The solution y and its derivative ˙y depend continuously on f, φ.
Assume that q defines a set of initial functions for (3.4) with respect to ω and Z and that
the derivative of every solution y(δ,q(z))(t) of (3.4) defined by any z = (δ,x) ∈ B has a
finite left limit at every point t provided that
(t,y(δ,q(z))(t)) ∈ ω.
Let, moreover, Z ∩W be a retract of W, but not a retract of Z. Then, there exists at least
one point z0 = (δ0,x0) ∈ Z ∩ω such that a solution y(δ0,q(z0))(t) exists on [t0 −r,t∗) and
(t,y(δ0,q(z0))(t)) ∈ ω
holds for all t ∈ [t0 −r,t∗).
Chapter 4
Applications to nonlinear systems
We start this chapter with a theorem describing the sufficient and necessary conditions
for the existence of at least one solution to a given RDFE in a predetermined domain.
Rn
≥0 Rn
>0 will denote the set of all component-wise nonnegative (positive ) vectors v in
Rn, i.e., v = (v1,...,vn) ∈ Rn
≥0 Rn
>0 if and only if vi ≥ 0 (vi > 0) for i = 1,...,n. For
u,v ∈ Rn, we write u ≤ v if v−u ∈ Rn
≥0; u v if v−u ∈ Rn
>0 and u < v if u ≤ v and u = v.
Let p∗, t∗ be constants satisfying p∗ = p(t∗,−1) for a given p -function. Define vector
valued functions ρ,δ ∈ C([ p∗,∞),Rn), satisfying ρ δ on [ p∗,∞), and continuously
differentiable on [t∗,∞). Put Ω := [t∗,∞)×C and
ω := {(t,y) : t ≥ p∗
, ρ(t) y δ(t)}.
Definition 8. A system of initial functions SE ,ω with respect to nonempty sets E and ω
where E ⊂ ω is defined as a continuous mapping ν : E → C such that a) and b) in the
following text hold:
a) For each z = (t,y) ∈ E ∩int ω and ϑ ∈ [−1,0] : (t +ϑ,ν(z)(p(t,ϑ))) ∈ ω.
b) For each z = (t,y) ∈ E ∩ ∂ω and ϑ ∈ [−1,0) : (t + ϑ,ν(z)(p(t,ϑ))) ∈ ω and,
moreover, (t,ν(z)(p(t,0))) = z.
We denote by S 1
E ,ω a system of initial functions SE ,ω if all functions ν(z), z = (t,y) ∈ E
are continuously differentiable on [−1,0).
The necessary and sufficient condition for the existence of a positive solution is given
by the next theorem.
Theorem 11. Let f ∈C(Ω,Rn) be locally Lipschitzian with respect to the second argument,
quasibounded, and, moreover:
(i ) For any i = 1,..., p (0 ≤ p ≤ n), t ≥ t∗ and π ∈ C([p(t,−1),t],Rn) such that
(θ,π(θ)) ∈ ω for all θ ∈ [p(t,−1),t), (t,π(t)) ∈ ∂ω, it follows (t,πt) ∈ Ω,
˙δi(t) < fi(t,πt) when πi(t) = δi(t) (4.1)
and
˙ρi(t) > fi(t,πt) when πi(t) = ρi(t). (4.2)
24
CHAPTER 4. APPLICATIONS TO NONLINEAR SYSTEMS 25
(ii ) For any i = p+1,...,n, t ≥ t∗ and π ∈ C([p(t,−1),t],Rn) such that (θ,π(θ)) ∈ ω
for all θ ∈ [p(t,−1),t), (t,π(t)) ∈ ∂ω, it follows (t,πt) ∈ Ω,
˙δi(t) > fi(t,πt) when πi(t) = δi(t) (4.3)
and
˙ρi(t) < fi(t,πt) when πi(t) = ρi(t). (4.4)
Then, there exists an uncountable set Y of solutions to the system (3.1) on the interval
[ p∗,∞) such that, for each y ∈ Y ,
ρ(t) y(t) δ(t), t ∈ [ p∗
,∞). (4.5)
The proof is given in [a3].
The above results on the existence of solutions of functional differential equations in
a prescribed area were used in two main directions. The first is to prove the existence
of a solution when modifying the first Ljapunov method. Here we assume a solution of
the perturbed equation in the form of a power series.These coefficients are solutions to a
system of linear equations. As the sequence thus obtained is not generally convergent, an
asymptotic expansion of this solution is constructed. Then, using the retract method, we
prove the existence of a sequence of solutions that are an asymptotic decomposition.
4.1 Asymptotic expansion of solution
The first method of Lyapunov is a well known technique used for studying the asymptotic
behavior of ordinary differential equations in the form of a linear system with perturbation.
This method uses the solution in the form of a convergent power series, for details see [10].
The results for equations in the implicit form [12] or for integro-differential equations [61]
were derived by modifying the first method of Lyapunov. The existence of solutions with a
certain asymptotic form were proved in the references cited using Wa˙zewski’s topological
method. For analogous representations of solutions to a retarded differential equation,
see [57], [a10]. The perturbation has a polynomial form in both cases. In this paper, we
study an equation in the form
˙y(t) = −a(t)y(t)+
∞
∑
|i|=2
ci(t)
n
∏
j=1
y(ξj(t))
ij
(4.6)
where i = (i1,...,in) is a multiindex, ij ≥ 0 are integers and |i| =
n
∑
j=1
ij. The continuous
functions ξj(t) satisfy ξj(t) ≥ t0 for all t ∈ [t0,∞) and the function ξ(t), which is defined as
ξ(t) = min
1≤i≤n
ξi(t), is nondecreasing for t ≥ t0. Therefore, all asymptotic relations such as
the Landau symbols o, O and the asymptotic equivalence ∼ will be considered for t → ∞.
This fact will not be pointed out in the sequel.
The function a(t) satisfies the following conditions:
C1 a(t) is continuous and positive on the interval [t0,∞) and 1/a(t) = O(1),
CHAPTER 4. APPLICATIONS TO NONLINEAR SYSTEMS 26
C2 (t −ξ(t))a(t) = o(A(t)) where the functions A(t), a(t) are defined as
A(t) =
t
t0
a(u)du, a(t) = max
u≤t
(a(u)).
Further conditions for the continuous functions ci(t) : [t0,∞) → R will be given later. In
order to apply the first method of Lyapunov to the equation (4.6) we assume the solution
in the form of a formal series
y(t,C) =
∞
∑
n=1
fn(t)ϕn
(t,C) (4.7)
where ϕ(t,C) is the solution of the homogeneous equation ˙y(t) = −a(t)y(t) given by the
formula
ϕ(t,C) = Cexp −A(t) ,
with f1(t) ≡ 1, and the functions fk(t) for k = 2,..,n being particular solutions to a certain
system of auxiliary differential equations. Using Wa˙zewski’s topological method in the
form used in [a3] for differential equations with unbounded delay and finite memory, we
prove the existence of a solution
yn(t,C) ∼ fn(t,C) =
n
∑
k=1
fk(t)ϕk
(t,C).
To facilitate the specification of the coefficients of the power series which is the product
of the power series raised to a power, we use the following notation: s = (s1,...,sn) is
an ordered n-tuple of sequences sj = sk
j
∞
k=1
of nonnegative integers with a finite sum
|sj| =
∞
∑
k=1
sk
j, denoting further
s! =
n
∏
j=1
∞
∏
k=1
sj!k
, i(s)! =
n
∏
j=1
|si|!
V(s) =
n
∑
j=1
∞
∑
k=1
ksk
j, i(s) = (|s1|,...,|sn|).
For any ordered n-tuple of sequences (of numbers or functions) C = (c1,...,cn) where
cj = {ck
j}∞
k=1, we denote C s
=
n
∏
j=1
∞
∏
k=1
ck
j
sk
j
where ck
j
0
= 1 for every ck
j. Then, it is
possible to write
n
∏
j=1
∞
∑
k=1
ck
jxk
ij
=
∞
∑
k=|i|
xk
∑
i(s)=i
V(s)=k
i(s)!
s!
C s
where the symbol ∑
i(s)=i
V(s)=k
denotes the sum over all s such that V(s) = k, i(s) = i and, for the empty set of s, this
symbol equals 0.
CHAPTER 4. APPLICATIONS TO NONLINEAR SYSTEMS 27
Substituting y(t) into equation (4.6) and matching the coefficients at identical powers
ϕk(t,C), an auxiliary system is obtained of linear differential equations
˙fk(t) = (k −1)a(t)fk(t)+
∞
∑
|i|=2
ci(t) ∑
i(s)=i
V(s)=k
i(s)!
s!
Fs
(4.8)
where F(t) is the n-tuple of sequences { fk(ξi(t))exp(k(A(t)−A(ξi(t))))}∞
k=1, i.e.,
F(t) = ...{ fk(ξi(t))exp(k(A(t)−A(ξi(t))))}∞
k=1,...).
V(s) = k ≥ 2 and |i(s)| ≥ 2 imply sl
k = 0 for l ≥ k. From this follows that the auxiliary
system (4.8) is recurrent.
Theorem 12. Let the functions ci(t) for all positive τ and i satisfy
lim
t→∞
ci(t)exp(−τA(t)) = 0.
Then, there exists a sequence { fk(t)}∞
k=1 of solutions of the auxiliary system (4.8)
fk(t) =
∞
t
−a(s)exp −
s
t
(k −1)a(u)du
∞
∑
|i|=2
ci(t) ∑
i(s)=i
V(s)=k
|i(s)|!
i(s)!
Fs
ds (4.9)
such that lim
t→∞
fk(t)exp(−τA(t)) = 0 for all positive τ.
Let . denote the maximum norm on C0[r∗,t0].
The next 2 Theorems are proved in [a11]. The first is a consequence of Theorem 11.
Theorem 13. Let the assumptions of Theorem 12 hold and let
lim
t→∞
(fk+1(t))−1
exp(−τA(t)) = 0
where τ < 1 is a constant. We denote r∗
= min
t≥t0
(ξ(t)). Then, for every C = 0 and
ψ ∈ C0[r∗,t0], ψ ≤ 1,ψ(t0) = 0, there exists a solution yC(t) of equation (4.6) such that
|yC(t)−yk(t)| ≤ σ|fk+1(t)ϕk+1
(t,C)| (4.10)
for t ∈ [tC,∞) where the functions fk(t) are solutions (4.9) of system (4.8), σ > 1 is a
constant. tC is a function of the parameter C and of σ,k.
Theorem 14. Let the assumptions of Theorem 12 be satisfied and let there exist a sequence
{Kk}∞
k=1, K0 = 1 such that the assumptions of Theorem 13 are satisfied for every Kk, i.e.,
lim
t→∞
(fKk
(t)−1)exp(−τA(t)) = 0. Then, there exists an asymptotic expansion of the solution
yC(t) in the form
yC(t) ≈
∞
∑
k=1
Fk(t), where Fk(t) =
Kk−1
∑
l=Kk−1
fl(t)ϕl
(t,C)
and fl(t) are solutions of (4.9).
These theorems are applicable to (1.2), (1.3) above and are used in the bellow illustrative
example
CHAPTER 4. APPLICATIONS TO NONLINEAR SYSTEMS 28
4.2 Example
Consider the equation
˙y(t) = −ycos(ty(ξ(t))) = −y(t)+
∞
∑
k=1
(−1)k+1 t2ky(t)(y(ξ(t)))2k
(2k)!
on the interval [1,∞) together with two various delays by choosing a pair of functions
ξ1(t) = t −r, ξ2(t) = t −lnt
• the first delay r1(t) = t −ξ1(t) = r = r > 0 is constant
• the second delay r2(t) = t −ξ2(t) = t −(t −lnt) = lnt is unbounded.
In this case, with a(t) = 1, we put y0 = 1 ⇒ A(t) = t −1, i = (i1,i2),
c(1,2k) = (−1)k+1 t2k
(2k)!
(for other multiindices ci = 0).
If we denote F = { fi(t)}∞
i=1, fi(ξ(t))ei(t−ξ(t))
∞
i=1
, the system of auxiliary differential
equations has the form
˙fk(t) = (k −1)fk(t)+
∞
∑
i=1
(−1)i+1 t2i
(2i)! ∑
i(s)=(1,2i)
V(s)=k
i(s)!
s!
Fs
.
By induction, we may prove that, for any delay, f2k = 0 holds. First, f2(t) = 0 is due to
˙f2(t) = f2(t).
From i(s1,s2) = (1,2l), it follows |s1| = 1 and |s2| = 2l. By the induction assumption
f2j = 0, from ({ fi(t)}∞
i=1)s1 = 0, it follows that V(s1) is odd. From the requirement
F(s1,s2) = 0 and the fact that 2k = V(s1,s2) = V(s1)+V(s2), we deduce that V(s2) is odd,
too. Because
V(s2) =
∞
∑
k=1
ksk
2 =
∞
∑
k=1
k +···+k
sk
2
can be interpreted as the sum of |V(s2)| = 2l numbers. We see that at least one number
is even (the sum of an even number of odd numbers is even) and every product on the
right-hand side of the auxiliary equation contains zero multiplicands and, for the function
f2k, we have
˙f2k(t) = f2k(t) ⇒ f2k = 0.
The asymptotic form of the solutions f2k+1 depends on the delay ri(t) = t −ξi(t) but the
property
f2k−1(t) ∼ f2k−1(ξ(t)) holds for both ri(t).
First, for r1(t), the solutions have the asymptotic form
f2k+1 = t2k
(c2k+1 +O(1/t)),
CHAPTER 4. APPLICATIONS TO NONLINEAR SYSTEMS 29
where c1 = 1 and c2k+1 are given by the recurrent formula
c2k+1 =
1
2k
∞
∑
i=1
(−1)i
(2i)! ∑
i(s)=(1,2i)
V(s)=2k+1
C s1C s2
r ,
where C = {ci}∞
i=1, Cr = {ci exp(ir)}∞
i=1.
Second, we have the equation exp(k(A(t) − A(ξ(t)))) = exp(klnt) = tk. It can be
proved by induction that the solutions f2k+1 have the asymptotic form
f2k+1 = −t2k+p(k−1)
(d(k −1)/2k +O(1/t)).
The constants d(k) and p(k) satisfy the recurrence formulas
d(k) = −d(k −1)/2k, p(k) = p(k −1)+2k,
otherwise
d(k) =
(−1)k−1
2k(k −1)!
and p(k) = (k +2)(k −1).
By Theorem 14, we obtain the existence of a pair of asymptotic expansions y1(t), y2(t) of
the solutions for two different delays r1(t), r2(t):
y1(t) ≈
∞
∑
k=1
t2(k−1)
c2k−1e(2k−1)t
C2k−1
y2(t) ≈
∞
∑
k=1
(−1)k−1t(k+2)(k−1)
2k(k −1)!
e(2k−1)t
C2k−1
.
Fore more details see [a11].
Remark 3. This example shows the fundamental dependence of the asymptotic properties
of the expansion on the magnitude of the delay. For a small delay (r1(t) → 0), the expansion
y1(t) converges to the expansion of the solution of an ordinary equation
˙y(t) = −ycos(ty(t)).
For a sufficiently large delay r2(t) = ln(t), the expansion y2(t) is the same as for the
equation
˙y(t) = −y(t)+t2
y(t)y2
(t −lnt)/2,
i.e., the expansions for the perturbation with an infinite sum and for the perturbation with
only the first summand are identical.
4.3 Positive solutions of nonlinear system
The next theorem easily follows from the more general Theorem 11 by putting ρ(t) = 0
and applying the next definition.
CHAPTER 4. APPLICATIONS TO NONLINEAR SYSTEMS 30
Definition 9. A functional g ∈ C(Ω,Rn) is called i-strongly decreasing (or i-strongly
increasing), i ∈ {1,2,...,n}, if, for each (t,ϕ) ∈ Ω and (t,ψ) ∈ Ω such that
ϕ(p(t,ϑ)) ψ(p(t,ϑ)), where ϑ ∈ [−1,0) and ϕi(p(t,0) = ψi(p(t,0)),
the inequality
gi(t,ϕ) > gi(t,ψ) (or gi(t,ϕ) < gi(t,ψ))
holds.
Let k = (k1,...,kn) 0 be a constant vector. Let λ(t) = (λ1(t),...,λn(t)) denote
a vector, defined and locally integrable on [ p∗,∞). Define an auxiliary operator
T(k,λ)(t) := ke
t
p∗ λ(s)ds
= k1e
t
p∗ λ1(s)ds
,...,kne
t
p∗ λn(s)ds
. (4.11)
Theorem 15. Let f ∈C(Ω,Rn) be locally Lipschitzian with respect to the second argument,
quasibounded and, moreover:
(i) f is i-strongly decreasing if i = 1,..., p and i-strongly increasing if i = p+1,...,n.
(ii) fi(t,0) ≤ 0 for i = 1,..., p and fi(t,0) ≥ 0 for i = p+1,...,n if (t,0) ∈ Ω.
Then, for the existence of a positive solution y = y(t) on [ p∗,∞) of the system of p RFDE’s
(3.1) (where p∗ = p(t∗,−1)), the existence of a positive constant vector k and
a locally integrable vector λ : [ p∗,∞) → Rn continuous on [ p∗,t∗)∪[t∗,∞) satisfying the
system of integral inequalities
µiλi(t) ≥
µi
ki
e− t
p∗ λi(s)ds
· fi (t,T(k,λ)t), i = 1,...,n (4.12)
for t ≥ t∗ with µi = −1 for i = 1,..., p and µi = 1 for i = p + 1,...,n is necessary and
sufficient.
For more details, see [a3]. Here and in [a4] applications to some examples may be
found.
The next theorem only gives a sufficient condition for the existence of a positive
solution. Let a constant vector k 0 and a vector λ(t) defined and locally integrable
on [ p∗,∞) are given. Then, an operator T is well defined by (4.11). Define for every
i ∈ {1,2,...,n} two types of subsets of the set C :
T i
:= φ ∈ C : 0 φ(ϑ) T(k,λ)t(ϑ), ϑ ∈ [−1,0]
except for φi(0) = kie
t
p∗ λi(s)ds
and
Ti := {φ ∈ C : 0 φ(ϑ) T(k,λ)t(ϑ), ϑ ∈ [−1,0] except for φi(0) = 0}.
CHAPTER 4. APPLICATIONS TO NONLINEAR SYSTEMS 31
Theorem 16. Let f ∈C(Ω,Rn) be locally Lipschitzian with respect to the second argument
and quasibounded. Let a constant vector k 0 and a vector λ(t) defined and locally
integrable on [ p∗,∞) are given. If, moreover, inequalities
µiλi(t) >
µi
ki
e− t
p∗ λi(s)ds
· fi (t,φ) (4.13)
hold for every i ∈ {1,2,...,n}, (t,φ) ∈ [t∗,∞)×T i and inequalities
µi fi(t,φ) > 0 (4.14)
hold for every i ∈ {1,2,...,n}, (t,φ) ∈ [t∗,∞) × Ti, where µi = −1 for i = 1,..., p and
µi = 1 for i = p + 1,...,n, then there exists a positive solution y = y(t) on [ p∗,∞) of the
system of p -RFDE’s (3.1).
These results together with Theorem 15 make it possible to formulate numerous consequences
particularly for linear applications. For more details, see [a5].
Chapter 5
Positive solutions of a linear system
Many resulte may be derived for a linear system. Consider a system
˙y(t) = L(t,yt)+h(t), (5.1)
where h ∈ C([t∗,∞),Rn), L ∈ C(Ω × C ,Rn) is a linear functional and yt is defined in
accordance with Definition 3. Then, the bellow Theorems 15 and 16 give corresponding
linenar analogies.
Theorem 17. Let L ∈ C(Ω×C ,Rn) and, moreover,:
(i) For i = 1,..., p, L is i-strongly decreasing and Li(t,0)+hi(t) ≤ 0 if (t,0) ∈ Ω and
(ii) for i = p+1,...,n, L is i-strongly increasing and Li(t,0)+hi(t) ≥ 0 for if (t,0) ∈ Ω.
Then, the existence of a positive solution y(t) on [ p∗,∞) of the system of p -RFDE’s (5.1)
(where p∗ = p(t∗,−1)) is equivalent with the existence of a positive constant vector k and
a locally integrable vector λ : [ p∗,∞) → Rn continuous on [ p∗,t∗)∪[t∗,∞) satisfying the
system of integral inequalities
µiλi(t) ≥
µi
ki
·e− t
p∗ λi(s)ds
·(Li (t,T(k,λ)t)+hi(t)), i = 1,...,n (5.2)
for t ≥ t∗ with µi = −1 for i = 1,..., p and µi = 1 for i = p+1,...,n.
Theorem 18. Let L ∈ C(Ω,Rn) be linear. Let a constant vector k 0 and a vector λ(t)
defined and locally integrable on [ p∗,∞) are given. If, moreover the inequalities
µiλi(t) >
µi
ki
·e− t
p∗ λi(s)ds
·(Li (t,φ)+hi(t)) (5.3)
hold for every i ∈ {1,2,...,n}, (t,φ) ∈ [t∗,∞)×T i and inequalities
µi (Li(t,φ)+hi(t)) > 0 (5.4)
hold for every i ∈ {1,2,...,n}, (t,φ) ∈ [t∗,∞) × Ti, where µi = −1 for i = 1,..., p and
µi = 1 for i = p+1,...,n. Then, there exists a positive solution y = y(t) on [ p∗,∞) of the
system p -RFDE’s (5.1).
32
CHAPTER 5. POSITIVE SOLUTIONS OF A LINEAR SYSTEM 33
The proofs of both theorems may be found in [58]. [a3] consideres the linear system
˙y = A(t)y(t)+B(t)y(τ(t)) (5.5)
where τ : [t∗,∞) → [ p∗,∞) is a continuous nondecreasing function and τ(t) < t. In this
case, p(t,ϑ) = t + ϑ · (t − τ(t)) and p∗ = τ(t∗). With respect to the n × n matrices
A(t) = (aij(t)) and B(t) = (bij(t)), we assume their continuity on [t∗,∞) and, moreover,
the validity of the inequalities:
aij(t) ≤ 0, bij(t) ≤ 0 if i = 1,..., p, j = 1,...,n, t ∈ [t∗
,∞), (5.6)
aij(t) ≥ 0, bij(t) ≥ 0 if i = p+1,...,n, j = 1,...,n, t ∈ [t∗
,∞), (5.7)
n
∑
j=1
bij(t) = 0 for every i = 1,...,n and t ∈ [t∗
,∞). (5.8)
Here the next Theorem is proved.
Theorem 19. For the existence of a solution y = y(t) of system (5.5), positive on [ p∗,∞),
the necessary and sufficient condition is that there exists a continuous vector λ ∈C([ p∗,∞),
Rn) such that λ(t) 0 fort ≥t∗, satisfying for i = 1,...,n the system of integral inequalities
λi(t) ≥ µi aii(t)+bii(t)e−µi
t
τ(t) λi(s)ds
+
µi
ki
·
n
∑
j=1,j=i
kje
t
p∗(µjλj(s)−µiλi(s))ds
aij(t)+bij(t)e−µj
t
τ(t) λj(s)ds
, (5.9)
on [t∗,∞) with a positive constant vector k and with µi = −1 for i = 1,..., p; µi = 1 for
i = p+1,...,n.
Remark 4. Earlier, sufficient conditions for the existence of bounded solutions of systems
and equations of the type (5.5) were given in [9, 8].
[a3] establishes sufficient conditions for the existence of positive solutions to the
following linear system
˙y(t) = −A(t)y(p(t,−1)) (5.10)
where A = {aij} is an n×n matrix with entries continuous on [t∗,∞) satisfying aij(t) ≥ 0,
i, j = 1,2,...,n and
n
∑
j=1
aij(t) > 0 for every i = 1,2,...,n,. The next theorem is proved
in [a3].
Theorem 20. For the existence of a positive solution y = y(t) on [ p∗,∞) (with p∗ =
p(t∗,−1)) of linear system (5.10) a sufficient condition is the existence of a positive constant
vector k and a locally integrable function λ∗ : [ p∗,∞) → R continuous on [ p∗,t∗)∪[t∗,∞)
and satisfying the integral inequality
λ∗
(t)e− t
p(t,−1) λ∗(q)dq
≥ max
i=1,2,...,n
1
ki
n
∑
j=1
kjaij(t) (5.11)
for t ≥ t∗.
CHAPTER 5. POSITIVE SOLUTIONS OF A LINEAR SYSTEM 34
Inequality (5.11) offers numerous possibilities of finding particular sufficient conditions.
We will consider two of them.
Theorem 21. Let a continuous nondecreasing function λ∗ : [ p∗,∞) → R satisfy the in-
equality
λ∗
(t)e−λ∗(t)·[t−p(t,−1)]
≥ max
i=1,2,...,n
1
ki
n
∑
j=1
kjaij(t) (5.12)
for t ≥ t∗, where k = (k1,k2,...,kn) is a suitable positive constant vector. Then, linear
system (5.10) has a positive solution y = y(t) on [ p∗,∞) (with p∗ = p(t∗,−1)).
Theorem 22. Let A be a constant matrix that cannot be decomposed. Then, for the existence
of a positive solution y = y(t) on [ p∗,∞) (with p∗ = p(t∗,−1)) of linear system (5.10), it is
sufficient if a locally integrable function λ∗ : [ p∗,∞) → R, continuous on [ p∗,t∗)∪[t∗,∞),
satisfies the inequality
λ∗
(t)e− t
p(t,−1) λ∗(q)dq
≥ ρ(A) (5.13)
for t ≥ t∗, where ρ(A) is the spectral radius of A.
5.1 A scalar equation with discrete delays
Let us study the conditions for the existence of a positive solution to a scalar equation with
discrete delays
˙y(t) = −
m
∑
q=1
cq(t)y(p(t,ϑq)) (5.14)
with −1 = ϑ1 < ϑ2 < ··· < ϑm = 0,functions cq, q = 1,2,...,m, continuous on [t∗,∞) that
are nonnegative if q = 1,2,...,m−1 and satisfy inequality ∑m−1
q=1 cq(t) > 0 for t ∈ [t∗,∞).
Theorem 23. For the existence of a positive solution y = y(t) on [ p∗,∞) (where p∗ =
p(t∗,−1)) of the equation (5.14) the existence is necessary and sufficient of a locally
integrable function λ∗ : [ p∗,∞) → R continuous on [ p∗,t∗) ∪ [t∗,∞) and satisfying the
integral inequality
λ∗
(t) ≥
m
∑
q=1
cq(t)e
t
p(t,ϑq) λ∗(s)ds
(5.15)
for t ≥ t∗.
Example 1. Consider equation (5.14) with m = 3, c3(t) ≡ 0. Let c1(t), c2(t) be positive
continuous functions, ϑ1 = −1, ϑ2 = −1/2, ϑ3 = 0 and let the p-function be defined as:
p(t,θ) =
t +2τθ for θ ∈ (−1/2,0],
2(t −τ)(θ +1)+
√
t(θ +1/2)(−2) for θ ∈ [−1,−1/2].
Then, equation (5.14) takes the form:
˙y(t) = −c1(t)y(
√
t)−c2(t)y(t −τ), (5.16)
CHAPTER 5. POSITIVE SOLUTIONS OF A LINEAR SYSTEM 35
where c1, c2 are positive continuous functions and the inequality (5.15) has the form:
λ(t) ≥ c1(t)exp t√
t λ(s)ds +c2(t)exp t
t−τ λ(s)ds .
We put λ(t) = 1/t. Then, we obtain
1
t
≥ c1(t)
t
t −τ
+c2(t)
t
√
t
.
This inequality (on the interval [p∗,∞)) is a sufficient condition for the existence of a
positive solution of equation (5.16) on interval [(p∗)2,∞). Also, the equalities
c1(t) = o
1
t
and c2(t) = o
1
t
√
t
for t → ∞
are sufficient conditions for the existence of an eventually positive solution of equation
(5.16).
Theorem 23 can serve as a source of various sufficient conditions including the wellknown
sufficient conditions given, e.g., in [19, 32]. It is possible to show several concrete
consequences of Theorem 23 concerning the equation
˙y(t) = −c(t)y(p(t,−1)) (5.17)
with a positive continuous function c. Obviously, equation (5.17) is a particular case
of (5.14) if m = 1.
Theorem 24. Let c be a positive continuous function on [ p∗,∞) and let the inequality
e·
t
p(t,−1)
c(s)ds ≤ 1 (5.18)
hold on [t∗,∞) (with p∗ = p(t∗,−1)). Then, (5.17) has a positive solution y = y(t) on
[ p∗,∞).
The following corollary follows directly from (5.18).
Corollary 1. Let all conditions of Theorem 24 be valid and let there exist a nondecreasing
function b(t), t ∈ [p∗,∞) such that c(t) ≤ b(t) holds on [p∗,∞) and
b(t) ≤
1
e·[t − p(t,−1)]
(5.19)
holds on [t∗,∞). Then, (5.17) has a positive solution y = y(t) on [ p∗,∞).
Theorem 25. Let c(t) be a positive continuous function on [t∗,∞) and let there exist a
positive constant K such that
c(t) ≤ Ke−K(t−p(t,−1))
(5.20)
on [t∗,∞). Then, (5.17) has a positive solution y = y(t) on [ p∗,∞) (with p∗ = p(t∗,−1)).
Remark 5. The results presented are sharp. This may be demonstrated, e.g., by the last
result. If p(t,−1) := t − τ with a positive constant τ, c(t) ≡ c = const and if K := 1/τ,
then (5.20) yields a classical result ([32, Theorem 2.2.3]) ensuring the existence of a
positive solution:
cτe ≤ 1.
CHAPTER 5. POSITIVE SOLUTIONS OF A LINEAR SYSTEM 36
5.2 Positive solutions to a scalar equation in the critical
case
In [a8], the oscillation is discussed of solutions to the equation
˙y(t) = −a(t)y(t −τ(t)), (5.21)
where t ∈ I := [t0,∞), t0 ∈ R, a: I → R+ := (0,∞) is a continuous function and τ : I → R+
is a continuous function such that t −τ(t) > t0 −τ(t0) if t > t0.
This study has been motivated by what can be found in the [13], [14], [17], [18], [54].
Using an example, it may be shown the that simple generalization of the results given
in [13], [14] does not describe the situation completely. In [a8] two criteria are derived.
The first one is based on the criterion for the case of a constant delay derived in [14].
Theorem 26. I) Let us assume that a(t) ≤ ak(t) with
ak(t) :=
1
eτ
+
τ
8et2
+
τ
8e(t lnt)2
(5.22)
+
τ
8e(t lnt ln2 t)2
+···+
τ
8e(t lnt ln2 t ...lnk t)2
if t → ∞ and for an integer k ≥ 0. Then, there exists a positive solution x = y(t) of (5.21)
with τ(t) ≡ τ = const. Moreover,
y(t) < νk(t) := e−t/τ
t lnt ln2 t ...lnk t
as t → ∞.
II) Let us assume that
a(t) > ak−2(t)+
θτ
8e(t lnt ln2 t ...lnk−1 t)2
(5.23)
if t → ∞, for an integer k ≥ 2 and a constant θ > 1. Then, all the solutions of (5.21) with
τ(t) ≡ τ = const oscillate.
Now we give two possible criteria. For the first, one we define a new auxiliary function
akτ(t) similarly to (5.22) replayicing constant τ by function τ(t).
Theorem 27. Let us assume that
a(t) ≤ akτ(t) and
t
t−τ(t)
ds/τ(s) ≤ 1
if t → ∞ for an integer k ≥ 0. Let moreover τ(t)lnt ln2 t ...lnk t = o(t) as t → ∞. Then,
there exists a positive solution x = y(t) of (5.21) satisfying
y(t) < t lnt ln2 t ...lnk t ·exp
t
t0−τ(t0)
−1
τ(s)
ds (5.24)
as t → ∞.
CHAPTER 5. POSITIVE SOLUTIONS OF A LINEAR SYSTEM 37
Theorem 28. Let us assume that
a(t) ≤
1
τ(t)
·exp −
t
t−τ(t)
ds
τ(s)
(5.25)
as t → ∞. Then, there exists a positive solution x = y(t) of (5.21). Moreover,
y(t) < exp −
t
t0−τ(t0)
ds
τ(s)
.
Analysis of both criteria To compare Theorem 27 with Theorem 28, we will investigate
equation (5.21), where
τ(t) := c+d/t (5.26)
and c, d are positive constants, i.e., we consider an equation
˙y(t) = −a(t)y(t −c−d/t). (5.27)
Application of the first criterion The delay (5.26) is decreasing, tends to c as t → ∞
and satisfies the inequality t
t−τ(t)
ds
τ(s) < 1. If
a(t) ≤ akτ(t) (5.28)
for an integer k ≥ 0 as t → ∞ then, by Theorem 27, equation (5.27) has a positive solution.
We will develop the first several terms of the asymptotic decomposition of akτ(t) with τ(t)
given by (5.26) if t → ∞ and rewrite condition (5.28) to get a sufficient condition for the
existence of a positive solution of (5.27) in the form
a(t) ≤ akτ(t) =
1
ec
−
d
ec2
·
1
t
+
1
e
·
d2
c3
+
c
8
·
1
t2
+o
1
t2
. (5.29)
Application of the second criterion We compute
t
t−τ(t)
ds
τ(s)
= 1+
d
ct
−
d
c2
ln
t
t −c
.
Now we are able to asymptotically decompose the right-hand side of inequality (5.25)
as t → ∞. We get
1
τ(t)
exp
t
t−τ(t)
−ds
τ(s)
=
1
ec
−
d
ec2
·
1
t
+
1
e
·
d2
c3
+
d
2c
·
1
t2
+o
1
t2
.
Finally, by the second criterion, a sufficient condition for the existence of a positive
solution of (5.27) is
a(t) ≤
1
ec
−
d
ec2
·
1
t
+
1
e
·
d2
c3
+
d
2c
·
1
t2
+o
1
t2
. (5.30)
CHAPTER 5. POSITIVE SOLUTIONS OF A LINEAR SYSTEM 38
Final comparison
Comparing the right-hand sides of expressions (5.29) and (5.30), we see that the first two
terms of both decompositions coincide. The quality of every criterion is expressed by the
coefficients of the term 1/t2, by the coefficient CI
2 in the case of expression (5.29) and the
coefficient CII
2 , i.e.,
CI
2 =
1
e
·
d2
c3
+
c
8
CII
2 =
1
e
·
d2
c3
+
d
2c
We conclude CI
2 < CII
2 if c2 < 4d and CI
2 > CII
2 if c2 > 4d. Thus, we can state.
Theorem 29. The first criterion is more general in the case of c2 > 4d; the second criterion
is more general if c2 < 4d.
Using this example we may show that Theorem 26 cannot be generalized by replacing
the constant delay τ by a nonconstant function τ(t). For more details, see [a8]. This result
has been used in some papers.
5.3 A scalar equation with distributed delay
In [a5] we consider the existence of a positive solution of a scalar equation having the
distributed delay
˙y(t) = −
ϑ∗
−1
c(t,ϑ)y(p(t,ϑ))dϑ (5.31)
with ϑ∗ ∈ (−1,0], and continuous c : [t∗,∞)×[−1,ϑ∗] → (0,∞). The main results of [a5]
are the following.
Theorem 30. For the existence of a positive solution y = y(t) on [ p∗,∞) (where p∗ =
p(t∗,−1)) of the equation (5.31), the existence is necessary and sufficient of a locally
integrable function λ∗ : [ p∗,∞) → R continuous on [ p∗,t∗) ∪ [t∗,∞) and satisfying the
integral inequality
λ∗
(t) ≥
ϑ∗
−1
c(t,ϑ)e
t
p(t,ϑ) λ∗(q)dq
dϑ (5.32)
for t ≥ t∗.
The following results are consequences of Theorem 30.
Theorem 31. Let there exist a positive constant K such that inequality
ϑ∗
−1
c(t,ϑ)dϑ ≤ Ke−K·[t−p(t,−1)]
(5.33)
holds on [t∗,∞). Then, equation (5.31) with a positive continuous function c on [t∗,∞)×
[−1,ϑ∗] has a positive solution y = y(t) on [ p∗,∞) (where p∗ = p(t∗,−1)).
CHAPTER 5. POSITIVE SOLUTIONS OF A LINEAR SYSTEM 39
Theorem 32. Let the difference t − p(t,−1) be a nonincreasing function on [t∗,∞). Then,
equation (5.31) with a positive continuous function c on [t∗,∞) × [−1,ϑ∗] has a positive
solution y = y(t) on [ p∗,∞) (where p∗ = p(t∗,−1)) if the inequality
ϑ∗
−1
c(t,ϑ)dϑ ≤
1
e·[t − p(t,−1)]
(5.34)
holds on [t∗,∞).
A straightforward consequence of inequality (5.34) is the following corollary.
Corollary 2. Let all conditions of Theorem 32 be valid and let there exist a function
b : [t∗,∞) × [−1,ϑ∗] → R, nondecreasing in ϑ on [−1,ϑ∗] for each t ∈ [t∗,∞), such that
c(t,ϑ) ≤ b(t,ϑ) on [t∗,∞)×[−1,ϑ∗]. If, moreover,
b(t,ϑ∗) ≤
1
e·[t − p(t,−1)](1+ϑ∗)
(5.35)
holds on [t∗,∞), then (5.31) has a positive solution y = y(t) on [ p∗,∞).
5.4 Existence of decreasing positive solutions to linear differential
equations of neutral type
Using the retract method, a new criterion is derived in [a7] for the existence of positive decreasing
solutions to linear differential equations of neutral type: linear neutral differential
equation
˙y(t) = −c(t)y(t −τ(t))+d(t)˙y(t −δ(t)) (5.36)
where c,d : [t0,∞) → [0,∞), t0 ∈ R and τ,δ : [t0,∞) → (0,r], r ∈ R, r > 0 are continuous
functions and c(t)+d(t) > 0, t ∈ [t0,∞).
Theorem 33. For the existence of a positive decreasing solution of (5.36) on [t0 −r,∞), the
necessary and sufficient condition is that there exists a continuous function λ : [t0 −r,∞) →
(0,∞) such that inequality
λ(t) ≥ c(t)exp
t
t−τ(t)
λ(s)ds +d(t)λ(t −δ(t))exp
t
t−δ(t)
λ(s)ds ,
holds for t ≥ t0.
Let the functions c(t), d(t) and delays τ(t), δ(t) in equation (5.36) be constant, i.e.,
c(t) ≡ c = const, d(t) ≡ d = const, τ(t) ≡ τ = const, δ(t) ≡ δ = const, then equation (5.36)
becomes
˙y(t) = −cy(t −τ)+d ˙y(t −δ). (5.37)
Corollary 3. For the existence of a positive decreasing solution of (5.37) on [t0 − r,∞),
the existence is sufficient of a positive constant λ such that inequality
λ ≥ ceλτ
+λdeλδ
(5.38)
holds.
CHAPTER 5. POSITIVE SOLUTIONS OF A LINEAR SYSTEM 40
For the choice of λ = 1/τ or λ = 1/δ in (5.38), we get
Corollary 4. For the existence of a positive decreasing solution of (5.37) on [t0 −r,∞) it
is sufficient that either inequality
1 > ceτ +deδ/τ
(5.39)
or inequality
1 > cδeτ/δ
+de (5.40)
holds.
Chapter 6
Exponential stability of linear delay
differential systems
6.1 Formulation of the problem
In [a1] the uniform exponential stability is studied of linear systems with time varying
coefficients
˙xi(t) = −
m
∑
j=1
rij
∑
k=1
ak
ij(t)xj(hk
ij(t)), i = 1,...,m (6.1)
where t ≥ 0, m and rij, i, j = 1,...,m are natural numbers, coefficients ak
ij : [0,∞) → R
and delays hk
ij : [0,∞) → R are measurable functions. For the scalar case (m = 1), the
system (6.1) reduces to a linear differential equation with several delays
˙x(t) = −
r
∑
k=1
ak(t)x(hk(t)). (6.2)
Equation (6.2) is studied in detail, e.g., in [3], [29], [30], [28], [42], and a review on stability
results can be found in [4]. For system (6.1), there are not so many results.
The following short overview of the existing results uses the notion of an M-matrix.
A square matrix is called a non-singular M-matrix if all its off-diagonal elements are
non-positive and its principal minors are positive. (In [5], equivalent definitions can be
found.)
An asymptotic stability conditions for the autonomous case of system (6.1) (when
ak
ij(t) ≡ ak
ij, hk
ij(t) ≡ t − τk
ij and ak
ij, τk
ij are constant) is considered in [31]. In particular,
for the system
˙xi(t) = −
m
∑
j=1
aijxj(t −τij), i = 1,...,m, (6.3)
where τij ≥ 0, the following holds (below, a+ denotes the positive part of a, i.e., a+ =
max{a,0}).
Theorem 1 (Corollary 4.3, [31]). Let
0 < aiiτii < 1+1/e, i = 1,...,m
41
CHAPTER 6. EXPONENTIAL STABILITY 42
and let the m×m matrix H with components
hij =



1−(aiiτii −1/e)+
1+(aiiτii −1/e)+
aii, i = j,
−|aij|, i = j,
i, j = 1,...,m be a non-singular M-matrix. Then, system (6.3) is asymptotically stable for
any selection of delays τij, i = j, i, j = 1,...,m.
In [55], the system (6.3) is also considered and the following derived.
Theorem 2 (Theorem 1.3, [55]). Let
0 ≤ aiiτii < 3/2, i = 1,...,m
and let the matrix G with components
gij =



−
1+aiiτii(3+2aiiτii)/9
1−aiiτii(3+2aiiτii)/9
|aij|, i = j,
aii, i = j,
be a nonsingular M-matrix. Then, system (6.3) is asymptotically stable for any selection
of delays τij, i = j, i, j = 1,...,m.
In [56], the authors consider the non-autonomous system
˙xi(t) = −
m
∑
j=1
aij(t)xj(hij(t)), i = 1,...,m, (6.4)
where t ∈ [t0,∞), t0 ∈ R, aij(t), hij(t) are continuous functions, hij(t) ≤ t, and hij(t) are
monotone increasing functions such that limt→∞ hij(t) = ∞, i, j = 1,...,m.
Theorem 3 (Theorem 2.2, [56]). Assume that, for t ≥ t0, there exist non-negative numbers
bij, i, j = 1,...,m, i = j such that |aij(t)| ≤ bijaii(t), i, j = 1,...,m, i = j, aii(t) ≥ 0 and
∞
aii(s)ds = ∞, di = limsup
t→∞
t
hii(t)
aii(s)ds < 3/2, i = 1,...m.
Let ˜B = (˜bij)m
i,j=1 be an m × m matrix with entries ˜bii = 1, i = 1,...,m and, for i = j,
i, j = 1,...,m,
˜bij =



−
2+d2
i
2−d2
i
bij, if di < 1,
−
1+2di
3−2di
bij, if di ≥ 1.
If ˜B is a nonsingular M-matrix, then system (6.4) is asymptotically stable.
Very interesting global asymptotic stability results have been obtained for nonlinear
systems of delay differential equations in the recent papers [45, 21, 22].
Paper [a1] considers general system (6.1) deriving the following result.
CHAPTER 6. EXPONENTIAL STABILITY 43
Theorem 34 ([a1). , Theorem 4] Let there be constants a0 and τ such that, for t ≥ t0,
a∗
i (t) :=
rii
∑
k=1
ak
ii(t) ≥ a0 > 0, 0 ≤ t −hk
ij(t) ≤ τ, i = 1,...,m (6.5)
and
max
i=1,...,m
esssup
t≥t0
1
a∗
i (t)




rii
∑
k=1
|ak
ii(t)|
t
max{0,hk
ii(t)}
m
∑
j=1
rij
∑
l=1
|al
ij(s)|ds
+
m
∑
j=1
j=i
rij
∑
k=1
|ak
ij(t)|



 < 1. (6.6)
Then, system (6.1) is uniformly exponentially stable.
6.2 Preliminaries
The linear system (6.1) for t ≥ t0 (assuming t0 ≥ 0) is considered with the initial condition
x(t) = ϕ(t), t ≤ t0, (6.7)
under the following assumptions:
(a1) Functions ak
ij : [0,∞) → R, i, j = 1,...,m, k = 1,...,rij are Lebesgue measurable
and essentially bounded functions.
(a2) Functions hk
ij : [0,∞) → R, i, j = 1,...,m, k = 1,...,rij are Lebesgue measurable
functions, hk
ij(t) ≤ t, and
limsup
t→∞
(t −hk
ij(t)) < ∞.
(a3) ϕ : (−∞,t0] → Rm is a Borel measurable bounded vector-function.
Definition 10. A locally absolutely continuous vector-function x: R → Rm is called a solution
to the problem (6.1), (6.7) for t ≥ t0 if its entries xi, i = 1,...,m satisfy equation (6.1)
for almost all t ∈ [t0,∞) and equality (6.7) holds for t ≤ t0.
Definition 11. Equation (6.1) is uniformly exponentially stable, if there exist constants
M > 0 and µ > 0 such that the solution x: R → Rm of problem (6.1), (6.7) satisfies the
inequality
|x(t)| ≤ Me−µ(t−t0)
sup
t≤t0
|ϕ(t)|, t ≥ t0
where M and µ do not depend on t0.
CHAPTER 6. EXPONENTIAL STABILITY 44
6.3 Statement of results
Let Ai, i = 1,...,m be functions defined as
Ai(t) :=
1
ai(t)




rii
∑
k=1
ak
ii(t)
t
max{t0,hk
ii(t)}
m
∑
j=1
rij
∑
l=1
|al
ij(s)|ds +
m
∑
j=1
j=i
rij
∑
k=1
|ak
ij(t)|




where
ai(t) :=
rii
∑
k=1
ak
ii(t). (6.8)
Theorem 35. Let
ai(t) ≥ a0 > 0, i = 1,...,m, t ≥ t0, (6.9)
max
i=1,...,m
esssup
t≥t0
1
ai(t)
m
∑
j=1
j=i
rij
∑
k=1
|ak
ij(t)| < 1 (6.10)
and
max
i=1,...,m
esssup
t≥t0
Ai(t) < 1+
1
e
. (6.11)
Then, the system (6.1) is uniformly exponentially stable.
PROOF Can be found in [a2]. It also includes the corollaries ( Corollary 1-10) of the
Theorem 35 that generalize the corollaries of the Theorem 34 in [A1]. Similarly, corollaries
of the Theorem 35 generalize corollaries of Theorem 34 in [a1].
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stability conditions for a system of linear delay differential equations. Applied
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(1). p. 1831 - 1848.
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functions via Lambert function, Discrete and Continuous Dynamical SystemsSeries
B 23-1 January 2018 special issue, doi:10.3934/dcdsb.2018008
[a7] J. DIBLÍK, Z. SVOBODA, Existence of Strictly Decreasing Positive Solutions of
Linear Differential Equations of Neutral Type, Discrete and Continuous Dynamical
Systems - S 8 special issue, accepted.
[a8] DIBLÍK J., SVOBODA Z., ŠMARDA Z., Explicit Criteria for Existence of Positive
Solutions for a Scalar Differential Equations with Variable Delay in Critical Case.
Computers and Mathematics with Applications. 2008. 56, 2008 (1). p. 556 - 564.
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BIBLIOGRAPHY 51
[a13] Z., Asymptotic unboundedness of the norms of delayed matrix sine and cosine
Electron. J. Qual. Theory Differ. Equ. No. 89, (2017) 15;
Simple uniform exponential stability conditions for a system
of linear delay differential equations
Leonid Berezansky a
, Josef Diblík b,⇑,1
, Zdeneˇk Svoboda b,1
, Zdeneˇk Šmarda b,2
a
Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
b
Brno University of Technology, Brno, Czech Republic
a r t i c l e i n f o
Keywords:
Uniform exponential stability
Linear delay differential system
Bohl–Perron theorem
a b s t r a c t
Uniform exponential stability of linear systems with time varying coefficients
_xiðtÞ ¼ À
Xm
j¼1
Xrij
k¼1
ak
ijðtÞxjðh
k
ijðtÞÞ; i ¼ 1; . . . ; m
is studied, where t P 0; m and rij; i; j ¼ 1; . . . ; m are natural numbers, ak
ij : ½0; 1Þ ! R and
h
k
ij : ½0; 1Þ ! R are measurable functions. New explicit result is derived with the proof
based on Bohl–Perron theorem. The resulting criterion has advantages over some previous
ones in that, e.g., it involves no M-matrix to establish stability. Several useful and easily
verifiable corollaries are deduced and examples are provided to demonstrate the advantage
of the stability result over known results.
Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction
In the paper uniform explicit exponential stability is investigated for the linear delay differential system with time varying
coefficients
_xiðtÞ ¼ À
Xm
j¼1
Xrij
k¼1
ak
ijðtÞxjðh
k
ijðtÞÞ; i ¼ 1; . . . ; m ð1Þ
where t P 0; m and rij; i; j ¼ 1; . . . ; m are natural numbers, coefficients ak
ij : ½0; 1Þ ! R and delays h
k
ij : ½0; 1Þ ! R are measurable
functions (additional assumptions will be formulated later).
For the scalar case (m ¼ 1), the system (1) reduces to a linear differential equation with several delays
_xðtÞ ¼ À
Xr
k¼1
akðtÞxðhkðtÞÞ: ð2Þ
http://dx.doi.org/10.1016/j.amc.2014.10.117
0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
⇑ Corresponding author.
E-mail addresses: brznsky@cs.bgu.ac.il (L. Berezansky), diblik@feec.vutbr.cz, diblik.j@fce.vutbr.cz (J. Diblík), svobodaz@feec.vutbr.cz (Z. Svoboda),
smarda@feec.vutbr.cz (Z. Šmarda).
1
This author was supported by the Grant P201/11/0768 of the Czech Grant Agency (Prague).
2
This author was supported by the Grant FEKT-S-14-2200 of Faculty of Electrical Engineering and Communication, Brno University of Technology.
Applied Mathematics and Computation 250 (2015) 605–614
Contents lists available at ScienceDirect
Applied Mathematics and Computation
journal homepage: www.elsevier.com/locate/amc
Eq. (2) is studied in detail, e.g., in [1–5], and a review on stability results can be found in [6]. For system (1), there are not so
many results.
In the following short overview of known results we use the notion of an M-matrix. For the reader’s convenience, we
recall that a square matrix is called a non-singular M-matrix if all its off-diagonal elements are non-positive and its principal
minors are positive. (In [7], equivalent definitions can be found.)
Asymptotic stability conditions for the autonomous case of system (1) (when ak
ijðtÞ  ak
ij; h
k
ijðtÞ  t À sk
ij and ak
ij; sk
ij are
constant) is considered in [8]. In particular, for the system
_xiðtÞ ¼ À
Xm
j¼1
aijxjðt À sijÞ; i ¼ 1; . . . ; m; ð3Þ
where sij P 0, the following result holds (below, aþ denotes the positive part of a, i.e., aþ ¼ maxfa; 0g).
Theorem 1 (Corollary 4.3, [8]). Let
0 < aiisii < 1 þ 1=e; i ¼ 1; . . . ; m
and let the m  m matrix H with components
hij ¼
1À aiisiiÀ1=eð Þþ
1þ aiisiiÀ1=eð Þþ
 
aii; i ¼ j;
Àjaijj; i – j;
(
i; j ¼ 1; . . . ; m be a non-singular M-matrix. Then, system (3) is asymptotically stable for any selection of delays
sij; i – j; i; j ¼ 1; . . . ; m.
In [9], the system (3) is also considered and the following result derived.
Theorem 2 (Theorem 1.3, [9]). Let
0 6 aiisii < 3=2; i ¼ 1; . . . ; m
and let the matrix G with components
gij ¼
À 1þaiisiið3þ2aiisiiÞ=9
1Àaiisiið3þ2aiisiiÞ=9
 
jaijj; i – j;
aii; i ¼ j;
(
be a nonsingular M-matrix. Then, system (3) is asymptotically stable for any selection of delays sij; i – j; i; j ¼ 1; . . . ; m.
In [10], the authors consider the non-autonomous system
_xiðtÞ ¼ À
Xm
j¼1
aijðtÞxjðhijðtÞÞ; i ¼ 1; . . . ; m; ð4Þ
where t 2 ½t0; 1Þ; t0 2 R; aijðtÞ; hijðtÞ are continuous functions, hijðtÞ 6 t, and hijðtÞ are monotone increasing functions such
that limt!1hijðtÞ ¼ 1; i; j ¼ 1; . . . ; m.
Theorem 3 (Theorem 2.2, [10]). Assume that, for t P t0, there exist non-negative numbers bij; i; j ¼ 1; . . . ; m, i – j such that
jaijðtÞj 6 bijaiiðtÞ; i; j ¼ 1; . . . ; m; i – j, aiiðtÞ P 0 and
Z 1
aiiðsÞds ¼ 1; di ¼ lim sup
t!1
Z t
hiiðtÞ
aiiðsÞds < 3=2; i ¼ 1; . . . m:
Let ~B ¼ ð~bijÞ
m
i;j¼1 be an m  m matrix with entries ~bii ¼ 1; i ¼ 1; . . . ; m and, for i – j; i; j ¼ 1; . . . ; m,
~bij ¼
À
2þd2
i
2Àd2
i
 
bij; if di < 1;
À 1þ2di
3À2di
 
bij; if di P 1:
8
><
>:
If ~B is a nonsingular M-matrix, then system (4) is asymptotically stable.
Very interesting global asymptotic stability results were obtained for nonlinear systems of delay differential equations in
the recent papers [15–17].
The aim of the paper is to extend Theorems 1–3 in the following directions. Instead of autonomous system (3) considered
in Theorems 1 and 2, we consider non-autonomous system (1). Unlike of assumptions of Theorem 3, we remove inequalities
jaijðtÞj 6 bijaiiðtÞ; i; j ¼ 1; . . . ; m; i – j and do not assume that hijðtÞ; i; j ¼ 1; . . . ; m are monotone increasing functions.
We will consider a more general system (1) and then, as a particular case, system (4) as well. We analyse systems with
measurable parameters unlike the systems with continuous parameters investigated in [10].
In Theorems 1–3, all conditions are formulated in such a way that special matrices constructed here are non-singular
M-matrices. We derive different stability conditions not assuming that a special matrix is an M-matrix and we show (in
606 L. Berezansky et al. / Applied Mathematics and Computation 250 (2015) 605–614
Section 5) that our conditions are in a sense the best possible conditions assuring the exponential stability for systems with
several delays.
Our approach is based on the Bohl–Perron theorem (see Lemma 1 below), and it is different from that applied in papers
[2,9,10].
The paper is organized as follows. Necessary auxiliary notions and results are collected in Section 2. Main result (Theorem 4)
and its detailed proof are given in Section 3. Section 4 contains ten simple corollaries to the main results for most interesting
classes of equations and systems. In the last Section 5 some conclusions related to derived results are formulated, their
advantages are demonstrated by two examples, and some problems deserving further analysis are described as well.
2. Preliminaries
In this paper, as a norm jxj of a vector x ¼ ðx1; . . . ; xmÞ 2 Rm
we use the following one: jxj ¼ maxi¼1;...;mjxij.
We consider the linear system (1) for t P t0 (assuming t0 P 0) with the initial condition
xðtÞ ¼ uðtÞ; t 6 t0; ð5Þ
under the following assumptions:
(a1) Functions ak
ij : ½0; 1Þ ! R, i; j ¼ 1; . . . ; m; k ¼ 1; . . . ; rij are Lebesgue measurable and essentially bounded functions.
(a2) Functions h
k
ij : ½0; 1Þ ! R, i; j ¼ 1; . . . ; m; k ¼ 1; . . . ; rij are Lebesgue measurable functions, h
k
ijðtÞ 6 t, and
lim sup
t!1
t À h
k
ijðtÞ
 
< 1:
(a3) u : ðÀ1; t0Š ! Rm
is a Borel measurable bounded vector-function.
The above formulated conditions (a1)–(a3) are assumed throughout the paper.
Remark 1. The initial vector-function u in (5) is defined on an interval ðÀ1; t0Š. By condition (a2), there exists a nonnegative
constant s such that t À h
k
ijðtÞ 6 s. Thus, in principle, the domain of the definition of the initial function u in (5) in the
following consideration can be restricted to the finite interval ½t0 À s; t0Š.
In following computations we often need to estimate differences of the form t À max 0; h
k
iiðtÞ
n o
or similar from above. We
obviously get
t À max 0; h
k
iiðtÞ
n o
6 s þ h
k
iiðtÞ À max 0; h
k
iiðtÞ
n o
6 s: ð6Þ
Definition 1. A locally absolutely continuous vector-function x : R ! Rm
is called a solution of the problem (1), (5) for t P t0
if its entries xi; i ¼ 1; . . . ; m satisfy Eq. (1) for almost all t 2 ½t0; 1Þ and equality (5) holds for t 6 t0.
Definition 2. Eq. (1) is uniformly exponentially stable, if there exist constants M > 0 and l > 0 such that the solution
x : R ! Rm
of problem (1), (5) satisfies the inequality
jxðtÞj 6 MeÀlðtÀt0Þ
sup
t6t0
juðtÞj; t P t0
where M and l do not depend on t0.
Along with the linear system (1), we will also consider a non-homogeneous system
_xiðtÞ ¼ À
Xm
j¼1
Xrij
k¼1
ak
ijðtÞxjðh
k
ijðtÞÞ þ fiðtÞ; t P t0; i ¼ 1; . . . ; m; ð7Þ
where f i : ½0; 1Þ ! R is a Lebesgue measurable locally essentially bounded function together with the initial condition
xðtÞ ¼ h; t 6 t0; ð8Þ
where h ¼ ð0; . . . ; 0Þ is an m-dimensional zero-vector. We omit the definition of a solution of problem (7), (8) because it is
similar to definition of a solution of problem (1), (5) given by Definition 1.
Let us introduce some functional spaces on a ray. Denote by Lm
1½t0; 1Þ the space of all essentially bounded functions
y : ½t0; 1Þ ! Rm
with the essential supremum norm
kykLm
1
¼ esssup
tPt0
jyðtÞj
and, by Cm
½t0; 1Þ, the space of all continuous m-dimensional bounded vector-functions on ½t0; 1Þ equipped with the supremum
norm.
L. Berezansky et al. / Applied Mathematics and Computation 250 (2015) 605–614 607
In the proof of the main result (Theorem 4 below), we will use the following Bohl–Perron type result which can be found,
e.g., in [11–13].
Lemma 1. If, for any f 2 Lm
1½t0; 1Þ, f ¼ ðf1; . . . ; f mÞ, the solution of initial problem (7), (8) belongs to Cm
½t0; 1Þ, then Eq. (1) is
uniformly exponentially stable.
We will also use the following elementary lemma.
Lemma 2. For arbitrary Lebesgue measurable function a : ½t0; 1Þ ! ½0; 1Þ and arbitrary x 2 L1
1½t0; 1Þ, the inequality
Z t
t0
e
À
Rt
s
aðsÞds
aðsÞxðsÞds







 6 esssup
tPt0
jxðtÞj; t 2 ½t0; 1Þ
holds.
Proof. We have
Z t
t0
e
À
Rt
s
aðsÞds
aðsÞxðsÞds







 6 esssup
tPt0
jxðtÞj
Z t
t0
e
À
Rt
s
aðsÞds
aðsÞds ¼ esssup
tPt0
jxðtÞj
Z t
t0
e
À
Rt
s
aðsÞds
 0
s
ds
¼ esssup
tPt0
jxðtÞj 1 À e
À
Rt
t0
aðsÞds
 
6 esssup
tPt0
jxðtÞj: Ã
3. Main results
In this part we formulate and prove the main result of the paper on uniform exponential stability of system (1). Define
auxiliary functions
aiðtÞ :¼
Xrii
k¼1
ak
iiðtÞ; i ¼ 1; . . . ; m; t 2 ½0; 1Þ:
Theorem 4 (Main result). Assume that, for t P t0,
aiðtÞ P a0 > 0; i ¼ 1; . . . ; m ð9Þ
and
max
i¼1;...;m
esssup
tPt0
1
aiðtÞ
Xrii
k¼1
jak
iiðtÞj
Z t
maxf0;hk
iiðtÞg
Xm
j¼1
Xrij
l¼1
jal
ijðsÞjds þ
Xm
j¼1
j–i
Xrij
k¼1
jak
ijðtÞj
2
6
4
3
7
5 < 1: ð10Þ
Then the system (1) is uniformly exponentially stable.
Proof. In the proof, we apply Lemma 1. Consider an initial value problem (7), (8). We transform system (7) to
_xiðtÞ ¼ ÀaiðtÞxiðtÞ þ
Xrii
k¼1
ak
iiðtÞ
Z t
hk
iiðtÞ
_xiðsÞds À
Xm
j¼1
j–i
Xrij
k¼1
ak
ijðtÞxjðh
k
ijðtÞÞ þ fiðtÞ; t P t0; i ¼ 1; . . . ; m ð11Þ
and, instead of problem (7), (8), we consider initial problem (11), (8). Note that all expressions in (11) are well-defined and,
by initial condition (8), system (11) is equivalent with
_xiðtÞ ¼ ÀaiðtÞxiðtÞ þ
Xrii
k¼1
ak
iiðtÞ
Z t
maxf0;hk
iiðtÞg
_xiðsÞds À
Xm
j¼1
j–i
Xrij
k¼1
ak
ijðtÞxjðh
k
ijðtÞÞ þ fiðtÞ; t P t0; i ¼ 1; . . . ; m: ð12Þ
Hence, (using (7) in the right-hand side of (12))
_xiðtÞ ¼ ÀaiðtÞxiðtÞ À
Xrii
k¼1
ak
iiðtÞ
Z t
maxf0;hk
iiðtÞg
Xm
j¼1
Xrij
l¼1
al
ijðsÞxjðh
l
ijðsÞÞds À
Xm
j¼1
j–i
Xrij
k¼1
ak
ijðtÞxjðh
k
ijðtÞÞ þ piðtÞ; t P t0; i ¼ 1; . . . ; m;
where
piðtÞ ¼ fiðtÞ þ
Xrii
k¼1
ak
iiðtÞ
Z t
maxf0;hk
iiðtÞg
fiðsÞds:
608 L. Berezansky et al. / Applied Mathematics and Computation 250 (2015) 605–614
Since xiðt0Þ ¼ 0, we get
xiðtÞ ¼ À
Z t
t0
e
À
Rt
s
aiðsÞds
Xrii
k¼1
ak
iiðsÞ
Z s
maxf0;hk
iiðsÞg
Xm
j¼1
Xrij
l¼1
al
ijðsÞxjðh
l
ijðsÞÞds þ
Xm
j¼1
j–i
Xrij
k¼1
ak
ijðsÞxjðh
k
ijðsÞÞ
2
6
4
3
7
5ds þ giðtÞ; t P t0;
i ¼ 1; . . . ; m; ð13Þ
where
giðtÞ ¼
Z t
t0
e
À
Rt
s
aiðsÞds
piðsÞds:
We show that gi; i ¼ 1; . . . ; m are essentially bounded functions. Using (9) and applying Lemma 2, we get
Z t
t0
e
À
Rt
s
aiðsÞds
piðsÞds







 6
Z t
t0
e
À
Rt
s
aiðsÞds
aiðsÞ
jpiðsÞj
aiðsÞ
ds 6 esssup
tPt0
jpiðtÞj
aiðtÞ
6
1
a0
esssup
tPt0
jpiðtÞj
6
1
a0
esssup
tPt0
jfiðtÞj þ esssup
tPt0
Xrii
k¼1
jak
iiðtÞj Á esssup
tP0
jfiðtÞj Á esssup
tPt0
t À max 0; h
k
iiðtÞ
n o 
!
< 1:
Motivated by the first expression in the right-hand side of (13), we consider in the space Lm
1½t0; 1Þ the operator
ðHxÞðtÞ ¼ ððH1xÞðtÞ; . . . ; ðHmxÞðtÞÞ;
where
ðHixÞðtÞ ¼ À
Z t
t0
e
À
Rt
s
aiðsÞds
Xrii
k¼1
ak
iiðsÞ
Z s
maxf0;hk
iiðsÞg
Xm
j¼1
Xrij
l¼1
al
ijðsÞxjðh
l
ijðsÞÞds þ
Xm
j¼1
j–i
Xrij
k¼1
ak
ijðsÞxjðh
k
ijðsÞÞ
2
6
4
3
7
5ds; t P t0; i ¼ 1; . . . ; m:
We have
jðHixÞðtÞj 6
Z t
t0
e
À
Rt
s
aiðsÞds
aiðsÞ
1
aiðsÞ
Xrii
k¼1
jak
iiðsÞj
Z s
maxf0;hk
iiðsÞg
Xm
j¼1
Xrij
l¼1
jal
ijðsÞjds þ
Xm
j¼1
j–i
Xrij
k¼1
jak
ijðsÞj
0
B
@
1
C
A
2
6
4
3
7
5ds Á xk kLm
1
:
Hence, for the norm of operator H : Lm
1½t0; 1Þ ! Lm
1½t0; 1Þ, we get (by Lemma 2 and inequality (10))
kHkLm
1
6 max
i¼1;...;m
esssup
tPt0
1
aiðtÞ
Xrii
k¼1
jak
iiðtÞj
Z t
maxf0;hk
iiðtÞg
Xm
j¼1
Xrij
l¼1
jal
ijðsÞjds þ
Xm
j¼1
j–i
Xrij
k¼1
jak
ijðtÞj
2
6
4
3
7
5 < 1:
Then, the operator equation x ¼ Hx þ g has a unique solution in the space Lm
1 and the solution of system (7) belongs to the
space Cm
½t0; 1Þ. By Lemma 1, system (1) is uniformly exponentially stable. h
4. Corollaries to the main result
Several useful corollaries on uniform exponential stability, mostly with simple conditions to be verified, are derived in
this part. Except for statements related to system (1) and to its particular cases (including system (4)), we consider the following
systems written in the vector–matrix form:
_XðtÞ þ BðtÞXðhðtÞÞ ¼ 0 ð14Þ
and
_XðtÞ þ AðtÞXðtÞ þ BðtÞXðhðtÞÞ ¼ 0; ð15Þ
where AðtÞ ¼ ðaijðtÞÞm
i;j¼1; BðtÞ ¼ ðbijðtÞÞ
m
i;j¼1 are m  m matrices with locally essentially bounded entries aij : ½0; 1Þ ! R; bij :
½0; 1Þ ! R, i; j ¼ 1; . . . ; m; XðtÞ ¼ ðx1ðtÞ; . . . ; xmðtÞÞT
is a vector-function with locally absolutely continuous entries and, for
the delay h : ½0; 1Þ ! R, condition (a2) holds, i.e., h is Lebesgue measurable, hðtÞ 6 t; t 2 ½0; 1Þ and
lim supt!1ðt À hðtÞÞ < 1. Particular cases of systems (14), (15), e.g.,
_XðtÞ þ BXðt À sÞ ¼ 0 ð16Þ
and
_XðtÞ þ AXðtÞ þ BXðt À sÞ ¼ 0; ð17Þ
where A ¼ ðaijÞm
i;j¼1
and B ¼ ðbijÞ
m
i;j¼1
are m  m constant matrices, s > 0, and aii P 0; bii P 0, i ¼ 1; . . . ; m, are considered, too.
L. Berezansky et al. / Applied Mathematics and Computation 250 (2015) 605–614 609
Corollary 1. Assume that aiiðtÞ P a0 > 0; t 2 ½t0; 1Þ; i ¼ 1; . . . ; m, and
max
i¼1;...;m
esssup
tPt0
Z t
maxf0;hiiðtÞg
Xm
j¼1
jaijðsÞjds þ
1
aiiðtÞ
Xm
j¼1
j–i
jaijðtÞj
2
6
4
3
7
5 < 1: ð18Þ
Then, system (4) is uniformly exponentially stable.
Proof. Put rij ¼ 1; ak
ijðtÞ ¼ aijðtÞ; h
k
ijðtÞ ¼ hijðtÞ; aiðtÞ ¼ aiiðtÞ; i; j ¼ 1; . . . ; m in Theorem 4. Hence, inequality (10) takes the
form
max
i¼1;...;m
esssup
tPt0
1
aiiðtÞ
aiiðtÞ
Z t
maxf0;hiiðtÞg
Xm
j¼1
jaijðsÞjds þ
Xm
j¼1
j–i
jaijðtÞj
2
6
4
3
7
5 < 1;
which is equivalent to (18). h
Corollary 2. Assume that, for t P t0, we have
Xrii
k¼1
ak
iiðtÞ P ai > 0; jak
ijðtÞj 6 ak
ij; t À h
k
ijðtÞ 6 sk
ij;
where i; j ¼ 1; . . . ; m; k ¼ 1; . . . ; rij; ai; ak
ij; sk
ij are constants, and
max
i¼1;...;m
1
ai
Xrii
k¼1
ak
iisk
ii
!
Xm
j¼1
Xrij
l¼1
al
ij
!
þ
Xm
j¼1
j–i
Xrij
k¼1
ak
ij
2
6
4
3
7
5 < 1: ð19Þ
Then, system (1) is uniformly exponentially stable.
Proof. We have for t P t0
1
aiðtÞ
Xrii
k¼1
jak
iiðtÞj
Z t
maxf0;hk
iiðtÞg
Xm
j¼1
Xrij
l¼1
jal
ijðsÞjds þ
Xm
j¼1
j–i
Xrij
k¼1
jak
ijðtÞj
2
6
4
3
7
5 6
1
ai
Xrii
k¼1
ak
ii
Xm
j¼1
Xrij
l¼1
al
ij
!
sk
ii þ
Xm
j¼1
j–i
Xrij
k¼1
ak
ij
2
6
4
3
7
5
¼
1
ai
Xrii
k¼1
ak
iisk
ii
!
Xm
j¼1
Xrij
l¼1
al
ij
!
þ
Xm
j¼1
j–i
Xrij
k¼1
ak
ij
2
6
4
3
7
5:
Hence, inequality (19) implies (10). h
Corollary 3. Assume that, for t P t0; aiiðtÞ P ai > 0, jaijðtÞj 6 aij; t À hijðtÞ 6 sij; i; j ¼ 1; . . . ; m where ai; aij, and sij are
constants, and
max
i¼1;...;m
sii
Xm
j¼1
aij þ
1
ai
Xm
j¼1
j–i
aij
2
6
4
3
7
5 < 1: ð20Þ
Then, system (4) is uniformly exponentially stable.
Proof. This follows directly from Corollary 1. h
Consider the linear autonomous system with constant delays
_xiðtÞ ¼ À
Xm
j¼1
Xrij
k¼1
ak
ijxjðt À sk
ijÞ; i ¼ 1; . . . ; m: ð21Þ
Corollary 4. Assume that condition (19) holds where ai :¼
Prii
k¼1
ak
ii > 0; i ¼ 1; . . . ; m. Then, autonomous system (21) is uniformly
exponentially stable.
Proof. This follows directly from Corollary 2 (we put rij ¼ 1). h
610 L. Berezansky et al. / Applied Mathematics and Computation 250 (2015) 605–614
Consider the linear autonomous system with constant delays
_xiðtÞ ¼ À
Xm
j¼1
aijxjðt À sijÞ; i ¼ 1; . . . ; m: ð22Þ
Corollary 5. Assume that aii > 0 and condition (20), where ai ¼ aii; i ¼ 1; . . . ; m, holds. Then, autonomous system (22) is
uniformly exponentially stable.
Proof. This follows directly from Corollary 3 (we put rij ¼ 1). h
Corollary 6. Assume that m ¼ 1 and, for t P t0, at least one of the following conditions hold:
1)
Pr
k¼1akðtÞ P a0 > 0,
esssup
tPt0
1
Pr
k¼1akðtÞ
Xr
k¼1
jakðtÞj
Z t
maxf0;hkðtÞg
Xr
l¼1
alðsÞds
" #
< 1: ð23Þ
2) aiðtÞ  ai;
Pr
k¼1ak > 0; t À hiðtÞ 6 si; i ¼ 1; . . . ; r, and
Xr
i¼1
jaijsi < 1: ð24Þ
Then, scalar Eq. (2) is uniformly exponentially stable.
Proof. Let condition 1) be true. Then, inequality (10) turns into inequality (23) for m ¼ 1. Let condition 2) be true. Since
aiðtÞ  ai, inequality (23) is transformed to
esssup
tPt0
Xr
k¼1
jakjðt À maxf0; hkðtÞgÞ < 1:
Because (in view of (6))
esssup
tPt0
Xr
k¼1
jakjðt À maxf0; hkðtÞgÞ 6 esssup
tPt0
Xr
k¼1
jakjsk
inequality (24) implies (23). h
The following two Corollaries 7 and 8 deal with exponential stability of systems (14) and (15).
Corollary 7. Assume that, for t P t0, at least one of the conditions hold:
(a) biiðtÞ P b0 > 0; i ¼ 1; . . . ; m, and
max
i¼1;...;m
esssup
tPt0
Z t
maxf0;hðtÞg
Xm
j¼1
jbijðsÞjds þ
1
biiðtÞ
Xm
j¼1
j–i
jbijðtÞj
2
6
4
3
7
5 < 1: ð25Þ
(b) biiðtÞ P ai > 0, jbijðtÞj 6 bij; t À hðtÞ 6 s; i; j ¼ 1; . . . ; m, and
max
i¼1;...;m
s
Xm
j¼1
bij þ
1
ai
Xm
j¼1
j–i
bij
2
6
4
3
7
5 < 1: ð26Þ
Then, system (14) is uniformly exponentially stable.
Proof. System (14) can be written in the form
_xiðtÞ ¼ À
Xm
j¼1
bijðtÞxjðhðtÞÞ; i ¼ 1; . . . ; m:
Now, the corollary directly follows from Corollaries 1 and 3. h
L. Berezansky et al. / Applied Mathematics and Computation 250 (2015) 605–614 611
Corollary 8. Assume that, for t P t0,
aiiðtÞ þ biiðtÞÞ P a0 > 0; i ¼ 1; . . . ; m
and
max
i¼1;...;m
esssup
tPt0
1
aiiðtÞ þ biiðtÞ
jbiiðtÞj
Z t
maxf0;hðtÞg
Xm
j¼1
ðjaijðsÞj þ jbijðsÞjÞds þ
Xm
j¼1
j–i
ðjaijðtÞj þ jbijðtÞjÞ
2
6
4
3
7
5 < 1: ð27Þ
Then, system (15) is uniformly exponentially stable.
Proof. System (15) can be written as
_xðtÞ ¼ À
Xm
j¼1
aijðtÞxjðtÞ À
Xm
j¼1
bijðtÞxjðhðtÞÞ; i ¼ 1; . . . ; m
and Theorem 4 used for the choice rii ¼ 2; a1
ijðtÞ ¼ aijðtÞ; a2
ijðtÞ ¼ bijðtÞ; h
1
ijðtÞ ¼ t; h
2
ijðtÞ ¼ hðtÞ, i; j ¼ 1; . . . ; m. Hence,
aiðtÞ ¼ aiiðtÞ þ biiðtÞ; i ¼ 1; . . . ; m and inequality (27) coincides with (10). h
The last two Corollaries 9 and 10 deal with systems (16) and (17) with constant coefficients.
Corollary 9. Assume that bii > 0; i ¼ 1; 2; . . . ; m, and
max
i¼1;...;m
s
Xm
j¼1
jbijj þ
1
bii
Xm
j¼1
j–i
jbijj
2
6
4
3
7
5 < 1: ð28Þ
Then, system (16) is uniformly exponentially stable.
Proof. This follows from Corollary 7 (b) where ai ¼ bii. h
Corollary 10. Assume that aii þ bii > 0; i ¼ 1; . . . ; m, and
1
aii þ bii
sjbiij
Xm
j¼1
ðjaijj þ jbijjÞ þ
Xm
j¼1
j–i
ðjaijj þ jbijjÞ
2
6
4
3
7
5 < 1; i ¼ 1; . . . ; m: ð29Þ
Then, system (17) is uniformly exponentially stable.
Proof. Estimating the left-hand side of inequality (27) in the case of system (17) and using (6), (29), we obtain
max
i¼1;...;m
esssup
tPt0
1
aiiðtÞ þ biiðtÞ
jbiiðtÞj
Z t
maxf0;hðtÞg
Xm
j¼1
ðjaijðsÞj þ jbijðsÞjÞds þ
Xm
j¼1
j–i
ðjaijðtÞj þ jbijðtÞjÞ
2
6
4
3
7
5
6 max
i¼1;...;m
1
aii þ bii
sjbiij
Xm
j¼1
ðjaijj þ jbijjÞ þ
Xm
j¼1
j–i
ðjaijj þ jbijjÞ
2
6
4
3
7
5 < 1:
Therefore, inequality (27) holds and Corollary 10 is a consequence of Corollary 8. h
5. Concluding remarks
There are many stability results for linear delay differential systems written in vector–matrix forms. See for
example a review paper [18] and papers [19,20] where different approaches to stability problems for delay systems were
applied.
Systems like (1) and (4) with several delays can be rewritten in vector–matrix forms. But these forms are usually not suitable
for obtaining stability conditions. On the other hand, systems given in vector–matrix forms can be rewritten as systems
such as (1) and (4). In Corollaries 7–10, we obtained explicit uniform exponential stability conditions for most interesting
systems given in vector–matrix forms by rewriting these systems in forms (1) and (4).
612 L. Berezansky et al. / Applied Mathematics and Computation 250 (2015) 605–614
In Theorem 4 and in its corollaries listed above, we generalized in many directions the known stability results for linear
delay differential systems. A remarkable feature, which we particularly underline, is that our approach does not require the
construction of any non-singular M-matrices. Some of these directions were mentioned in Introduction.
Compare now our results with Theorems 1–3. In Theorems 1 and 2, only autonomous systems are considered. In
Theorem 3 the non-autonomous case is treated but in a less general setting than in our Theorem 4. To compare Theorem 3
with Theorem 4, consider the following two examples.
Example 1. Consider for t P 0 the following system
_x1ðtÞ ¼ Àx1ðt À j sin t j=2Þ þ ax2ðt À s1Þ;
_x2ðtÞ ¼ bx1ðt À s2Þ À x2 t À j cos tj=2ð Þ:
ð30Þ
The conditions of Corollary 1 are satisfied if
Z t
tÀj sin tj=2
ð1 þ jajÞds þ jaj < 1;
Z t
tÀj cos tj=2
ð1 þ jbjÞds þ jbj < 1:
Hence, if jaj < 1=3; jbj < 1=3, system (30) is uniformly exponentially stable. Theorem 3 is not applicable to system (30) since
the delay functions are not monotone increasing.
Example 2. Consider for t P 0 the following system
_x1ðtÞ ¼ Àx1ðt À 0:1Þ þ 0:1x1ðt À 0:2Þ þ ax2ðt À sðtÞÞ;
_x2ðtÞ ¼ bx1ðt À sðtÞÞ À x2ðt À 0:1Þ þ 0:1x2ðt À 0:2Þ;
ð31Þ
where sðtÞ ¼ t if t 2 ½0; 1Þ and sðt þ 1Þ ¼ sðtÞ. The conditions of Theorem 4 are satisfied if
1
0:9
Z t
tÀ0:1
ð1:1 þ jajÞds þ 0:1
Z t
tÀ0:2
ð1:1 þ jajÞds þ jaj
 
< 1;
1
0:9
Z t
tÀ0:1
ð1:1 þ jbjÞds þ 0:1
Z t
tÀ0:2
ð1:1 þ jbjÞds þ jbj
 
< 1:
Hence, if jaj < 0:68; jbj < 0:68, system (31) is uniformly exponentially stable.
Theorem 3 fails for system (31) since the first equation has two terms with x1, the second equation has two terms with x2
and, also, since the delay function hðtÞ ¼ t À sðtÞ is not continuous.
In the scalar case, the system (1) is reduced to a differential equation with several delays (2). Corollary 6 gives explicit
exponential stability conditions for Eq. (2). The second part of Corollary 6 was obtained before in [5]. Moreover, by this paper,
the constant 1 in the right-hand side of (24) is the best possible. Therefore, the constant 1 in the inequality (10) in Theorem 4
and in all its corollaries (in the right-hand sides of (18)–(20) and (23)–(29)) is the best possible one as well.
Together with the delay differential systems considered in this paper, one can consider other linear functional–differential
systems, in particular, differential systems with distributed delay and integro-differential systems. Since the Bohl–Perron
theorem is known for these systems as well [14], one can obtain stability results for these systems similar to Theorem 4.
At the end of this section, we will formulate several open problems. The Bohl–Perron theorem is formulated for systems
with bounded delays. Thus, our stability conditions were obtained only for such systems. It is a mathematical challenge to
obtain explicit asymptotic stability conditions or explicit exponential stability conditions for systems (1) with unbounded
delays.
Consider a linear delay differential equation of the second-order
€xðtÞ ¼
Xm
k¼1
akðtÞ_xðgkðtÞÞ þ
Xn
k¼1
bkðtÞxðhkðtÞÞ;
where ak; bk; gk; hk : ½0; 1Þ ! R. For this equation, there are only few stability results. It would be interesting to obtain
exponential stability results for this equation and for equations of higher-order as well by reducing them to systems of delay
differential equations of first-order and applying the known stability results.
Definition 2 on exponential stability assumes the existence of two positive constants M and l. It would be interesting to
replace the stability conditions obtained in Theorems 1–4 by explicit estimates of these constants.
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[2] I. Györi, F. Hartung, J. Turi, Preservation of stability in delay equations under delay perturbations, J. Math. Anal. Appl. 220 (1998) 290–312.
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(2002) 1297–1304.
[10] Joseph W.-H. So, X.H. Tang, Xingfu Zou, Global attractivity for non-autonomous linear delay systems, Funkcial. Ekvac. 47 (1) (2004) 25–40.
[11] A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York, London, 1966.
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N.V. Azbelev, L. Berezansky, P.M. Simonov, A.V. Chistyakov, The stability of linear systems with aftereffect: II, Differ. Equ. 27 (4) (1991) 383–388;
N.V. Azbelev, L. Berezansky, P.M. Simonov, A.V. Chistyakov, The stability of linear systems with aftereffect: III, Differ. Equ. 27 (10) (1991) 1165–1172;
N.V. Azbelev, L. Berezansky, P.M. Simonov, A.V. Chistyakov, The stability of linear systems with aftereffect: IV, Differ. Equ. 29 (2) (1993) 153–160.
[13] N.V. Azbelev, P.M. Simonov, Stability of Differential Equations with Aftereffect, Stability and Control: Theory, Methods and Applications, vol. 20, Taylor
& Francis, London, 2003.
[14] M. Gil’, Stability of Vector Differential Delay Equations, Frontiers in Mathematics, Birkhäuser/Springer, Basel AG, Basel, 2013.
[15] E. Liz, A. Ruiz-Herrera, Attractivity, multistability and bifurcation in delayed Hopfield’s model with non-monotonic feedback, J. Differ. Equ. 255 (11)
(2013) 4244–4266.
[16] T. Faria, Asymptotic behaviour for a class of delayed cooperative models with patch structure, Discrete Contin. Dyn. Syst. Ser. B 18 (6) (2013)
1567–1579.
[17] T. Faria, J.J. Oliveira, General criteria for asymptotic and exponential stabilities of neural network models with unbounded delays, Appl. Math. Comput.
217 (23) (2011) 9646–9965.
[18] J.-P. Richard, Time-delay systems: an overview of some recent advances and open problems, Automatica 39 (10) (2003) 1667–1694.
[19] K. Liu, V. Suplin, E. Fridman, Stability of linear systems with general sawtooth delay, IMA J. Math. Control Inform. 27 (4) (2010) 419–436.
[20] V. Kolmanovskii, L. Shaikhet, About one application of the general method of Lyapunov functionals construction, Int. J. Robust Nonlinear Control 13 (9)
(2003) 805–818.
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Electronic Journal of Qualitative Theory of Differential Equations
Proc. 10th Coll. Qualitative Theory of Diff. Equ. (July 1–4, 2015, Szeged, Hungary)
2016, No. 5, 1–18; doi: 10.14232/ejqtde.2016.8.5 http://www.math.u-szeged.hu/ejqtde/
New exponential stability conditions for linear
delayed systems of differential equations
Leonid Berezansky1
, Josef Diblík 2
,
Zdenˇek Svoboda2
and Zdenˇek Šmarda2
1Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
2Brno University of Technology, Technická 10, 61600 Brno, Czech Republic
Appeared 11 August 2016
Communicated by Tibor Krisztin
Abstract. New explicit results on exponential stability, improving recently published
results by the authors, are derived for linear delayed systems
˙xi(t) = −
m
∑
j=1
rij
∑
k=1
ak
ij(t)xj(hk
ij(t)), i = 1, . . . , m
where t ≥ 0, m and rij, i, j = 1, . . . , m are natural numbers, ak
ij : [0, ∞) → R are measurable
coefficients, and hk
ij : [0, ∞) → R are measurable delays. The progress was achieved
by using a new technique making it possible to replace the constant 1 by the constant
1 + 1/e on the right-hand sides of crucial inequalities ensuring exponential stability.
Keywords: exponential stability, linear delayed differential system, estimate of fundamental
function, Bohl–Perron theorem.
2010 Mathematics Subject Classification: 34K20.
1 Introduction
The objective of the present investigation is to derive easily verifiable explicit exponential
stability conditions for the following non-autonomous linear delay differential system
˙xi(t) = −
m
∑
j=1
rij
∑
k=1
ak
ij(t)xj(hk
ij(t)), i = 1, . . . , m (1.1)
where t ≥ 0, m is a natural number, rij, i, j = 1, . . . , m are natural numbers, the coefficients
ak
ij : [0, ∞) → R and delays hk
ij : [0, ∞) → R are measurable functions.
The equation
˙x(t) = −
r
∑
k=1
ak(t)x(hk(t)), (1.2)
Corresponding author. Email: diblik.j@fce.vutbr.cz
2 L. Berezansky, J. Diblík, Z. Svoboda and Z. Šmarda
which is a special scalar case of (1.1), has been studied, e.g., in [6, 12, 14, 15, 20, 25]. A review
on stability results to equation (1.2) can be found in [7]. Below, we cite some selected results
from the above papers or give extracts of them.
From [20, Theorem 1.2], we get the following corollary.
Theorem 1.1. Let there be constants a0, Ak and τk, k = 1, 2, . . . , r such that
0 ≤ ak(t) ≤ Ak,
r
∑
k=1
ak(t) ≥ a0 > 0, 0 ≤ t − hk(t) ≤ τk, t ≥ 0.
If, moreover,
r
∑
k=1
Akτk ≤ 1, (1.3)
then the equation (1.2) is uniformly asymptotically stable and the constant 1 on the right-hand side
of (1.3) is the best one possible.
A corollary deduced from [20, Theorem 1.1] follows.
Theorem 1.2. Let there be constants Ak and τk, k = 1, 2, . . . , r such that
ak(t) ≡ Ak > 0, 0 ≤ t − hk(t) ≤ τk, t ≥ 0.
If, moreover,
r
∑
k=1
Akτk <
3
2
, (1.4)
then the equation (1.2) is uniformly asymptotically stable and the constant 3/2 on the right-hand side
of (1.4) is the best one possible.
From [25, Corollary 2.4] we get the following theorem.
Theorem 1.3. Let ak(t) and hk(t), k = 1, . . . , r, t ≥ 0 be continuous functions and
ak(t) ≥ 0,
∞
0
r
∑
k=1
ak(t)dt = ∞, 0 < h1(t) ≤ h2(t) ≤ · · · ≤ hr(t) ≤ t.
If, moreover,
lim sup
t→∞
r
∑
k=1
t
h1(t)
ak(s)ds <
3
2
,
then the equation (1.2) is asymptotically stable.
The following result reproduces [15, Proposition 4.4].
Theorem 1.4. Let ak(t) ≡ ak > 0, k = 1, 2, . . . , r and let a constant α ∈ [0, 1] exist such that
α
e
r
∑
i=1
ai
≤ max
k
(t − hk(t)), t ≥ t0
and
r
∑
i=1
ai lim sup
t→∞
(t − hi(t)) < 1 +
α
e
.
Then, the equation (1.2) is uniformly asymptotically stable.
New exponential stability conditions for linear delayed systems 3
Now we give a corollary of [7, Lemma 3.1].
Theorem 1.5. Let ak(t) be Lebesgue measurable essentially bounded functions and let there be constants
a0 and τk, k = 1, 2, . . . , r such that
ak(t) ≥ 0,
∞
t0
r
∑
k=1
ak(s)ds = ∞, 0 ≤ t − hk(t) ≤ τk, t ≥ t0.
If, moreover,
lim sup
t→∞
r
∑
k=1
ak(t)
∑r
i=1 ai(t)
t
hk(t)
r
∑
i=1
ai(s)ds < 1 +
1
e
, (1.5)
then the equation (1.2) is uniformly exponentially stable.
Except for the paper [15], the above mentioned papers consider stability problems for
scalar equations only. In [15], linear systems with constant matrices are treated. Unfortunately,
there are no results on the stability of general systems of the form (1.1), which can be reduced
to Theorems 1.1–1.5 in the scalar case. To illustrate this claim, consider several known results.
In [24], the authors consider the non-autonomous system
˙xi(t) = −
m
∑
j=1
aij(t)xj(hij(t)), i = 1, . . . , m (1.6)
where t ∈ [t0, ∞), t0 ∈ R, aij(t), hij(t) are continuous functions, hij(t) ≤ t, hij(t) are monotone
increasing and such that limt→∞ hij(t) = ∞, i, j = 1, . . . , m.
Theorem 1.6 ([24, Theorem 2.2]). Assume that, for t ≥ t0, there exist non-negative numbers bij,
i, j = 1, . . . , m, i = j such that |aij(t)| ≤ bijaii(t), i, j = 1, . . . , m, i = j, aii(t) ≥ 0 and
∞
aii(s)ds = ∞, di = lim sup
t→∞
t
hii(t)
aii(s)ds < 3/2, i = 1, . . . m.
Let ˜B = (˜bij)m
i,j=1 be an m × m matrix with entries ˜bii = 1, i = 1, . . . , m and, for i = j, i, j = 1, . . . , m,
˜bij =



−
2 + d2
i
2 − d2
i
bij, if di < 1,
−
1 + 2di
3 − 2di
bij, if di ≥ 1.
If ˜B is a nonsingular M-matrix, then system (1.6) is asymptotically stable.
This theorem can be viewed as a certain generalization of Theorems 1.2 and 1.3 to systems
but only for the case of “one delay” (rij = 1, i, j = 1, . . . , m).
Paper [13] gives a generalization of Theorem 1.4 to linear systems with constant coefficients
and delays.
In our recent paper [8], we considered general system (1.1) deriving the following result.
Theorem 1.7 ([8, Theorem 4]). Let there be constants a0 and τ such that, for t ≥ t0,
a∗
i (t) :=
rii
∑
k=1
ak
ii(t) ≥ a0 > 0, 0 ≤ t − hk
ij(t) ≤ τ, i = 1, . . . , m (1.7)
4 L. Berezansky, J. Diblík, Z. Svoboda and Z. Šmarda
and
max
i=1,...,m
ess sup
t≥t0
1
a∗
i (t)




rii
∑
k=1
|ak
ii(t)|
t
max{0,hk
ii(t)}
m
∑
j=1
rij
∑
l=1
|al
ij(s)|ds +
m
∑
j=1
j=i
rij
∑
k=1
|ak
ij(t)|



 < 1. (1.8)
Then, the system (1.1) is uniformly exponentially stable.
Requiring that all assumptions of Theorem 1.5 and Theorem 1.7 are valid simultaneously,
condition (1.8) in Theorem 1.7 turns, in the case of equation (1.2) where ak(t) ≥ 0, into
ess sup
t≥t0
1
∑r
k=1 ak(t)
r
∑
k=1
ak(t)
t
max{0,hk(t)}
r
∑
l=1
al(s)ds < 1
and, for t0 sufficiently large, coincides with the left-hand side of inequality (1.5).
Nevertheless, Theorem 1.7 is not an extension of Theorem 1.5 to system (1.1) since the
right-hand side in the inequality (1.8) is equal to 1 instead of 1 + 1/e on the right-hand side
of inequality (1.5) in Theorem 1.5.
The aim of the paper is to improve all the results of [8] and replace the constant 1 by the
constant 1 + 1/e not only on the right-hand side of inequality (1.8), but in all explicit stability
conditions derived in [8]. The only limitation in this paper in comparison with paper [8] is
the condition
ak
ii(t) ≥ 0, i = 1, . . . , m, k = 1, . . . , rii. (1.9)
Since this condition does not necessarily hold for equations considered in [8], all results of
this paper and in [8] are independent.
Our approach is based on estimates of the fundamental solution for scalar delay differential
equations and on the Bohl–Perron type result. Some ideas and schemes of [8] are utilized
as well.
2 Preliminaries
Let t0 ≥ 0. We consider an initial problem
x(t) = ϕ(t), t ≤ t0 (2.1)
for (1.1) where ϕ = (ϕ1, . . . , ϕm)T : (−∞, t0] → Rm is a vector-function. Throughout the rest of
the paper, we assume (a1)–(a3) where
(a1) ak
ij : [0, ∞) → R, i, j = 1, . . . , m, k = 1, . . . , rij are Lebesgue measurable and essentially
bounded functions, ak
ii(t) ≥ 0;
(a2) hk
ij : [0, ∞) → R, i, j = 1, . . . , m, k = 1, . . . , rij are Lebesgue measurable functions, hk
ij(t) ≤
t, and t − hk
ij(t) ≤ K, t ≥ 0 where K is a positive constant;
(a3) ϕ: (−∞, t0] → Rm is a Borel measurable bounded vector-function.
For a vector x = (x1, . . . , xm)T ∈ Rm, we define |x| := maxi=1,...,m |xi|.
New exponential stability conditions for linear delayed systems 5
Remark 2.1. The function ϕ in (2.1) is defined on (−∞, t0]. By (a2), there exists a positive
constant K such that t − hk
ij(t) ≤ K, i, j = 1, . . . , m, k = 1, . . . , rij. Thus, the domain of the
definition of the initial function ϕ in (2.1) in the following consideration can be, in principle,
restricted to the finite interval [t0 − K, t0]. In the following computations, it is often necessary
to estimate differences t − max{t0, hk
ii(t)} (or similar) from above. Obviously,
t − max{t0, hk
ii(t)} ≤ K.
Definition 2.2. A locally absolutely continuous vector-function x: R → Rm is called a solution
of the problem (1.1), (2.1) for t ≥ t0, if its components xi(t), i = 1, . . . , m satisfy (1.1) for almost
all t ∈ [t0, ∞) and (2.1) holds for t ≤ t0.
Definition 2.3. Equation (1.1) is called uniformly exponentially stable if there exist constants
M > 0 and µ > 0 such that the solution x: R → Rm of (1.1), (2.1) satisfies
|x(t)| ≤ M e−µ(t−t0)
sup
t≤t0
|ϕ(t)|, t ≥ t0
where M and µ do not depend on t0.
A non-homogeneous system
˙xi(t) = −
m
∑
j=1
rij
∑
k=1
ak
ij(t)xj(hk
ij(t)) + fi(t), i = 1, . . . , m (2.2)
where fi : [0, ∞) → R is a Lebesgue measurable locally essentially bounded function together
with the initial problem
x(t) = θ, t ≤ t0, (2.3)
where θ = (0, . . . , 0)T ∈ Rm, will be used together with homogeneous system (1.1).
In what follows, Lm
∞[t0, ∞) denotes the space of all essentially bounded real vectorfunctions
y: [t0, ∞) → Rm with the essential supremum norm
y Lm
∞
= ess sup
t≥t0
|y(t)|.
As Cm[t0, ∞) we denote the space of all continuous m-dimensional bounded real vectorfunctions
on [t0, ∞) equipped with the supremum norm.
The proof of our main result uses the Bohl–Perron type result ([1–5,11,16]).
Theorem 2.4. If the solution of initial problem (2.2), (2.3) belongs to Cm[t0, ∞) for any f ∈ Lm
∞[t0, ∞),
f = (f1, . . . , fm)T, then equation (1.1) is uniformly exponentially stable.
Note that, without loss of generality, we can assume f (t) ≡ θ on the interval [t0, t1] for
some t1 > t0 in Lemma 2.4.
Consider the scalar homogeneous initial problem
˙x(t) = −
r
∑
k=1
ak(t)x(hk(t)), t ≥ s ≥ t0, (2.4)
x(t) = 0, t < s, x(s) = 1, (2.5)
where ak : [0, ∞) → R, k = 1, . . . , r are Lebesgue measurable and essentially bounded functions,
hk : [0, ∞) → R, k = 1, . . . , r are Lebesgue measurable functions, hk(t) ≤ t.
6 L. Berezansky, J. Diblík, Z. Svoboda and Z. Šmarda
Definition 2.5. A solution x = X(t, s) of (2.4), (2.5) is called the fundamental function of (1.1).
The associated non-homogeneous equation to (2.4) is
˙x(t) = −
r
∑
k=1
ak(t)x(hk(t)) + f (t), t ≥ t0. (2.6)
We will need the following representation formula (see, e.g. [1–5]) for solution of (2.6) (with a
locally Lebesgue integrable right-hand side f) satisfying the initial problem
x(t) = 0, t ≤ t0. (2.7)
Theorem 2.6. The solution of initial problem (2.6), (2.7) is given by the formula
x(t) =
t
t0
X(t, s)f (s)ds. (2.8)
The following lemma is taken from [12].
Theorem 2.7. Let ak(t) ≥ 0 and
t
mink{hk(t)}
r
∑
k=1
ak(s)ds ≤
1
e
where t ≥ t0, k = 1, . . . , r. Then, the fundamental function X(t, s) of (2.4) satisfies X(t, s) > 0 for
t ≥ s ≥ t0.
We will finish this section by an auxiliary result from [6]. In its formulation, X(t, s) is the
fundamental function of (2.4).
Theorem 2.8. Let ak(t) ≥ 0, X(t, s) > 0, t ≥ s ≥ t0, t − hk(t) ≤ K, t ≥ t0, k = 1, . . . , r. Then,
0 ≤
t
t0
X(t, s)
r
∑
k=1
ak(s) ξ(s)ds ≤ 1, t ≥ t0,
where ξ is the characteristic function of the interval [t0 + K, ∞).
3 Main result
The main result (Theorem 3.1 below) gives sufficient conditions for the uniform exponential
stability to system (1.1). We underline that this theorem is a significant improvement to Theorem
1.7 because almost the same expression is estimated by the constant 1 + 1/e on the
right-hand side of inequality (3.4) rather than by the constant 1 on the right-hand side of
inequality (1.8).
Let Ai, i = 1, . . . , m be functions defined as
Ai(t) :=
1
ai(t)




rii
∑
k=1
ak
ii(t)
t
max{t0,hk
ii(t)}
m
∑
j=1
rij
∑
l=1
|al
ij(s)|ds +
m
∑
j=1
j=i
rij
∑
k=1
|ak
ij(t)|




where
ai(t) :=
rii
∑
k=1
ak
ii(t). (3.1)
New exponential stability conditions for linear delayed systems 7
Theorem 3.1 (Main result). Let
ai(t) ≥ a0 > 0, i = 1, . . . , m, t ≥ t0, (3.2)
max
i=1,...,m
ess sup
t≥t0
1
ai(t)
m
∑
j=1
j=i
rij
∑
k=1
|ak
ij(t)| < 1 (3.3)
and
max
i=1,...,m
ess sup
t≥t0
Ai(t) < 1 +
1
e
. (3.4)
Then, the system (1.1) is uniformly exponentially stable.
Proof. Define auxiliary functions Hk
i : [t0, ∞) → R, i = 1, . . . , m, k = 1, . . . , rii as follows:
i) If
t
hk
ii(t)
m
∑
j=1
rij
∑
l=1
|al
ij(s)|ds ≤
1
e
, (3.5)
then
Hk
i (t) := hk
ii(t).
ii) If
t
hk
ii(t)
m
∑
j=1
rij
∑
l=1
|al
ij(s)|ds >
1
e
, (3.6)
then Hk
i (t) is a unique solution of an implicit equation
t
Hk
i (t)
m
∑
j=1
rij
∑
l=1
|al
ij(s)|ds =
1
e
.
Consider the problem (2.2), (2.3) assuming that
fi(t) ≡ 0 if t ∈ [t0, t0 + K], i = 1, . . . , m. (3.7)
Condition (3.7) implies that for the solution of the problem (2.2), (2.3) we have xi(t) = 0,
i = 1, . . . , m if t ∈ [t0, t0 + K].
System (2.2) can be transformed to
˙xi(t) = −
rii
∑
k=1
ak
ii(t)xi(Hk
i (t)) +
rii
∑
k=1
ak
ii(t)
Hk
i (t)
hk
ii(t)
˙xi(s)ds
−
m
∑
j=1
j=i
rij
∑
k=1
ak
ij(t)xj(hk
ij(t)) + fi(t), t ≥ t0, i = 1, . . . , m. (3.8)
It is easy to see that (due to (2.3)) system (3.8) is equivalent with
˙xi(t) = −
rii
∑
k=1
ak
ii(t)xi(Hk
i (t)) +
rii
∑
k=1
ak
ii(t)
Hk
i (t)
max{t0,hk
ii(t)}
˙xi(s)ds
−
m
∑
j=1
j=i
rij
∑
k=1
ak
ij(t)xj(hk
ij(t)) + fi(t), t ≥ t0, i = 1, . . . , m. (3.9)
8 L. Berezansky, J. Diblík, Z. Svoboda and Z. Šmarda
Moreover, utilizing (2.2), (3.9), it can be transformed to
˙xi(t) = −
rii
∑
k=1
ak
ii(t)xi(Hk
i (t))
−
rii
∑
k=1
ak
ii(t)
Hk
i (t)
max{t0,hk
ii(t)}
m
∑
j=1
rij
∑
l=1
al
ij(s)xj(hl
ij(s))ds
−
m
∑
j=1
j=i
rij
∑
k=1
ak
ij(t)xj(hk
ij(t)) + pi(t), t ≥ t0, i = 1, . . . , m (3.10)
where
pi(t) = fi(t) +
rii
∑
k=1
ak
ii(t)
Hk
i (t)
max{t0,hk
ii(t)}
fi(s)ds.
By assumption (a2), the definition of Hk
i (note that hk
ii(t) ≤ Hk
i (t) ≤ t), and (3.7) we get
pi(t) ≡ 0 if t ≤ t0 + K.
Let Xi(t, s), i = 1, . . . , m be the fundamental function (see Definition 2.5) of the scalar initialvalue
problem
˙xi(t) = −
rii
∑
k=1
ak
ii(t)xi(Hk
i (t)), t ≥ t0,
xi(t) = 0, t ≤ t0.
By virtue of (a1), the definition of Hk
i (t), i = 1, . . . , m and Lemma 2.7, we have Xi(t, s) > 0,
t ≥ s ≥ t0, i = 1, . . . , m. Using formula (2.8) in Lemma 2.6, from (3.10), we get
xi(t) = −
t
t0
Xi(t, s)




rii
∑
k=1
ak
ii(s)
Hk
i (s)
max{t0,hk
ii(s)}
m
∑
j=1
rij
∑
l=1
al
ij(τ)xj(hl
ij(τ))dτ
+
m
∑
j=1
j=i
rij
∑
k=1
ak
ij(s)xj(hk
ij(s))



 ds + gi(t), t ≥ t0, i = 1, . . . , m (3.11)
where
gi(t) =
t
t0
Xi(t, s)pi(s)ds
and
pi(t) = gi(t) ≡ 0 if t ≤ t0 + K.
Next, we explain why gi, i = 1, . . . , m are essentially bounded functions. By (a1), properties
New exponential stability conditions for linear delayed systems 9
of fi and Hk
i , i = 1, . . . , m, definition (1.7), Remark 2.1, and Lemma 2.8, we deduce
ess sup
t≥t0
|gi(t)|
= ess sup
t≥t0
t
t0
Xi(t, s)pi(s)ds
= ess sup
t≥t0+K
t
t0
Xi(t, s)pi(s)ds
≤ ess sup
t≥t0+K
t
t0
Xi(t, s)ai(s)
|pi(s)|
ai(s)
ds ≤ ess sup
t≥t0+K
|pi(t)|
ai(t)
≤
1
a0
ess sup
t≥t0+K
|pi(t)|
≤
1
a0
ess sup
t≥t0+K
|fi(t)| + ess sup
t≥t0+K
rii
∑
k=1
ak
ii(t) ess sup
t≥t0+K
|fi(t)| · ess sup
t≥t0+K
(Hk
i (t) − max{t0, hk
ii(t)})
< ∞.
System (3.11) can be written in an operator form
xi(t) = (Gix)(t) + gi(t), t ≥ t0, i = 1, . . . , m
where
(Gix)(t) = −
t
t0
Xi(t, s)




rii
∑
k=1
ak
ii(s)
Hk
i (s)
max{t0,hk
ii(s)}
m
∑
j=1
rij
∑
l=1
al
ij(τ)xj(hl
ij(τ))dτ
+
m
∑
j=1
j=i
rij
∑
k=1
ak
ij(s)xj(hk
ij(s))



 ds, t ≥ t0, i = 1, . . . , m
or as
x = Gx + g (3.12)
where
G: Lm
∞ → Lm
∞, (Gx)(t) = ((G1x)(t), . . . , (Gmx)(t))T
and g(t) = (g1(t), . . . , gm(t))T. Estimate the norm G Lm
∞
of the operator G. Since xi(t) ≡ 0, if
t ∈ [t0, t0 + K], i = 1, . . . , m, then
|(Gix)(t)| ≤
t
t0+H
Xi(t, s)ai(s)Ai(s)ds · x L∞
, i = 1, . . . , m
where
Ai(t) :=
1
ai(t)




rii
∑
k=1
ak
ii(t)
Hk
i (t)
max{t0,hk
ii(t)}
m
∑
j=1
rij
∑
l=1
|al
ij(s)|ds +
m
∑
j=1
j=i
rij
∑
k=1
|ak
ij(t)|



 .
Hence, by Lemma 2.8,
G Lm
∞
≤ max
i=1,...,m
ess sup
t≥t0
Ai(t) (3.13)
10 L. Berezansky, J. Diblík, Z. Svoboda and Z. Šmarda
If (3.5) holds, then Hk
i (t) = hk
ii(t), i = 1, . . . , m, k = 1, . . . , rii and, consequently,
Ai(t) ≤
1
ai(t)




m
∑
j=1
j=i
rij
∑
k=1
|ak
ij(t)|



 .
By (3.3) we get
max
i=1,...,m
ess sup
t≥t0
Ai(t) ≤ max
i=1,...,m
ess sup
t≥t0
1
ai(t)




m
∑
j=1
j=i
rij
∑
k=1
|ak
ij(t)|



 < 1. (3.14)
If (3.6) is valid, then
t
Hk
i (t)
m
∑
j=1
rij
∑
l=1
|al
ij(s)|ds =
1
e
.
Hence
1
ai(t)
rii
∑
k=1
ak
ii(t)
Hk
i (t)
max{t0,hk
ii(t)}
m
∑
j=1
rij
∑
l=1
|al
ij(s)|ds
=
1
ai(t)
rii
∑
k=1
ak
ii(t)
t
max{t0,hk
ii(t)}
m
∑
j=1
rij
∑
l=1
|al
ij(s)|ds −
t
Hk
i (t)
m
∑
j=1
rij
∑
l=1
|al
ij(s)|ds
=
1
ai(t)
rii
∑
k=1
ak
ii(t)
t
max{t0,hk
ii(t)}
m
∑
j=1
rij
∑
l=1
|al
ij(s)|ds −
1
e
=
1
ai(t)
rii
∑
k=1
ak
ii(t)
t
max{t0,hk
ii(t)}
m
∑
j=1
rij
∑
l=1
|al
ij(s)|ds −
1
e
. (3.15)
In this case, using (3.15) and (3.4), we get
max
i=1,...,m
ess sup
t≥t0
Ai(t) ≤ max
i=1,...,m
ess sup
t≥t0
Ai(t) −
1
e
< 1. (3.16)
Finally, from (3.13), (3.14) and (3.16), we deduce G Lm
∞
< 1. Therefore, the operator equation
(3.12) has a unique solution x ∈ Lm
∞ This solution solves the system (2.2) and belongs to
the space Cm[t0, ∞). By Lemma 2.4, system (1.1) is uniformly exponentially stable.
4 Corollaries to the main result
The purpose of this part is to consider some special cases of the system (1.1) and from Theorem
3.1, deduce simple corollaries on uniform exponential stability. In the proofs, we verify
the assumptions of Theorem 3.1 for the case considered. It is often obvious and we omit the
unnecessary details.
Corollary 4.1. Assume that
aii(t) ≥ a0 > 0, i = 1, . . . , m, t ≥ t0, (4.1)
New exponential stability conditions for linear delayed systems 11
max
i=1,...,m
ess sup
t≥t0
1
aii(t)
m
∑
j=1
j=i
|aij(t)| < 1 (4.2)
and
max
i=1,...,m
ess sup
t≥t0




t
max{t0,hii(t)}
m
∑
j=1
|aij(s)|ds +
1
aii(t)
m
∑
j=1
j=i
|aij(t)|



 < 1 +
1
e
. (4.3)
Then, the system
˙xi(t) = −
m
∑
j=1
aij(t)xi(hij(t))), i = 1, . . . , m (4.4)
is uniformly exponentially stable.
Proof. Let rij = 1, ak
ij(t) = aij(t), hk
ij(t) = hij(t), ai(t) = aii(t), i, j = 1, . . . , m. Then, the
system (1.1) reduces to (4.4) and we can apply Theorem 3.1 since assumptions (3.2), (3.3)
and (3.4) are, in the particular case, reduced to assumptions (4.1), (4.2) and (4.3).
Corollary 4.2. Assume that, for t ≥ t0, we have ak
ii(t) ≥ 0,
rii
∑
k=1
ak
ii(t) ≥ αi > 0, |ak
ij(t)| ≤ ak
ij, t − hk
ij(t) ≤ τk
ij
where i, j = 1, . . . , m, k = 1, . . . , rij, αi, ak
ij, τk
ij are constants,
max
i=1,...,m
1
αi
m
∑
j=1
j=i
rij
∑
k=1
ak
ij < 1, (4.5)
and
max
i=1,...,m
1
αi




rii
∑
k=1
ak
iiτk
ii
m
∑
j=1
rij
∑
l=1
al
ij +
m
∑
j=1
j=i
rij
∑
k=1
ak
ij



 < 1 +
1
e
. (4.6)
Then, the system (1.1) is uniformly exponentially stable.
Proof. We have for t ≥ t0
Ai(t) ≤
1
αi




rii
∑
k=1
ak
ii
m
∑
j=1
rij
∑
l=1
al
ij τk
ii +
m
∑
j=1
j=i
rij
∑
k=1
ak
ij



 =
1
αi




rii
∑
k=1
ak
iiτk
ii
m
∑
j=1
rij
∑
l=1
al
ij +
m
∑
j=1
j=i
rij
∑
k=1
ak
ij




and (4.6) implies (3.4).
Corollary 4.3. Assume that aii(t) ≥ αi > 0, |aij(t)| ≤ aij, t − hij(t) ≤ τij for i, j = 1, . . . , m and
t ≥ t0 where αi, aij, and τij are constants and
max
i=1,...,m
1
αi
m
∑
j=1
j=i
aij < 1, max
i=1,...,m



τii
m
∑
j=1
aij +
1
αi
m
∑
j=1
j=i
aij



 < 1 +
1
e
. (4.7)
Then, the system (4.4) is uniformly exponentially stable.
12 L. Berezansky, J. Diblík, Z. Svoboda and Z. Šmarda
Proof. This result follows from Corollary 4.1.
Now we give stability conditions for the following linear autonomous system with constant
delays
˙xi(t) = −
m
∑
j=1
rij
∑
k=1
ak
ijxj(t − τk
ij), i = 1, . . . , m. (4.8)
Corollary 4.4. Assume that ak
ii ≥ 0, conditions (4.5) and (4.6) hold where
αi :=
rii
∑
k=1
ak
ii > 0, i = 1, . . . , m.
Then, the autonomous system (4.8) is uniformly exponentially stable.
Proof. This follows directly from Corollary 4.2.
Consider the linear autonomous system with constant delays
˙xi(t) = −
m
∑
j=1
aijxj(t − τij), i = 1, . . . , m. (4.9)
Corollary 4.5. Assume that aii > 0 and inequalities (4.7) hold where αi = aii, i = 1, . . . , m. Then,
the autonomous system (4.9) is uniformly exponentially stable.
Proof. This follows directly from Corollary 4.3.
Corollary 4.6. Assume that m = 1, ak(t) ≥ 0, k = 1, . . . , r and, for t ≥ t0, at least one of the
following conditions hold (a0, ai and τi, i = 1, . . . , r are constants):
1) ∑r
k=1 ak(t) ≥ a0 > 0,
ess sup
t≥t0
1
∑r
k=1 ak(t)
r
∑
k=1
ak(t)
t
max{t0,hk(t)}
r
∑
l=1
al(s)ds < 1 +
1
e
. (4.10)
2) ai(t) ≡ ai, ∑r
i=1 ai > 0, t − hi(t) ≤ τi, i = 1, . . . , r, and
r
∑
i=1
aiτi < 1 +
1
e
. (4.11)
Then, the scalar equation (1.2) is uniformly exponentially stable.
Proof. Let condition 1) be true. Then, inequality (3.4) turns into inequality (4.10) for m = 1.
Let condition 2) be true. Since ai(t) ≡ ai, inequality (4.10) is transformed to
ess sup
t≥t0
r
∑
k=1
ak(t − max{t0, hk(t)}) < 1 +
1
e
.
Since
ess sup
t≥t0
r
∑
k=1
ak(t − max{t0, hk(t)}) ≤ ess sup
t≥t0
r
∑
k=1
akτk
inequality (4.11) implies (4.10).
New exponential stability conditions for linear delayed systems 13
Now we consider two particular cases of system (1.1),
˙X(t) = −B(t)X(h(t)) (4.12)
and
˙X(t) = −A(t)X(t) − B(t)X(h(t)) (4.13)
where A(t) = (aij(t))m
i,j=1, B(t) = (bij(t))m
i,j=1 are m × m matrices with Lebesgue measurable
and locally essentially bounded entries
aij : [0, ∞) → R, bij : [0, ∞) → R, i, j = 1, . . . , m
and X(t) = (x1(t), . . . , xm(t))T. Assume that, for the delay h: [0, ∞) → R, the relevant adaptation
of condition (a2) holds, i.e., h is Lebesgue measurable, h(t) ≤ t and t − h(t) ≤ K, t ∈ [0, ∞)
and lim supt→∞(t − h(t)) < ∞.
The following two Corollaries 4.7 and 4.8 deal with the exponential stability of systems
(4.12), (4.13).
Corollary 4.7. Assume that, for t ≥ t0, at least one of the conditions hold (b0, τ, αi and b∗
ij, i, j =
1, . . . , r are constants):
a) bii(t) ≥ b0 > 0, i = 1, . . . , m,
max
i=1,...,m
ess sup
t≥t0
1
bii(t)
m
∑
j=1
j=i
|bij(t)| < 1,
and
max
i=1,...,m
ess sup
t≥t0




t
max{t0,h(t)}
m
∑
j=1
|bij(s)|ds +
1
bii(t)
m
∑
j=1
j=i
|bij(t)|



 < 1 +
1
e
.
b) bii(t) ≥ αi > 0, |bij(t)| ≤ b∗
ij, t − h(t) ≤ τ, i, j = 1, . . . , m,
max
i=1,...,m
1
αi
m
∑
j=1
j=i
b∗
ij < 1, max
i=1,...,m



τ
m
∑
j=1
b∗
ij +
1
αi
m
∑
j=1
j=i
b∗
ij



 < 1 +
1
e
.
Then, the system (4.12) is uniformly exponentially stable.
Proof. System (4.12) can be written in the form
˙xi(t) = −
m
∑
j=1
bij(t)xj(h(t)), i = 1, . . . , m.
Now, the corollary directly follows from Corollaries 4.1 and 4.3.
Corollary 4.8. Assume that, for t ≥ t0,
aii(t) ≥ 0, bii(t) ≥ 0, aii(t) + bii(t) ≥ a0 > 0, i = 1, . . . , m,
14 L. Berezansky, J. Diblík, Z. Svoboda and Z. Šmarda
where a0 is a constant,
max
i=1,...,m
ess sup
t≥t0
1
aii(t) + bii(t)
m
∑
j=1
j=i
(|aij(t)| + |bij(t)|) < 1,
and
max
i=1,...,m
ess sup
t≥t0
1
aii(t) + bii(t)



bii(t)
t
max{t0,h(t)}
m
∑
j=1
(|aij(s)| + |bij(s)|)ds +
m
∑
j=1
j=i
(|aij(t)| + |bij(t)|)




< 1 +
1
e
. (4.14)
Then, the system (4.13) is uniformly exponentially stable.
Proof. We can write system (4.13) as
˙x(t) = −
m
∑
j=1
aij(t)xj(t)) −
m
∑
j=1
bij(t)xj(h(t)), i = 1, . . . , m
and use Theorem 3.1 for the choice rii = 2, a1
ij(t) = aij(t), a2
ij(t) = bij(t), h1
ij(t) = t, h2
ij(t) = h(t),
i, j = 1, . . . , m. Hence, ai(t) = aii(t) + bii(t), i = 1, . . . , m and inequality (4.14) coincides
with (3.4).
Consider particular cases of systems (4.12), (4.13)
˙X(t) = −BX(t − τ) (4.15)
and
˙X(t) = −AX(t) − BX(t − τ) (4.16)
where A = (aij)m
i,j=1 and B = (bij)m
i,j=1 are constant matrices, τ > 0, and aii ≥ 0, bii ≥ 0,
i = 1, . . . , m.
Corollary 4.9. Assume that bii > 0, i = 1, 2, . . . , m, and
max
i=1,...,m
1
bii
m
∑
j=1
j=i
|bij| < 1, max
i=1,...,m



τ
m
∑
j=1
|bij| +
1
bii
m
∑
j=1
j=i
|bij|



 < 1 +
1
e
.
Then, the system (4.15) is uniformly exponentially stable.
Proof. This follows from Corollary 4.7 (b) where αi = bii.
Corollary 4.10. Assume that aii ≥ 0, bii ≥ 0, aii + bii > 0,
1
aii + bii
m
∑
j=1
j=i
(|aij| + |bij|) < 1,
and
1
aii + bii



τbii
m
∑
j=1
(|aij| + |bij|) +
m
∑
j=1
j=i
(|aij| + |bij|)



 < 1 +
1
e
(4.17)
for i = 1, . . . , m. Then, the system (4.16) is uniformly exponentially stable.
New exponential stability conditions for linear delayed systems 15
Proof. Estimating the left-hand side of inequality (4.14) in the case of system (4.16) and using
(4.17), we obtain
max
i=1,...,m
ess sup
t≥t0
1
aii(t) + bii(t)



bii(t)
t
max{t0,h(t)}
m
∑
j=1
(|aij(s)| + |bij(s)|)ds +
m
∑
j=1
j=i
(|aij(t)| + |bij(t)|)




≤ max
i=1,...,m
1
aii + bii



τbii
m
∑
j=1
(|aij| + |bij|) +
m
∑
j=1
j=i
(|aij| + |bij|)



 < 1 +
1
e
.
Therefore, inequality (4.14) holds and Corollary 4.10 is a consequence of Corollary 4.8.
5 Concluding remarks
First we will compare the stability results obtained in the paper with some known result. Let
system (1.1) be of the form
˙x1(t) = −a11(t)x1(h11(t)) − a12(t)x2(h12(t)),
˙x2(t) = −a21(t)x1(h21(t)) − a22(t)x2(h22(t)).
(5.1)
Here, m = 2 and rij = 1, i, j = 1, 2. Assume that there are constants αi, Aij, τij, i, j = 1, 2
such that 0 < αi ≤ aii(t), |aij(t)| ≤ Aij and t − hij(t) ≤ τij ≤ K and, for a constant q ∈ (0, 1),
|a12(t)| ≤ qa11 and |a21(t)| ≤ qa22, t ∈ [t0, ∞). Then, (3.2) and (3.3) hold. Inequality (3.4) holds
if
(A11 + A12)τ11 +
A12
α1
< 1 +
1
e
,
(A22 + A21)τ22 +
A21
α2
< 1 +
1
e
.
(5.2)
By Theorem 3.1, system (5.1) is uniformly exponential stable. The above assumptions are
valid, e.g., for the choice
aii(t) ≡ Aii = αi = 0.1, aij(t) ≡ Aij = 0.099, i = j, τij = 1.89 (5.3)
in (5.1) if i, j = 1, 2.
Apply Theorem 1.6 if t − hij(t) ≡ τij ≤ K, aii(t) ≡ Aii = αi > 0, aij(t) ≡ Aij if i = j,
i, j = 1, 2 in (5.1). Let 0 < a12 = b12a11 and 0 < a21 = b21a22, t ∈ [t0, ∞). We get di = Aiiτii,
i = 1, 2. If di < 1, then
˜b12 = −
2 + A2
11τ2
11
2 − A2
11τ2
11
A12
A11
,
˜b21 = −
2 + A2
22τ2
22
2 − A2
22τ2
22
A21
A22
.
Theorem 1.6 implies (recall that a square matrix is a nonsingular M-matrix if its inverse is a
positive matrix)) the following result. If
Aiiτii < 1, ˜b12
˜b21 < 1,
then system (5.1) is asymptotically stable.
16 L. Berezansky, J. Diblík, Z. Svoboda and Z. Šmarda
Let (5.3) is set in (5.1). Then,
Aiiτii = 0.189 < 1, ˜b12
˜b21
.
= 1.053 < 1
and Theorem 1.6 is not applicable.
It is not difficult to derive examples when conditions (5.2) hold, but stability conditions of
another known results are not valid.
The stability conditions derived in the paper are written in the form of inequalities with the
right-hand sides which are equal the constant 1 + 1/e. As we mentioned in the introduction,
the purpose of this paper was to improve all the results of [8] with the extra condition (1.9).
The first open problem is to remove this condition in all statements of this paper.
Nevertheless, there is another challenge for a possible continuation of investigations. Analysing
some stability results (e.g. [18, Theorem 5.9]) where in the inequalities considered,
the constant 3/2 plays a significant role as a non-improvable bound, an open problem arises,
if we can expect that our results can be improved by replacing the constant 1 + 1/e by the
constant 3/2 in the inequalities used. An alternative problem is to prove or disprove that,
for the general case of variable coefficients and delays, the constant 1 + 1/e is the best one
possible.
For further results on the stability of linear delay differential systems, we refer, e.g., to
the review paper [23] and to [19, 21]. Recent results on global asymptotic stability for delay
differential systems can be found in [9,10,17,22].
Another research challenge is the following. In this paper and in all known papers on the
stability of linear delay differential systems, the conditions sufficient for stability involve only
diagonal delays. It will be interesting to obtain stability conditions such that all delays are
utilized in the relevant inequalities.
As noted in [8], only few necessary stability conditions are known for systems. One of
the interesting problems is the following. To prove or disprove the following conjecture: if
system (1.1) is asymptotically stable, then the sum of the diagonal elements is nonnegative,
i.e.,
m
∑
i=1
rii
∑
k=1
ak
ii(t) ≥ 0, t ≥ t0.
Finally, we recall a problem tacitly mentioned in the introduction – for system (1.1), derive
stability results that could be reduced to Theorems 1.1–1.5 in the scalar case.
Acknowledgements
The second and the third author were supported by the Grant 201/11/0768 of the Czech Grant
Agency (Prague). The fourth author was supported by the grant FEKT-S-14-2200 of Faculty of
Electrical Engineering and Communication, Brno University of Technology.
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Journal of Computational and Applied Mathematics 147 (2002) 315–331
www.elsevier.com/locate/cam
An existence criterion of positive solutions of p-type retarded
functional di erential equations
Josef DiblÃka; ∗, ZdenÄek Svobodab
a
Department of Mathematics, Faculty of Electrical Engineering and Communication,
Brno University of Technology, TechnickÃa 8, 616 00 Brno, Czech Republic
b
Department of Mathematics, Military Academy of Brno, PS 13, 612 00 Brno, Czech Republic
Received 20 February 2001; received in revised form 4 February 2002
Abstract
The conditions of existence of a positive solution (i.e., a solution with positive coordinates on a considered
interval) of systems of retarded functional equations in the case of unbounded delay with ÿnite memory
are discussed. A general criterion for nonlinear case is given as well as its application to a linear system.
Illustrative special cases are considered too. c 2002 Elsevier Science B.V. All rights reserved.
MSC: 34K20; 34K25
Keywords: Positive solution; Delayed equation; Retract principle; p-Function
1. Introduction
The main aim of this paper is to give conditions for the existence of solutions with positive
coordinates for systems of retarded functional di erential equations (RFDEs) with unbounded delay
and with ÿnite memory. Before the formulation of the results of this paper let us give a short
survey of the known results. In [11], a criterion concerning the existence of positive solution for the
equation
˙x(t) + p(t)x(t − (t)) = 0 (1)
is given, where p; ∈ C([t0; ∞); R+), (t) 6 t; limt→∞ (t − (t)) = ∞ and R+ = [0; ∞). A function
x is called a solution of Eq. (1) corresponding to an initial point t1 ¿ t0 (or with respect to t1) if x
is deÿned and is continuous on [T1; ∞), T1 =inft¿t1 {t − (t)}, di erentiable on [t1; ∞), and satisÿes
(1) for t ¿ t1.
∗
Corresponding author.
E-mail addresses: diblik@feec.vutbr.cz (J. DiblÃk), zsvoboda@scova.vabo.cz (Z. Svoboda).
0377-0427/02/$ - see front matter c 2002 Elsevier Science B.V. All rights reserved.
PII: S 0377-0427(02)00439-9
316 J. DiblÃk, Z. Svoboda / Journal of Computational and Applied Mathematics 147 (2002) 315–331
Theorem A (Erbe et al. [11, p. 29]). Eq. (1) has a positive solution with respect to t1 if and only
if there exists a continuous function (t) on [T1; ∞) such that (t) ¿ 0 for t ¿ t1 and
(t) ¿ p(t)e
t
t− (t)
(s) ds
; t ¿ t1:
This result was generalized for nonlinear systems of RFDEs in [6]. Some results in this direction
are formulated in [13] and in [1] as well. Positive solutions of Eq. (1) in the critical case were
studied e.g., in [5,7–11]. Unfortunately, results of [6] hold for systems with bounded retardation
only. In the present paper we investigate the problem of existence of positive solutions (i.e., problem
of existence of solutions having all its coordinates positive on considered intervals) for nonlinear
systems of RFDEs with unbounded delay but with ÿnite memory in the sense given in [16]. Let us
recall this notion.
Deÿnition 1 (Lakshmikamthan et al. [16, p. 8]). The function p ∈ C[R × [ − 1; 0]; R] is called a
p-function if it has the following properties:
(i) p(t; 0) = t;
(ii) p(t; −1) is a nondecreasing function of t;
(iii) there exists a ¿ − ∞ such that p(t; #) is an increasing function for # for each t ∈ ( ; ∞).
Remark 1. Let us note that conditions (i) and (iii) imply property (iv) (introduced as an additional
property in [16; p. 8]): t = p(t; 0) ¿ p(t; −1) for t ∈ ( ; ∞). In the following; we will suppose that
t is su ciently large; i.e.; that (iii) holds on the considered intervals.
In the theory of RFDEs the symbol yt, which expresses “taking into account”, the history of the
process y(t) considered, is used. With the aid of p-functions the symbol yt is deÿned as follows:
Deÿnition 2 (Lakshmikamthan et al. [16, p. 8]). Let t0 ∈ R; A ¿ 0 and y ∈ C([p(t0; −1); t0 + A);
Rn
). For any t ∈ [t0; t0 + A); we deÿne yt by yt(#) = y(p(t; #)); −1 6 # 6 0 and write yt ∈ C ≡
C[[ − 1; 0]; Rn
].
In this paper we investigate the system
˙y(t) = f(t; yt); (2)
where f ∈ C[[t0t0 + A) × C; Rn
]. This system is called the system of p-type retarded functional
di erential equations (p-RFDEs). The function y ∈ C([p(t0; −1); t0 + A); Rn
) ∩ C1
([t0; t0 + A); Rn
)
satisfying (2) on [t0; t0 + A) is called a solution of this system p-RFDEs on [[p(t0; −1); t0 + A).
Remark 2. System (2) with yt deÿned in accordance with Deÿnition 2 is called a system with
unbounded delay with ÿnite memory. Note that the frequently used symbol “yt” (e.g.; in accordance
with [14; p. 38]; yt(s) = y(t + s); where − 6 s 6 0; ¿ 0; = const) for equation with bounded
delay is a partial case of the above deÿnition of yt. Indeed; in this case we can put p(t; #) ≡
t + #.
J. DiblÃk, Z. Svoboda / Journal of Computational and Applied Mathematics 147 (2002) 315–331 317
Suppose that is an open subset of R × C and the function f ∈ C( ; Rn
). If (t0; ) ∈ , then
there exists a solution y = y(t0; ) of the system p-RFDEs (2) through (t0; ) (see [16, p. 25]).
Moreover, this solution is unique if f(t; ) is locally Lipschitzian with respect to [16, p. 30]
and is continuable in the usual sense of extended existence if f is quasibounded (see [16, p. 41]).
Suppose that the solution y = y(t0; ) of p-RFDEs (2) through (t0; ) ∈ , deÿned on [t0; A], is
unique. Then the property of the continuous dependence holds too (see [16, p. 33]), i.e., for every
¿ 0, there exists a ( ) ¿ 0 such that (s; ) ∈ , |s − t0| ¡ and − ¡ imply
yt(s; ) − yt(t0; ) ¡ ; for all t ∈ [ ; A];
where y(s; ) is the solution of the system p-RFDEs (2) through (s; ), = max{s; t0} and · is
the supremum norm in Rn
. Note that these results can be adapted easily for the case (which will
be used in the sequel) when has the form = [p∗; ∞) × C where p∗ ∈ R and the cross-section
{(˜t; ’) ∈ } is an open set for every ˜t ∈ [p∗; ∞).
The paper is organized as follows. In Section 2, a general nonlinear case is considered and the
main result of the paper is presented together with its nonlinear applications. Applications to a linear
system and scalar linear equations are given in Section 3. Proofs of the results (and corresponding
auxiliary material) are collected in Section 4. The method used in the proof of the main result also
permits to conclude that positive solutions of nonlinear equations exist on half-inÿnity interval. This
is an additional advantage of the results presented.
2. Nonlinear case
With Rn
¿0 (Rn
¿0) we denote the set of all component-wise nonnegative (positive) vectors v in Rn
,
i.e., v =(v1; : : : ; vn) ∈ Rn
¿0(Rn
¿0) if and only if vi ¿ 0 (vi ¿ 0) for i =1; : : : ; n. For u; v ∈ Rn
we write
u 6 v if v − u ∈ Rn
¿0; u
v if v − u ∈ Rn
¿0 and u ¡ v if u 6 v and u = v.
2.1. General nonlinear case
Let p∗, t∗ be constants satisfying p∗ = p(t∗; −1) for a given p-function. Let us introduce vectors
; ∈ C([p∗; ∞); Rn
) ∩ C1
([t∗; ∞); Rn
) satisfying 
 on [p∗; ∞). Let us suppose ⊆ (t0; ∞) × C
with t0 6 t∗ and let us put
!:={(t; y): t ¿ p∗
; (t)
y
 (t)}:
Theorem 1. Suppose f ∈ C( ; Rn
) is locally Lipschitzian with respect to the second argument,
quasibounded and, moreover:
(i) For any i=1; : : : ; p (with p ∈ {0; 1; : : : ; n}); t ¿ t∗ and ∈ C([p(t; −1); t]; Rn
) such that (Â; (Â))
∈ ! for all  ∈ [p(t; −1); t); (t; (t)) ∈ @! it follows (t; t) ∈ ;
i(t) ¡ fi(t; t) when i(t) = i(t) (3)
and
i(t) ¿ fi(t; t) when i(t) = i(t): (4)
318 J. DiblÃk, Z. Svoboda / Journal of Computational and Applied Mathematics 147 (2002) 315–331
(ii) For any i = p + 1; : : : ; n; t ¿ t∗ and ∈ C([p(t; −1); t]; Rn
) such that (Â; (Â)) ∈ ! for all
 ∈ [p(t; −1); t); (t; (t)) ∈ @! it follows (t; t) ∈ ;
i(t) ¿ fi(t; t) when i(t) = i(t) (5)
and
i(t) ¡ fi(t; t) when i(t) = i(t): (6)
Then there exists an uncountable set Y of solutions of system (2) on the interval [p∗; ∞) such
that for each y ∈ Y
(t)
y(t)
 (t); t ∈ [p∗
; ∞): (7)
Remark 3. The number p in the formulation of Theorem 1 and in the following can also be equal
to 0 or n. In such cases the corresponding conditions (either (i) or (ii)) are omitted. Note that in
the case (t) ¿ 0 we deal; as follows from (7); with positive solutions.
Deÿnition 3. We say that the functional g ∈ C( ; R) is strongly decreasing (strongly increasing)
with respect to the second argument on if for each (t; ’) ∈ and (t; ) ∈ such that
’(p(t; #))
 (p(t; #)); # ∈ [ − 1; 0);
the inequality
g(t; ’) ¿ g(t; ) (or g(t; ’) ¡ g(t; ))
holds.
Let k0 be constant vectors, i = −1 for i = 1; : : : ; p and i = 1 for i = p + 1; : : : ; n. Let
(t) = ( 1(t); : : : ; n(t)) denote a vector, having continuous entries on [p∗; ∞). Deÿne
T(k; )(t) ≡ k e
t
p∗ (s) ds
= (k1 e 1
t
p∗ 1(s) ds
; : : : ; kn e n
t
p∗ n(s) ds
):
Theorem 2 (Main result). Suppose =[t∗; ∞)×C; f ∈ C( ; Rn
) is locally Lipschitzian with respect
to the second argument, quasibounded and; moreover:
(i) f(t; 0) ≡ 0 if t ¿ t∗.
(ii) The functional fi is strongly decreasing if i=1; : : : ; p and strongly increasing if i=p+1; : : : ; n
with respect to the second argument on .
Then for the existence of a positive solution y = y(t) on [p∗; ∞) of the system p-RFDEs (2) a
necessary and su cient condition is that there exists a vector ∈ C([p∗; ∞); Rn
); such that 0
on [t∗; ∞); satisfying the system of integral inequalities
i(t) ¿
i
ki
e− i
t
p∗ i(s) ds
fi(t; T(k; )t); i = 1; : : : ; n (8)
for t ¿ t∗; with a positive constant vector k and with i=−1 for i=1; : : : ; p; i=1 for i=p+1; : : : ; n.
J. DiblÃk, Z. Svoboda / Journal of Computational and Applied Mathematics 147 (2002) 315–331 319
2.2. Nonlinear applications
Consider a nonlinear integro-di erential equation
˙y(t) = a
t
0
L(s)y(t − s) ds + by2
(t); t ¿ ˜¿ 0 (9)
with continuous function L : [0; ∞) → R+
= (0; ∞), a ∈ {−1; 1}; b ∈ R and sign b = sign a. Note that
similar classes of equations are used for describing the dynamics of a single species of population
(see e.g. [12]). The following theorem (the proof of which is a consequence of the main result)
holds:
Theorem 3. For the existence of a positive solution y = y(t) on [0; ∞) of Eq. (9) satisfying Eq.
(9) on the prescribed interval [˜; ∞); the existence of the function ∈ C([0; ∞); R); positive on
[˜; ∞) and satisfying here the integral inequality
(t) ¿
t
0
L(s)e−a
t
t−s
(u) du
ds + abk ea
t
0
(s) ds
(10)
with a positive constant k; is a necessary and su cient condition.
Remark 4. As an addition to this theorem note that every positive solution y = y(t) of Eq. (9)
with a = 1 and with any b ∈ R; which is deÿned on interval [0; A]; remains positive on its maximal
interval of existence [0; B] ⊆ [0; ∞) with B ¿ A (see the proof of Theorem 3).
Theorem 4. Consider the equation of type (9) with a = −1; i.e.; the equation
˙y(t) = −
t
0
L(s)y(t − s) ds + by2
(t); t ¿ ˜¿ 0; (11)
where b ¡ 0 and suppose L(t) 6 l e− t
; t ∈ [0; ∞) with positive constants l; and with ¿ 2
√
l.
Then there exists a positive solution y = y(t) of Eq. (11) on [0; ∞); satisfying Eq. (11) on [˜; ∞).
3. Linear case
The main result can be applied easily to various classes of linear delayed systems and can serve
as a source for various new criteria.
3.1. Linear delayed system
Let us consider the linear system
˙y = A(t)y(t) + B(t)y( (t)); (12)
where : [t∗; ∞) → [p∗; ∞) is a continuous nondecreasing function and (t) ¡ t. In this case,
p(t; #)=t +#·(t − (t)) and p∗ = (t∗). With respect to n×n matrices A(t)=(aij(t)), B(t)=(bij(t))
320 J. DiblÃk, Z. Svoboda / Journal of Computational and Applied Mathematics 147 (2002) 315–331
we suppose their continuity on [t∗; ∞) and, moreover, the validity of inequalities:
aij(t) 6 0; bij(t) 6 0 if i = 1; : : : ; p; j = 1; : : : ; n; t ∈ [t∗
; ∞); (13)
aij(t) ¿ 0; bij(t) ¿ 0 if i = p + 1; : : : ; n; j = 1; : : : ; n; t ∈ [t∗
; ∞); (14)
n
j=1
bij(t) = 0 for every i = 1; : : : ; n and t ∈ [t∗
; ∞): (15)
Theorem 5. For the existence of a solution y=y(t) of system (12); positive on [p∗; ∞); a necessary
and su cient condition is that there exists a continuous vector ∈ C([p∗; ∞); Rn
) such that (t)0
for t ¿ t∗; satisfying the system of integral inequalities
i(t) ¿ i(aii(t) + bii(t)e− i
t
(t) i(s) ds
)
+
i
ki
n
j=1;j=i
kj e
t
p∗ ( j j(s)− i i(s)) ds
(aij(t) + bij(t)e− j
t
(t) j(s) ds
); i = 1; : : : ; n (16)
on [t∗; ∞) with a positive constant vector k and with i=−1 for i=1; : : : ; p; i=1 for i=p+1; : : : ; n.
Remark 5. Let us remark that su cient conditions for the existence of bounded solutions of systems
and equations of type (12) were given in [3;4].
3.2. Scalar linear applications
Let us consider the scalar linear equation with delay
˙y(t) = −
t
(t)
K(t; s)y(s) ds; (17)
where K : [t∗; ∞) × [p∗; ∞) → R+
is a continuous function, and : [t∗; ∞) → [p∗; ∞) is a nondecreasing
function with (t) ¡ t.
Theorem 6. Eq. (17) has a positive solution y = y(t) on [p∗; ∞) if and only if there exists a
function ∈ C([p∗; ∞); R); such that (t) ¿ 0 for t ¿ t∗ and
(t) ¿
t
(t)
K(t; s)e
t
s
(u) du
ds (18)
on the interval [t∗; ∞).
Inequality (18) can be used for ÿnding su cient conditions for the existence of a positive solution
of Eq. (17). Let us give two of them.
In the case when (t) ≡ p∗ ¡ t∗ and K(t; s) ≡ c(t) for every t ∈ [t∗; ∞), Eq. (17) takes the form
˙y(t) = −c(t)
t
p∗
y(s) ds: (19)
J. DiblÃk, Z. Svoboda / Journal of Computational and Applied Mathematics 147 (2002) 315–331 321
Theorem 7. For the existence of a solution of Eq. (19); positive on [p∗; ∞); the inequality
c(t) 6
2
e (t−p∗) − 1
; t ∈ [t∗
; ∞) (20)
with a positive constant is a su cient condition.
In the case when (t) ≡ t − l, l ∈ R+
and K(t; s) ≡ c(t) for every t ∈ [t∗; ∞), Eq. (17) takes the
form
˙y(t) = −c(t)
t
t−l
y(s) ds: (21)
Theorem 8. For the existence of a solution of Eq. (21); positive on [t∗ − l; ∞); the inequality
c(t) 6 M; t ∈ [t∗
; ∞) (22)
is su cient for M = (2 − )=l2
= const with a constant being the positive root of the equation
2 − = 2e− . (The approximate values are ˙=1:5936 and M ˙=0:6476=l2
.)
4. Auxiliary material and proofs
4.1. Retract principle and Lyapunov-type principle
The proof of Theorem 1 is made with the aid of the retract principle. This principle, well known
and often used in the theory of ordinary di erential equations (see e.g. [15]), goes back to Wa˙zewski
[19]. For RFDEs with bounded retardation, this principle was modiÿed e.g. in Rybakowski [18]. Here,
we use Rybakowski’s modiÿed result (Lemma 1 below) which concerns the existence of at least
one curve in a given family of curves, with graph lying in an open set. Then this lemma is applied
to systems of p-RFDEs (Lemma 3 below). Except this, the inverse principle is used (Lemmas 2
and 4). This principle has the origin in the theory of Lyapunov stability and for retarded functional
di erential equations, it was developed by Razumikhin (e.g., [17]).
If a set A ⊂ R × Rn
is given, then int A, A and @A denote, as usual, the interior, the closure, and
the boundary of A, respectively.
Deÿnition 4. Let be a topological space; let a subset ˜ ⊂ R × be open in R × ; and let x be
a mapping; associating with every ( ; ) ∈ ˜ a function x( ; ) : D ; → Rn
where D ; is an interval
in R. Assume (1)–(3)
(1) ∈ D ; .
(2) If t ∈ int D ; ; then there is open neighbourhood O( ; ) of ( ; ) in ˜ such that t ∈ D ; holds
for all ( ; ) ∈ O( ; ).
(3) If ( ; ); ( ; ) ∈ ˜; and t ∈ D ; ; t ∈ D ; ; then
lim
( ; ;t )→( ; ;t)
x( ; )(t ) = x( ; )(t):
322 J. DiblÃk, Z. Svoboda / Journal of Computational and Applied Mathematics 147 (2002) 315–331
If all these conditions are satisÿed; then ( ; ˜; x) is called a system of curves in Rn
.
Studying the proof of Theorem 2.1 in [18, p. 119], we get the following formulation of it, suitable
for our applications.
Lemma 1 (Retract principle). Let ( ; ˜; x) be a system of curves in Rn
. Let ˜!; W; Z be sets.
Assume that conditions (1)–(4) below hold:
(1) (a) ˜! ⊂ [p∗; ∞) × Rn
where p∗ ∈ R and the cross-section {(˜t; y) ∈ ˜!} is an open set for every
˜t ∈ [p∗; ∞); W ⊂ @ ˜!;
(b) z ⊂ ˜! ∪ W; Z ∩ W is a retract of W; but not a retract of Z.
(2) There is a continuous map q : B → ; where B = Z ∩ (Z ∪ W); such that for any z =
( ; y) ∈ B: ( ; q(z)) ∈ ˜; and if also z ∈ W; then x( ; q(z))( ) = y.
(3) Let A be the set of all z=( ; y) ∈ Z ∩ ˜! such that for ÿxed ( ; y) ∈ A there is a t ¿ ; t ∈ D ;q(z)
and (t; x( ; q(z))(t)) ∈ ˜!.
Assume that for every z = ( ; y) ∈ A there is a t(z); t(z) ¿ ; such that:
(a) t(z) ∈ D ;q(z) and for all t; 6 t ¡ t(z): (t; x( ; q(z))(t)) ∈ ˜!;
(b) (t(z); x( ; q(z))(t(z))) ∈ W;
(c) for any ¿ 0; there is a t = t( ; z); t(z) ¡ t 6 t(z) + ; such that t ∈ D ;q(z) and (t; x( ; q(z))
(t)) ∈ ˜!.
(4) For any z=( ; y) ∈ W ∩B; and all ¿ 0; there is a t=t( ; z); ¡ t 6 + such that t ∈ D ;q(z)
and (t; x( ; q(z))(t)) ∈ ˜!.
Then there is a z0 = ( 0; y0) ∈ Z ∩ ˜! such that for every t ∈ D 0;q(z0):
(t; x( 0; q(z0))(t)) ∈ ˜!:
Proof of the following lemma is obvious and is therefore omitted.
Lemma 2 (Lyapunov principle). Let ( ; ˜; x) be a system of curves in Rn
and ˜! be a set. Assume
that conditions (1)–(4) below hold:
(1) ˜! ⊂ [p∗; ∞) × Rn
where p∗ ∈ R and the cross-section {(˜t; y) ∈ ˜!} is an open set for every
˜t ∈ [p∗; ∞).
(2) There is a continuous map q : B → ; where B= ˜!∩{(t∗; y); t∗ ∈ R; t∗ =const; t∗ ¿ p∗; y ∈ Rn
};
such that for any z = (t∗; y) ∈ B: (t∗; q(z)) ∈ ˜; and if also z ∈ @ ˜!; then x(t∗; q(z))(t∗) = y.
(3) For every z = (t∗; y) ∈ B ∩ ˜! with property (t; x(t∗; q(z))(t)) ∈ ˜! for all t within an interval
t∗ ¡ t ¡ t(z) and (t(z); x(t∗; q(z))(t(z))) ∈ @ ˜!; t(z) ∈ Dt∗;q(z) there is a such that t(z) +
∈ Dt∗;q(z) and (t; x(t∗; q(z))(t)) ∈ ˜! for all t; t(z) ¡ t ¡ t(z) + .
(4) For any z = (t∗; y) ∈ B ∩ @ ˜!; and all ¿ 0; there is a t = t( ; z); ¡ t 6 + such that
t ∈ Dt∗;q(z) and (t; x(t∗; q(z))(t)) ∈ ˜!.
Then for every z0 = (t∗; y0) ∈ B ∩ ˜! and every t ∈ Dt∗;q(z0):
(t; x(t∗
; q(z0))(t)) ∈ ˜!: (23)
J. DiblÃk, Z. Svoboda / Journal of Computational and Applied Mathematics 147 (2002) 315–331 323
4.2. Regular polyfacial set, retract, and Lyapunov methods for p-RFDEs
Let = C. Let ˜ be open in R × C, f ∈ C( ˜; Rn
) and through each ( ; ) ∈ ˜ there exists a
unique solution y( ; ) of (2) deÿned on maximal interval [ ; a), ¡ a 6 ∞. Let D ; =[ ; a). Then
( ; ˜; y) is a system of curves in Rn
in the sense of Deÿnition 4.
Let li; mj, i = 1; : : : ; p, j = 1; : : : ; s, p + s ¿ 0 be real-valued C1
-functions deÿned on R × Rn
. The
set
˜! = {(t; y) ∈ [p∗
; ∞) × Rn
; li(t; y) ¡ 0; mj(t; y) ¡ 0; for all i; j}
will be called a polyfacial set.
Deÿnition 5. A polyfacial set ˜! is called regular with respect to Eq. (2) if ( ); (ÿ); ( ) below hold:
( ) If (t; t) ∈ R × C and if (p(t; #); t(#)) ∈ ˜! for all # ∈ [ − 1; 0); then (t; t) ∈ ˜.
(ÿ) For all i = 1; : : : ; p; all (t; y) ∈ @ ˜! for which li(t; y) = 0 and for all t ∈ C for which t(0) = y
and (p(t; #); t(#)) ∈ ˜! for all # ∈ [ − 1; 0); it follows that
Dli(t; y) ≡
n
r=1
@li
@yr
(t; y)fr(t; t) +
@li
@t
(t; y) ¿ 0:
( ) For all j = 1; : : : ; s; all (t; y) ∈ @ ˜! which mj(t; y) = 0 and for all t ∈ C which t(0) = y and
(p; (t; #); t(#)) ∈ ˜! for all # ∈ [ − 1; 0); it follows that
Dmj(t; y) ≡
n
r=1
@mj
@yr
(t; y)fr(t; t) +
@mj
@t
(t; y) ¡ 0:
The following lemma concerning the existence of a solution of Eq. (2) with graph remaining in
the set ˜! on its maximal existence interval, will play a crucial role in the proof of Theorem 1.
Lemma 3 (Retract method). Let p ¿ 0. Let ˜! be a nonempty polyfacial set; regular with respect to
Eq. (2); let the function f ∈ C( ˜; Rn
) be locally Lipschitzian with respect to the second argument;
and
W = {(t; y) ∈ @ ˜!: mj(t; y) ¡ 0; j = 1; : : : ; s}: (24)
Let Z be a subset of ˜! ∪ W and let mapping q : B = Z ∩ (Z ∪ W) → C be continuous and such that
if z = ( ; y) ∈ B; then ( ; q(z)) ∈ ˜; and:
(1) if z ∈ Z ∩ ˜!; then (p( ; #); q(z)(p( ; #))) ∈ ˜! for # ∈ [ − 1; 0];
(2) if z ∈ W ∩ B; then ( ; q(z)( )) = z and (p( ; #); q(z)(p( ; #))) ∈ ˜! for # ∈ [ − 1; 0).
Let; moreover; Z ∩ W be a retract of W; but not a retract of Z. Then there exists a z0 =
( 0; y0) ∈ Z ∩ ˜! such that (t; y( 0; q(z0))(t)) ∈ ˜! for every t ∈ D 0;q(z0).
Proof. We prove the lemma using Lemma 1. Conditions (1) and (2) of Lemma 1 are obviously
satisÿed. Let us verify conditions (3) and (4).
324 J. DiblÃk, Z. Svoboda / Journal of Computational and Applied Mathematics 147 (2002) 315–331
Veriÿcation of condition (3): Let z = ( ; y) ∈ A, and let t(z) be the smallest of all t ¿ such
that t ∈ D ;q(z) and (t; y( ; q(z))(t)) ∈ ˜!. Since ( ; y( ; q(z))( )) = ( ; q(z)(p(t; 0))) ∈ ˜!, it follows
that ¡ t(z) ¡ ∞. Obviously, (t(z); y( ; q(z))(t(z))) ∈ @ ˜! and moreover for 6 t ¡ t(z) it holds:
(t; y( ; q(z))(t)) ∈ ˜!, hence (3a) is satisÿed.
Let t ≡ yt(z)( ; q(z)). Obviously t ∈ C. Then (t(z)); t) ∈ ˜, and (t(z); (t(z)))=(t(z); y( ; q(z))
(t(z))) ∈ @ ˜!, and
(p(t(z); #); (p(t(z); #))) ∈ ˜!; for # ∈ [ − 1; 0):
To prove condition (3b) suppose, on the contrary, that (t(z); (p(t(z); 0))) ∈ W. Since (t(z);
(p(t(z); 0))) ∈ @ ˜! it follows mj0 (t(z); (p(t(z); 0))) = 0 for some j0 ∈ {1; : : : ; s}. Hence, inequality
( ) in Deÿnition 5 is satisÿed. Since y( ; q(z))(t) is di erentiable in t for t ¿ , this inequality
becomes
Dmj0 (t; y( ; p(z))(t))|t=t(z) ¡ 0;
i.e., for some ¿ 0 and all 0 ¡ h ¡ :
mj0 (t(z) − h; y( ; q(z))(t(z) − h))
¿ mj0 (t(z); y( ; q(z))(t(z))) = mj0 (t(z); (p(t(z); 0)) = mj0 (t(z); t(0)) = 0:
Hence, (t(z) − h; y( ; q(z))(t(z) − h)) ∈ ˜!. This is a contradiction to (3a). Then (t(z); t(0)) ∈ W
and, therefore, (3b) is satisÿed.
It follows that li0 (t(z); (t(z); 0)) = 0 for some i0 ∈ {1; : : : ; p}. Applying (ÿ) of Deÿnition 5, we
get
Dli0 (t; y( ; p(z))(t))|t=t(z) ¿ 0;
hence, for some ¿ 0 and all 0 ¡ h ¡ :
li0 (t(z) + h; y( ; q(z))(t(z) + h)) ¿ li0 (t(z); y( ; q(z))(t(z))) = li0 (t(z); (p(t(z); 0)) = 0:
Hence (t(z) + h; y( ; q(z))(t(z) + h)) ∈ ˜! and (3c) is satisÿed.
Veriÿcation of condition (4): If z=( ; y) ∈ W ∩B, then there is a i0 ∈ {1; : : : ; p} such that li0 ( ; y)=
0. Let = q(z), then ( + #; (p( ; #))) ∈ ˜!, for all # ∈ [ − 1; 0). Hence, the derivative from the
right
Dli0 (t; y( ; p(z))(t))|t= +0 ¿ 0:
This implies the existence of some ¿ 0 such that for all 0 ¡ h ¡ :
li0 ( + h; y( ; q(z))( + h)) ¿ li0 ( ; y( ; q(z))( )) = li0 ( ; (p( ; 0))) = 0;
i.e., ( + h; y( ; q(z))( + h)) ∈ ˜! for 0 ¡ h; . So, condition (4) of Lemma 1 holds and the Lemma
1 is valid in the described situation. From its conclusion, the conclusion of Lemma 3 follows.
Lemma 4 (Lyapunov method). Let p=0. Let ˜! be a nonempty polyfacial set; regular with respect
to Eq. (2) and let the function f ∈ C( ˜; Rn
) be locally Lipschitzian with respect to the second
argument. Let mapping q : B → C; B = ˜! ∩ {(t∗; y); t∗ ∈ R; t∗ = const; y ∈ Rn
} be continuous and
such that if z = (t∗; y) ∈ B; then (t∗; q(z)) ∈ ˜; and:
J. DiblÃk, Z. Svoboda / Journal of Computational and Applied Mathematics 147 (2002) 315–331 325
(1) If z ∈ ˜!; then (p(t∗; #); q(z)(p(t∗; #))) ∈ ˜! for # ∈ [ − 1; 0].
(2) If z ∈ @ ˜!. then (t∗; q(z)(t∗)) = z and (p(t∗; #); q(z)(p(t∗; #))) ∈ ˜! for # ∈ [ − 1; 0).
Then for every z0 = (t∗; y0) ∈ B ∩ ˜! and every t ∈ Dt∗;q(z0):
(t; y(t∗
; q(z0))(t)) ∈ ˜!: (25)
Proof. We prove the lemma using Lemma 2. Conditions (1) and (2) of Lemma 2 are obviously
satisÿed. Let us verify condition (3).
Suppose, on the contrary, that for a z = (t∗; y) ∈ B with property (t; y(t∗; q(z))(t)) ∈ ˜! for all,
t; t∗ ¡ t ¡ t(z) and (t(z); y(t∗; q(z))(t(z)) ∈ @ ˜!, t(z) ∈ Dt∗;q(z0) there is a ∗ ¿ 0 such that t(z) +
∗ ∈ Dt∗;q(z0) and (t; y(t∗; q(z))(t)) ∈ ˜! for all t, t(z) ¡ t ¡ t(z) + ∗.
Let us put t ≡ yt(z)(t∗; q(z)). Then t ∈ C, (t(z)); t) ∈ ˜; (t(z); (t(z))) = (t(z); y(t∗; q(z))
(t(z))) ∈ @ ˜!, and
(p(t(z); #); (p(t(z); #))) ∈ ˜! for # ∈ [ − 1; 0):
Since (p(t(z); 0); (p(t(z); 0)))=(t(z); (t(z))) ∈ @ ˜!, it follows mj0 (t(z); (p(t(z); 0)))=0 for some
j0 ∈ {1; : : : ; s}. Hence inequality ( ) in Deÿnition 5 is satisÿed and, similarly as in the proof of Lemma
3, the inequality
Dmj0 (t; y(t∗
; p(z))(t))|t=t(z) ¡ 0
leads to a contradiction. Thus, condition (3) of Lemma 2 holds.
Let us verify condition (4). If z = (t∗; y) ∈ B ∩ @ ˜!, then there is a j0 ∈ {1; : : : ; s} such that
mj0 (t∗; y)=0. Let =q(z). Then (t∗ +#; (p(t∗; #))) ∈ ˜!, for all # ∈ [−1; 0). Hence, the derivative
from the right
Dmj0 (t; y(t∗
; p(z))(t))|t=t∗+0 ¡ 0:
This inequality implies the validity of condition (4) of Lemma 2. From its conclusion (see formula
(23)) we have
(t; y(t∗
; q(z0))(t)) ∈ ˜!
for every z0=(t∗; y0) ∈ B ∩ ˜! and every t ∈ Dt∗;q(z0). The stronger inequality (25) can be proved by the
method used above, since if (t0
; y(t∗; q(z0))(t0
)) ∈ @ ˜! with t0
∈ Dt∗;q(z0), then mj0 (t0
; y(t∗; q(z0))(t0
))=
0 for some j0 ∈ {1; : : : ; s}. This fact again leads to a contradiction. The lemma is proved.
Proof of Theorem 1. Suppose p ¿ 0. Let us deÿne the auxiliary functions
li(t; y) ≡ li(t; yi) ≡ (yi − i(t))(yi − i(t)); i = 1; : : : ; p;
mj(t; y) ≡ mj(t; yp+j) ≡ (yp+j − p+j(t))(yp+j − p+j(t)); j = 1; : : : ; r
with p + r = n. Then
! = {(t; y): t ¿ p∗
; li(t; y) ¡ 0; mj(t; y) ¡ 0; i = 1; : : : ; p; j = 1; : : : ; r}:
326 J. DiblÃk, Z. Svoboda / Journal of Computational and Applied Mathematics 147 (2002) 315–331
At ÿrst we show that the set ! is a regular polyfacial set with respect to system (2). Condition
( ) of Deÿnition 5 holds obviously (we suppose that ˜! ≡ ! and ˜ ≡ is put here). Let us compute
Dli(t; y) = (yi − i(t))(fi(t; t) − i(t)) + (yi − i(t))(fi(t; t) − i(t));
where i = 1; : : : ; p and
Dmj(t; y) = (yp+j − p+j(t))(fp+j(t; t) − p+j(t))
+ (yp+j − p+j(t))(fp+j(t; t) − p+j(t));
where j = 1; : : : ; r. In view of (3) and (4) we get for (t; y) ∈ @! and i = 1; : : : ; p
Dli(t; y)|yi= i(t) = ( i(t) − i(t))(fi(t; t) − i(t))|yi= i(t) ¿ 0;
Dli(t; y)|yi= i(t) = ( i(t) − i(t))(fi(t; t) − i(t))|yi= i(t) ¿ 0
and in view of (5) and (6) we get for (t; y) ∈ @! and j = 1; : : : ; r
Dmj(t; y)|yp+j= p+j(t) = ( p+j(t) − p+j(t))(fp+j(t; t) − p+j(t))|yp+j= p+j(t) ¡ 0;
Dmj(t; y)|yp+j= p+j(t) = ( p+j(t) − p+j(t))(fp+j(t; t) − p+j(t))|yp+j= p+j(t) ¡ 0:
So, conditions (ÿ) and ( ) of Deÿnition 5 are valid and ! is a regular polyfacial set with respect
to system (2).
Let us show now that Lemma 1 (where ˜! ≡ ! and ˜ ≡ is put) holds. Deÿne the set
Z ≡ {t∗
; y1; : : : ; yp; y0
p+1; : : : ; y0
n) : li(t∗
; yi) 6 0; i = 1; : : : ; p;
mj(t∗
; y0
p+j) ¡ 0; y0
p+j = const; j = 1; : : : ; r}
and a mapping of the set
W = {(t; y) ∈ @!: mj(t; y) ¡ 0; j = 1; : : : ; r}
(see formula (24)) into Z ∩ W:
W (t; y1; : : : ; yp; yp+1; : : : ; yn) → (t∗
; ˜y1; : : : ; ˜yp; y0
p+1; : : : ; y0
n) ∈ Z ∩ W
with
˜yi = i(t∗
) + (yi − i(t))
i(t∗) − i(t∗)
i(t) − i(t)
; i = 1; : : : ; p:
This mapping is continuous (points of the set Z ∩ W are mapped into itself) and, consequently,
Z ∩W is a retract of W. The set Z ∩W is not a retract of Z because in the case p ¿ 1 the boundary
of p-dimensional ball is not its retract (see e.g. [2]) and in the case p = 1, the set Z ∩ W consists
of two disjoint nonempty subsets and, consequently, is not a retract of Z.
It is easy to deÿne the mapping q : B = Z ∩ (Z ∪ W) → C, for z = (t∗; y1; : : : ; yn) ∈ B as
qi(z)(#) = i(p(t∗
; #)) + h(#)
i(t∗) − yi
i(t∗) − i(t∗)
( i(p(t∗
; #)) − i(p(t∗
; #))); (26)
where h is any function such that h ∈ C([ − 1; 0]; (0; 1]), h(t) = 1 ⇔ t = 0. This mapping is continuous
and for all # ∈ [ − 1; 0) the inequality (p(t∗; #))
q(z)(#)
 (p(t∗; #)) holds. Moreover
(t∗; q(z)(0)) = z.
J. DiblÃk, Z. Svoboda / Journal of Computational and Applied Mathematics 147 (2002) 315–331 327
All the assumptions of Lemma 3 are fulÿlled. Then there exists a point z0 = (t∗; y0) ∈ Z ∩ !
such that the graph of the corresponding solution y(t∗; q(z0))(t) of system (2) belongs to the set
! for each t ∈ Dt∗;q(z0). Since in ! existence and unicity of every initial problem is guaranteed, we
conclude Dt∗;q(z0) = [t∗; ∞), i.e. inequalities (7) hold on [t∗; ∞). Taking into account the properties
of initial functions and quasiboundedness of f, we conclude that inequalities (7) hold even on the
larger interval [p∗; ∞).
Let p =0. In this case the proof can be simpliÿed without using the topological principle. Putting
˜! ≡ !, ˜ ≡ and deÿning the mapping q : B → C with B= ˜!∩{(t∗; y); t∗ ∈ R, t∗ =const, y ∈ Rn
}
by formula (26) we see that all assumptions of Lemma 4 are valid. From its conclusion (and from
the last steps of the previous part of the proof) we conclude that inequalities (7) hold. The theorem
is proved.
Proof of Theorem 2. Necessity: Let y(t) ∈ C([p∗; ∞); Rn
) be a positive solution of system (2)
on [t∗; ∞). It can be shown easily that for every i = 1; : : : ; n there exist continuous function
i ∈ C([p∗; ∞); R) such that
yi(t) = ki e i
t
p∗ i(s) ds
; t ∈ [p∗
; ∞) (27)
i.e.;
y(t) = k e
t
p∗ (s) ds
; = ( 1; 2; : : : ; n); t ∈ [p∗
; ∞)
with ki = yi(p∗) ¿ 0. System (2) turns; by means of (27); into the following system of integrofunctional
equations
i(t) =
i
ki
e− i
t
p∗ i(s) ds
fi(t; T(k; )t); t ¿ t∗
; i = 1; : : : ; n: (28)
The necessity of condition (8) is now in view of (27); (28); (i) and (ii) obvious since inequalities
(8) hold and
i(t) ≡
yi(t)
iyi(t)
=
fi(t; T(k; )t)
iyi(t)
¿ 0; t ¿ t∗
; i = 1; : : : ; n:
Su ciency: This part of the proof follows immediately from Theorem 1 for (t) ≡ 0 and (t) ≡
k exp(
t
p∗ (s) ds). Indeed, in this case inequality (3) holds since, for i = 1; : : : ; p and i(t) = i(t)
(in view of (8) and condition (ii) of Theorem 2), we get for t ¿ t∗:
i(t) − fi(t; t) = ki i i(t)e i
t
p∗ i(s) ds
− fi(t; t)
= − ki i(t)e− t
p∗ i(s) ds
− fi(t; t) 6 [in view of (8)] 6 fi(t; T(k; )t) − fi(t; t)
¡ [in view of (ii) since T(k; )t(#) ¿ t(#) for # ∈ [ − 1; 0) and Ti(k; )(0) = t(0)]
¡ fi(t; t) − fi(t; t) = 0:
Inequality (4) holds too since, for i = 1; : : : ; p; i(t) = i(t) = 0 in view of conditions (i) and (ii) of
Theorem 2, we get for t ¿ t∗
i(t) − fi(t; t) = −fi(t; t) ¿ 0:
Inequalities (5) and (6) can be veriÿed in a similar manner. Theorem 2 is proved.
328 J. DiblÃk, Z. Svoboda / Journal of Computational and Applied Mathematics 147 (2002) 315–331
Proof of Theorem 3. We show that the proof is a consequence of Theorem 2 if n = 1 and
f(t; yt) ≡ a
t
0
L(s)y(t − s) ds + by2
(t)
is put in its formulation. In our case t∗ = ˜; p∗ = 0 and we can put = (0; ∞) × C. Inequality (10)
follows from Inequality (8). The functional f(t; yt) (which is obviously quasibounded) is for a = 1
(and b ¿ 0) strongly increasing with respect to the second argument on and for a = −1 (and
b ¡ 0) strongly decreasing with respect to the second argument on in the sense of Deÿnition 3.
In the ÿrst case we put = 1; in the second one = −1. Except this; if a = 1 and b ∈ R is arbitrary;
every solution y(t) which is positive for 0 6 t 6 A remains positive for every t ¿ A on its maximal
interval of existence [0; B). Obviously; supposition y(t1) = 0 for a t1 ∈ (A; B) and y(t) ¿ 0 on [0; t1)
leads to a contradiction; since
˙y(t1) =
t1
0
L(s)y(t1 − s) ds ¿ 0
and; consequently; for t ¡ t1 (if t is su ciently close to t1) we get y(t) ¡ 0. This contradicts the
supposition of positively of y(t) on [0; t1).
Proof of Theorem 4. This proof uses Theorem 3. Inequality (10) will hold if there exists a positive
(t) satisfying the inequalities
(t) ¿ l
t
0
e− s+
t
t−s
(u) du
ds − bk e− t
0
(s) ds
¿
t
0
L(s)e
t
t−s
(u) du
ds − bk e− t
0
(s) ds
on [˜; ∞) with a positive constant k. Supposing (t) ≡ = const; = we get
¿
l
−
(1 − e( − )t
) − bk e− t
; t ∈ [˜; ∞): (29)
Suppose − ¿ 0. Then Inequality (29) holds if
¿
l
−
− bk
or
f( ) ≡ −
l
−
¿ − bk:
Since the right side of this inequality can be made su ciently small (due to the positive number k
which can be chosen su ciently small); it is enough to take = ∗ such that f( ∗) ¿ 0. Since the
equation
f ( ) ≡ 1 −
l
( − )2
= 0
has the roots 1;2 = ±
√
l; we can put ∗ = −
√
l. Then ∗ ¿ 0; − ∗ ¿ 0 and f( ∗)= −2
√
l ¿ 0.
The theorem is proved.
Proof of Theorem 5. Theorem 5 follows from Theorem 2 if = [t∗; ∞) × C;
f(t; yt) = (f1(t; yt); : : : ; fn(t; yt))
J. DiblÃk, Z. Svoboda / Journal of Computational and Applied Mathematics 147 (2002) 315–331 329
and
f(t; yt) ≡
n
j=1
[aij(t)yj(t) + bij(t)yj( (t))]; i = 1; : : : ; n
is put in its formulation. Note that the system of integral inequalities (8) turns into the system of
inequalities (16). Conditions (13) and (15) ensure that the functional fi(t; yt) is strongly decreasing
on for i = 1; : : : ; p and conditions (14) and (15) ensure that the functional fi(t; yt) is strongly
increasing on for i=p+1; : : : ; n in the sense of Deÿnition 3. Condition (15) gives a guarantee that
in every row of matrix B there is at least one nonzero element for every t ∈ [t∗; ∞). This property
is su cient for the validity of Deÿnition 3. The theorem is proved.
Proof of Theorem 6. We will use Theorem 2 again. The functional
f(t; yt) = −
t
(t)
K(t; s)y(s) ds
is strongly decreasing with respect to the second argument on = [t∗; ∞) × C in the sense of
Deÿnition 3. Integral inequality (8) with n = i = 1; = −1 takes the form
(t) ¿ e
t
p∗ (u) du
t
(t)
K(t; s)e− t
p∗ (u) du
ds; t ∈ [t∗
; ∞):
From this inequality; inequality (18) follows. The theorem is proved.
Proof of Theorem 7. In the case considered; inequality (18) takes the form
(t) ¿ c(t)
t
p∗
e
t
s
(u) du
ds; t ∈ [t∗
; ∞): (30)
We will look for a constant solution of this inequality; i.e.; we put (t) ≡ = const. Then
¿ c(t)
t
p∗
e (t−s)
ds = c(t)e t e− s
−
t
p∗
= c(t)e t
−
1
(e− t
− e− p∗
) =
c(t)
(e (t−p∗
)
− 1); t ∈ [t∗
; ∞)
or
c(t) 6
2
e (t−p∗) − 1
; t ∈ [t∗
; ∞):
It is now clear that the value = ¿ 0 satisÿes Inequality (30) and Inequality (20) is a consequence
of Theorem 6. The theorem is proved.
Proof of Theorem 8. In the case considered; p∗ = t∗ − l and Inequality (18) takes the form
(t) ¿ c(t)
t
t−l
e
t
s
(u) du
ds; t ∈ [t∗
; ∞):
330 J. DiblÃk, Z. Svoboda / Journal of Computational and Applied Mathematics 147 (2002) 315–331
Supposing (t) ≡ = const; we get
¿ c(t)
t
t−l
e (t−s)
ds = c(t)e t e− s
−
t
t−l
= c(t)e t
−
1
(e− t
− e− (t−l)
) =
c(t)
(e l
− 1); t ∈ [t∗
; ∞)
or
c(t) 6
2
e l − 1
=
1
l2
( l)2
e l − 1
≡
1
l2
g( l); t ∈ [t∗
; ∞):
Let us look for the maximum of the function
g(x) =
x2
ex − 1
in (0; ∞). Since g(0+
) = g(+∞) = 0 and g(x) ¿ 0 for x ∈ (0; ∞) this maximum exists. Since
g (x) =
x
(ex − 1)2
[ex
(2 − x) − 2];
the maximum is reached in the point x = satisfying the equation
e =
2
2 −
and g( ) = (2 − ). So inequality (22) is a consequence of inequality (18). Easy numerical computation
shows that ˙=1:5936; g( ) ˙=0:6476. Theorem 8 is now a consequence of Theorem 6.
Acknowledgement
This research was supported by the Grant 201=99=0295 of the Grant Agency of Czech Republic
and by the Project ME 423 of Ministry of Education of Czech Republic.
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[9] Y. Domshlak, I.P. Stavroulakis, Oscillation of ÿrst-order delay di erential equations in a critical case, Appl. Anal.
61 (1996) 359–371.
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[10] ÃA. Elbert, I.P. Stavroulakis, Oscillation and non-oscillation criteria for delay di erential equations, Proc. Amer. Math.
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[13] I. Gyori, G. Ladas, Oscillation Theory of Delay Di erential Equations, Clarendon Press, Oxford, 1991.
[14] J.K. Hale, S.M.V. Lunel, Introduction to Functional Di erential Equations, Springer, New York, 1993.
[15] P. Hartman, Ordinary Di erential Equations, 2nd Edition, Birkhauser, Basel, 1982.
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Nonlinear Analysis 63 (2005) e813–e821
www.elsevier.com/locate/na
Positive solutions of retarded functional
differential equations
Josef Diblíka,∗
, Zdenˇek Svobodab
aDepartment of Mathematics, Faculty of Electrical Engineering and Computer Science, Brno University of
Technology, Technická 8, 616 00 Brno, Czech Republic
bDepartment of Mathematics, University of Defence, Kounicova 65, 612 00 Brno, Czech Republic
Abstract
For systems of retarded functional differential equations with unbounded delay and with finite
memory sufficient conditions of existence of positive solutions on an interval of the form [t0, ∞) are
derived.
᭧ 2005 Elsevier Ltd. All rights reserved.
MSC: 34K20; 34K25
Keywords: Positive solution; Delayed equation; p-function
1. Introduction
In this paper we give sufficient conditions for the existence of positive solutions (i.e. a solution
with positive coordinates on a considered interval) for systems of retarded functional
differential equations (RFDEs) with unbounded delay and with finite memory. At first let
us give short explanation emphasized above terms. Let us recall basic notions of RFDEs
with unbounded delay but with finite memory. A function p ∈ C[R × [−1, 0], R] is called
a p-function if it has the following properties [14, p. 8]:
(i) p(t, 0) = t.
(ii) p(t, −1) is a nondecreasing function of t.
∗ Corresponding author.
E-mail addresses: diblik.j@fce.vutbr.cz (J. Diblík), zdenek.svoboda@unob.cz (Z. Svoboda).
0362-546X/$ - see front matter ᭧ 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.na.2005.01.006
e814 J. Diblík, Z. Svoboda / Nonlinear Analysis 63 (2005) e813–e821
(iii) There exists a − ∞ such that p(t, ϑ) is an increasing function for ϑ for each
t ∈ ( , ∞). (Throughout the following text we suppose t ∈ ( , ∞).)
In the theory of RFDEs the symbol yt , which expresses “taking into account’’, the history
of the process y(t) considered, is used. With the aid of p-functions the symbol yt is defined
as follows:
Definition 1 (Lakshmikamthan et al. [14, p. 8]). Let t0 ∈ R, A > 0 and y ∈ C([p(t0, −1),
t0 + A), Rn
). For any t ∈ [t0, t0 + A), we define
yt (ϑ) := y(p(t, ϑ)), −1 ϑ 0
and write
yt ∈ C := C[[−1, 0], Rn
].
1.1. System with unbounded delay with finite memory
In this paper we investigate existence of positive solutions of the system
˙y(t) = f (t, yt ), (1)
where f ∈ C([t0, t0 +A)×C, Rn
), A>0, and yt is defined in accordance with Definition 1.
This system is called the system of p-type retarded functional differential equations (pRFDE’s)
or a system with unbounded delay with finite memory.
Definition 2. A function y is said to be a solution of (1) on [p(t0, −1), t0 + A) if
y ∈ C([p(t0, −1), t0 + A), Rn
)
and y(t) satisfies (1) on [t0, t0 + A).
Suppose that is an open subset of R × C and the function f : → Rn
is continuous.
If (t0, 
) ∈ , then there exists a solution y = y(t0, 
) of the system p-RFDEs (1) through
(t0, 
) (see [14, p. 25]). Moreover, this solution is unique if f (t, 
) is locally Lipschitzian
with respect to second argument 
 [14, p. 30] and is continuable in the usual sense of
extended existence if f is quasibounded [14, p. 41]. (Recall the definition of quasiboundedness.
If D is any set in Rn
we will let CD := C[[−1, 0], D]. We say that the functional f is
quasibounded if f is bounded on every set of the form [t0, ] × CD where t0 < < A and D
is a closed bounded set.)
Suppose that the solution y = y(t0, 
) of p-RFDEs (1) through (t0, 
) ∈ , defined
on [t0, A], is unique. Then the property of the continuous dependence holds too (see [14,
p. 33]), i.e. for every > 0, there exists a ( ) > 0 such that (s, ) ∈ , |s − t0| < and
− 
 < imply
yt (s, ) − yt (t0, 
) < for all t ∈ [ , A],
J. Diblík, Z. Svoboda / Nonlinear Analysis 63 (2005) e813–e821 e815
where y(s, ) is the solution of the system p-RFDEs (1) through (s, ), = max{s, t0} and
· is the supremum norm in Rn
. Note that these results can be adapted easily for the case
(which will be used in the sequel) when has the form = [p∗, ∞) × C where p∗ ∈ R.
1.2. Problem of existence of positive solutions
In this paper we are concerned with the problem of existence of positive solutions (i.e.
problem of existence of solutions having all its coordinates positive on considered intervals)
for nonlinear systems of RFDEs with unbounded delay but with finite memory. Let us cite
some known results for retarded functional differential equations. Results in this direction
are formulated in the books [11–13] and in the papers [1,2], too. Positive solutions in the
critical case were studied, e.g. in [3,5–10]. Some known scalar results concerning existence
of positive solutions were extended for nonlinear systems of RFDEs with bounded retardation
in [4] and for nonlinear systems of RFDEs with unbounded delay and with finite
memory in [6].
2. Auxiliary lemma
With Rn
0 (Rn
>0) we denote the set of all component-wise nonnegative (positive ) vectors
v in Rn
, i.e., v = (v1, . . . , vn) ∈ Rn
0 (Rn
>0) if and only if vi 0 (vi > 0) for i = 1, . . . , n.
For u, v ∈ Rn
we write u v if v − u ∈ Rn
0; u>v if v − u ∈ Rn
>0 and u < v if u v and
u = v.
Let p∗, t∗ be constants satisfying p∗ = p(t∗, −1) for a given p-function. Define vector
valued functions , ∈ C([p∗, ∞), Rn
), satisfying > on [p∗, ∞), and continuously
differentiable on [t∗, ∞). Let us put := [t∗, ∞) × C and
:= {(t, y) : t p∗
, (t)>y> (t)}.
Definition 3. A system of initial functions SE, with respect to nonempty sets E and
where E ⊂ is defined as a continuous mapping : E → C such that (a) and (b) in the
following text hold:
(a) For each z = (t, y) ∈ E ∩ int and ϑ ∈ [−1, 0] : (t + ϑ, (z)(p(t, ϑ))) ∈ .
(b) For each z=(t, y) ∈ E∩ j and ϑ ∈ [−1, 0) : (t +ϑ, (z)(p(t, ϑ))) ∈ and, moreover,
(t, (z)(p(t, 0))) = z.
We define as S1
E, a system of initial functions SE, if all functions (z), z = (t, y) ∈ E
are continuously differentiable on [−1, 0).
The next lemma deals with sufficient conditions for existence of solutions of system (1),
the graphs of which remain in the set . The proof of this lemma is based on the retract
method and the Lyapunov method and can be found in [6, Theorem 1]. Since this result
will be used in the following, we modify slightly its original formulation underlying the
necessary (for our purposes) fact that every set of initial functions contain at least one
initial function generating solution with desired properties. This claim is a consequence of
the proof of cited result.
e816 J. Diblík, Z. Svoboda / Nonlinear Analysis 63 (2005) e813–e821
Lemma 1. Suppose f ∈ C( , Rn
) is locally Lipschitzian with respect to the second argument,
quasibounded and moreover:
(i) For any i = 1, . . . , p (with p ∈ {0, 1, . . . , n}), t t∗ and ∈ C([p(t, −1), t], Rn
) such
that ( , ( )) ∈ for all ∈ [p(t, −1), t), (t, (t)) ∈ j it follows (t, t ) ∈ ,
i(t) < fi(t, t ) when i(t) = i(t) (2)
and
i(t) > fi(t, t ) when i(t) = i(t). (3)
(ii) For any i = p + 1, . . . , n, t t∗ and ∈ C([p(t, −1), t], Rn
) such that ( , ( )) ∈
for all ∈ [p(t, −1), t), (t, (t)) ∈ j it follows (t, t ) ∈ ,
i(t) > fi(t, t ) when i(t) = i(t) (4)
and
i(t) < fi(t, t ) when i(t) = i(t). (5)
Then at every set of initial functions SE, with
E := {(t, y) : t = t∗
, (t) y (t)}
there exist at least one = ∗ ∈ SE, defined by a z∗ = (t∗, y∗) ∈ E ∩ int such that for
corresponding solution y(t∗, ∗(z∗)) we have
(t, y(t∗
, ∗
(z∗
))(t)) ∈ (6)
for every t p∗.
3. Existence of positive solutions
Let
k := (k1, . . . , kn)?0
be a constant vector and
(t) := ( 1(t), . . . , n(t))
denote a vector, defined and locally integrable on [p∗, ∞). Define an auxiliary operator
T (k, )(t) := ke
t
p∗ (s) ds
= (k1e
t
p∗ 1(s) ds
, k2e
t
p∗ 2(s) ds
, . . . , kne
t
p∗ n(s) ds
). (7)
Let a constant vector k?0 and a vector (t) be defined and locally integrable on [p∗, ∞).
Then the operator T is well defined by (7). Define for every i ∈ {1, 2, . . . , n} two type of
subsets of the set C:
T i
:= {
 ∈ C : 0>
(ϑ)>T (k, )t (ϑ), ϑ ∈ [−1, 0]
except for 
i(0) = kie
t
p∗ i(s) ds
}
J. Diblík, Z. Svoboda / Nonlinear Analysis 63 (2005) e813–e821 e817
and
T i := {
 ∈ C : 0>
(ϑ)>T (k, )t (ϑ), ϑ ∈ [−1, 0] except for 
i(0) = 0}.
Theorem 1. Suppose f ∈ C( , Rn
) is locally Lipschitzian with respect to the second
argument and quasibounded. Let a constant vector k?0 and a vector (t) be defined and
locally integrable on [p∗, ∞). If, moreover, inequalities
i i(t) > i
ki
e−
t
p∗ i(s) ds
· fi(t, 
) (8)
hold for every i ∈ {1, 2, . . . , n}, (t, 
) ∈ [t∗, ∞) × T i
and inequalities
ifi(t, 
) > 0 (9)
hold for every i ∈ {1, 2, . . . , n}, (t, 
) ∈ [t∗, ∞) × Ti, where i = −1 for i = 1, . . . , p
and i = 1 for i = p + 1, . . . , n, then there exists a positive solution y = y(t) on [p∗, ∞)
of the system p-RFDEs (1).
Proof. We will employ Lemma 1. Put (t) := 0, (t) := T (k, )(t). Let us suppose
i ∈ {1, . . . , p}. It is easy to conclude that inequality (2) is equivalent to
i(t) < fi(t, 
) when 
 ∈ T i
(10)
if the function is changed by the function 
 ∈ T i
and inequality (3) is equivalent to
i(t) > fi(t, 
) when 
 ∈ Ti (11)
if the function is changed by the function 
 ∈ T i. Similarly, for i ∈ {p + 1, . . . , n} we
conclude that inequality (5) is equivalent to
i(t) > fi(t, 
) when 
 ∈ T i
(12)
if the function is changed by the function 
 ∈ Ti
and inequality (4) is equivalent to
i(t) < fi(t, 
) when 
 ∈ Ti (13)
if the function is changed by the function 
 ∈ Ti. Let us verify that above inequalities
are valid. For t t∗ and i ∈ {1, . . . , p} (i.e. i = −1) we get:
fi(t, 
) − i(t) = i( i(t) − fi(t, 
)) = i(ki i(t)e
t
p∗ i(s) ds
− fi(t, 
))
= kie
t
p∗ i(s) ds
i i(t) − i
ki
e−
t
p∗ i(s) ds
fi(t, 
)
> [in view of (8)] > kie
t
p∗ i(s) ds
( i i(t) − i i(t)) = 0.
e818 J. Diblík, Z. Svoboda / Nonlinear Analysis 63 (2005) e813–e821
Similarly, for t t∗ and i ∈ {p + 1, . . . , n} (i.e. i = 1) we get
i(t) − fi(t, 
) = i( i(t) − fi(t, 
)) = i(ki i(t)e
t
p∗ i(s) ds
− fi(t, 
))
= kie
t
p∗ i(s) ds
i i(t) − i
ki
e−
t
p∗ i(s) ds
fi(t, 
)
> [in view of (8)] > kie
t
p∗ i(s) ds
( i i(t) − i i(t)) = 0.
Therefore inequalities (10) and (12) hold. Inequalities (11) and (13) are valid, too since,
due to (9)
i(t) − fi(t, 
) = ifi(t, 
) > 0 if i = 1, 2, . . . , p (i.e. i = −1)
and
fi(t, 
) − i(t) = ifi(t, 
) > 0 if i = p + 1, p + 2, . . . , n (i.e. i = 1).
All conditions of Lemma 1 are satisfied. From its conclusion we immediately get the
desired statement. Theorem 1 is proved.
Remark 1. Let us underline that ifTheorem 1 hold, then indicated positive solution y=y(t)
satisfies on [p∗, ∞] inequalities
0>y(t)> (t)
with corresponding given .
3.1. A nonlinear example
The following example demonstrates that results can be successfully applied to nonlinear
systems. Let us show that the system
y1(t) = −1
2 [y4
1(t1/2
) + y2
1(t) · y2(t)],
y2(t) = y2(t) − y1(t) · y2(t1/2
) · y3(t),
y3(t) = y2
1(t1/2
) · y2
3(t1/2
) (14)
has a positive solution on interval [2, ∞). Define
p(t, ϑ) := t + (t −
√
t)ϑ, ϑ ∈ [−1, 0].
Then system (14) can be rewritten as
y1(t) = f1(t, yt ) := −1
2 [y4
1(p(t, −1)) + y2
1(p(t, 0)) · y2(p(t, 0))],
y2(t) = f2(t, yt ) := y2(p(t, 0)) − y1(p(t, 0) · y2(p(t, −1)) · y3(p(t, 0)),
y3(t) = f3(t, yt ) := y2
1(p(t, −1)) · y2
3(p(t, −1)).
J. Diblík, Z. Svoboda / Nonlinear Analysis 63 (2005) e813–e821 e819
Let us verify that Theorem 1 can be used. If we put
p∗
= 2 = p(t∗
, −1),
t∗
= 4,
k = (k1, k2, k3) = (1/4, 1, 1/2),
= ( 1, 2, 3) = (−1/t, 0, 1/t),
1 = 2 = −1,
3 = 1,
then
T (k, )(t) := ke
t
2 (s) ds
=
1
4
· e−
t
2 ds/s
, 1,
1
2
· e
t
2 ds/s
=
1
2t
1,
t
4
.
Let us verify inequalities (8) and (9). If i = 1 and 
 ∈ T1
then
1
k1
e−
t
p∗ 1(s) ds
· f1(t, 
)
= −2t · f1(t, 
) < t ·
1
2
√
t
4
+
1
2t
2
=
3
8t
<
1
t
= 1 1(t),
if i = 2 and 
 ∈ T2
then
2
k2
e−
t
p∗ 2(s) ds
· f2(t, 
)
= −
2
t
· f2(t, 
) = −
2
t
· 1 − 
2(−1) ·
1
2t
·
t
4
<
2
t
· −1 +
1
8
= −
7
4t
< 0 = 2 2(t)
and if i = 3 and 
 ∈ T3
then
3
k3
e−
t
p∗ 3(s) ds
· f3(t, 
)
=
4
t
· f3(t, 
) <
4
t
·
1
2
√
t
2
·
√
t
4
2
=
1
16t
<
1
t
= 3 3(t)
and inequalities (8) on interval [4, ∞) hold.
Inequalities (9) hold on interval [4, ∞) since if i = 1 and 
 ∈ T1 then
1
k1
· f1(t, 
) = −4f1(t, 
) = 2[
4
1(−1) + 
2
1(0) · 
2(0)] > 0,
if i = 2 and 
 ∈ T2 then
2
k2
· f2(t, 
) = −f2(t, 
)
= −[
2(0) − 
1(0) · 
2(−1) · 
3(0)] = 
1(0) · 
2(−1) · 
3(0) > 0
e820 J. Diblík, Z. Svoboda / Nonlinear Analysis 63 (2005) e813–e821
and if i = 3 and 
 ∈ T3 then
3
k3
· f3(t, 
) =
1
2
f3(t, 
) =
1
2
[
2
1(−1) · 
2
3(−1)] > 0.
All conditions of Theorem 1 are valid. Therefore a positive solution
y = y(t) = (y1(t), y2(t), y3(t)),
of system (14) exists on [2, ∞). Taking into account Remark 1 we conclude that on the
interval considered inequalities
0 < y1(t) < 1/2t,
0 < y2(t) < 1,
0 < y3(t) < t/4
hold.
Acknowledgements
This research was supported by the Grant IAA1163401 of the GrantAgency of theASCR
and by the Council of Czech Government MSM 2622000 13. The authors are grateful to
the referee for his/her suggestions to improve the presentation of this paper.
References
[1] L. Berezanski, E. Braverman, On oscillation of a logistic equation with several delays, J. Comput. Appl.
Math. 113 (2000) 255–265.
[2] J. ˇCermák, The asymptotic bounds of solutions of linear delay systems, J. Math. Anal. Appl. 225 (1998)
373–388.
[3] J. Diblík, Positive and oscillating solutions of differential equations with delay in critical case, J. Comput.
Appl. Math. 88 (1998) 185–202.
[4] J. Diblík,A criterion for existence of positive solutions of systems of retarded functional differential equations,
Nonlinear Anal. TMA 38 (1999) 327–339.
[5] J. Diblík, N. Koksch, Positive solutions of the equation ˙x(t) = −c(t)x(t − ) in the critical case, J. Math.
Anal. Appl. 250 (2000) 635–659.
[6] J. Diblík, Z. Svoboda, An existence criterion of positive solutions of p-type retarded functional differential
equations, J. Comput. Appl. Math. 147 (2002) 315–331.
[7] Y. Domshlak, On oscillation properties of delay differential equations with oscillating coefficients, Funct.
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[8] Y. Domshlak, I.P. Stavroulakis, Oscillation of first-order delay differential equations in a critical case, Appl.
Anal. 61 (1996) 359–371.
[9] Á. Elbert, I.P. Stavroulakis, Oscillation and non-oscillation criteria for delay differential equations, Proc.Am.
Math. Soc. 123 (1995) 1503–1510.
[10] L.H. Erbe, Q. Kong, B.G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker,
New York, 1995.
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[11] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer
Academic Publishers, Dordrecht, 1992.
[12] I. Györi, G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, Oxford, 1991.
[13] J.K. Hale, S.M.V. Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993.
[14] V. Lakshmikamthan, L. Wen, B. Zhang, Theory of Differential Equations with Unbounded Delay, Kluwer
Academic Publishers, Dordrecht, 1994.
Nonlinear Analysis 64 (2006) 1831–1848
www.elsevier.com/locate/na
Positive solutions of p-type retarded functional
differential equations
Josef Diblíka,∗
, Zdenˇek Svobodab
aBrno University of Technology, 616 00 Brno, Czech Republic
bDepartment of Mathematics, Military Academy of Brno, PS 13, 612 00 Brno, Czech Republic
Received 30 April 2004; accepted 19 July 2005
Abstract
For systems of retarded functional differential equations with unbounded delay and with finite
memory sufficient and necessary conditions of existence of positive solutions on an interval of the form
[t0, ∞) are derived. A general criterion is given together with corresponding applications (including
a linear case, too). Examples are inserted to illustrate the results.
᭧ 2005 Elsevier Ltd. All rights reserved.
MSC: 34K20; 34K25
Keywords: Positive solution; Delayed equation; p-function
1. Introduction
In this paper is given a criterion for the existence of positive solutions (i.e., solutions with
positive coordinates on a considered interval) for systems of retarded functional differential
equations (RFDEs) with unbounded delay and with finite memory. At first let us give short
explanation of emphasized above terms.
∗ Corresponding author. Tel.: +420 54 11 47 618; fax: +420 54 11 43 392.
E-mail addresses: diblik.j@fce.vutbr.cz (J. Diblík), zsvoboda@scova.vabo.cz (Z. Svoboda).
0362-546X/$ - see front matter ᭧ 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.na.2005.07.020
1832 J. Diblík, Z. Svoboda / Nonlinear Analysis 64 (2006) 1831–1848
1.1. Time delay expressed by p-functions
Let us recall basic notions of RFDEs with unbounded delay but with finite memory. A
function p ∈ C[R × [−1, 0], R] is called a p-function if it has the following properties [13,
p. 8]:
(i) p(t, 0) = t.
(ii) p(t, −1) is a nondecreasing function of t.
(iii) there exists a − ∞ such that p(t, ϑ) is an increasing function for ϑ for each
t ∈ ( , ∞). (Throughout the following text we suppose t ∈ ( , ∞).)
In the theory of RFDEs the symbol yt , which expresses “taking into account”, the history
of the process y(t) considered, is used. With the aid of p-functions the symbol yt is defined
as follows:
Definition 1 (Lakshmikamthan et al. [13, p. 8]). Let t0 ∈ R, A > 0 and y ∈ C([p(t0, −1),
t0 + A), Rn
). For any t ∈ [t0, t0 + A), we define
yt (ϑ) := y(p(t, ϑ)), −1 ϑ 0
and write
yt ∈ C := C[[−1, 0], Rn
].
Note that the frequently used symbol “yt ” (e.g., in [12, p. 38], yt (s) := y(t + s), where
− s 0, > 0, = const) in the theory of delayed functional differential equations for
equations with bounded delays is a partial case of the above definition. Indeed, in this case
we can put p(t, ϑ) := t + ϑ, ϑ ∈ [−1, 0].
1.2. System with unbounded delay with finite memory
In this paper we investigate existence of positive solutions of the system
˙y(t) = f (t, yt ), (1)
where f ∈C([t0, t0+A)×C, Rn
), A>0, and yt is defined in accordance with Definition 1.
This system is called the system of p-type retarded functional differential equations (pRFDEs)
or a system with unbounded delay with finite memory.
Definition 2. The function y ∈ C([p(t0, −1), t0+A), Rn
)∩C1([t0, t0+A), Rn
) satisfying
(1) on [t0, t0 + A) is called a solution of (1) on [ p(t0, −1), t0 + A).
Suppose that is an open subset of R × C and the function f : → Rn
is continuous.
If (t0, 
) ∈ , then there exists a solution y = y(t0, 
) of the system p-RFDEs (1) through
(t0, 
) (see [13, p. 25]). Moreover this solution is unique if f (t, 
) is locally Lipschitzian
with respect to second argument 
 [13, p. 30] and is continuable in the usual sense of
extended existence if f is quasibounded [13, p. 41]. Suppose that the solution y = y(t0, 
)
J. Diblík, Z. Svoboda / Nonlinear Analysis 64 (2006) 1831–1848 1833
of p-RFDEs (1) through (t0, 
) ∈ , defined on [t0, A], is unique. Then the property of
the continuous dependence holds too (see [13, p. 33]), i.e., for every > 0, there exists a
( ) > 0 such that (s, ) ∈ , |s − t0| < and − 
 < imply
yt (s, ) − yt (t0, 
) < for all t ∈ [ , A],
where y(s, ) is the solution of the system p-RFDEs (1) through (s, ), = max{s, t0} and
· is the supremum norm in Rn
. Note that these results can be adapted easily for the case
(which will be used in the sequel) when has the form = [ t∗, ∞) × C where t∗ ∈ R.
1.3. Problem of existence of positive solutions
In this paper we are concerned with the problem of existence of positive solutions (i.e.,
problem of existence of solutions having all its coordinates positive on considered intervals)
for nonlinear systems of RFDEs with unbounded delay but with finite memory. Let us cite
some known results for RFDEs. For the scalar equation
˙x(t) + p(t)x(t − (t)) = 0 (2)
with p, ∈ C([t0, ∞), R+), (t) t, limt→∞ (t − (t)) = ∞ and R+ = [0, ∞) a criterion
for existence of a positive solution is given in the book [10]. Namely, (2) has a positive
solution with respect to t1 if and only if there exists a continuous function (t) on [T1, ∞)
with T1 = inft t1 {t − (t)}, such that (t) > 0 for t t1 and
(t) p(t)e
t
t− (t) (s) ds
, t t1. (3)
(A function x is called a solution of (2) with respect to an initial point t1 t0 if x is defined and
is continuous on [T1, ∞), differentiable on [t1, ∞), and satisfies (2) for t t1.) Results in
this direction are formulated in the book [11] and in the papers [1,2], too. Positive solutions
of (2) in the critical case were studied e.g. in [3,5–10]. The cited criterion was generalized
for nonlinear systems of RFDEs with bounded retardation in [4] and for nonlinear systems
of RFDEs with unbounded delay and with finite memory in [6]. These generalizations
are in a sense “direct” generalizations since in their formulations existence of a positive
(vector) functions playing a similar role as in (3) is supposed. Results of presented paper
concern the problem of existence of positive solutions for nonlinear systems of RFDEs with
unbounded delay and with finite memory. In general, assumption of positivity of indicated
functions is not necessary. This is illustrated by a nonlinear example in Section 3.3. The
paper is organized as follows. Auxiliary material is placed in Section 2, nonlinear results in
Section 3, and linear applications in Section 4.
2. Auxiliary lemma
With Rn
0 (Rn
>0) we denote the set of all component-wise nonnegative (positive) vectors
v in Rn
, i.e., v = (v1, . . . , vn) ∈ Rn
0(Rn
>0) if and only if vi 0 (vi > 0) for i = 1, . . . , n.
For u, v ∈ Rn
we write u v if v − u ∈ Rn
0; u>v if v − u ∈ Rn
>0 and u < v if u v and
u = v.
1834 J. Diblík, Z. Svoboda / Nonlinear Analysis 64 (2006) 1831–1848
Let p∗, t∗ be constants satisfying p∗ = p(t∗, −1) for a given p-function. Define vectors
, ∈ C([ p∗
, ∞), Rn
) ∩ C1
([t∗
, ∞), Rn
)
satisfying > on [ p∗, ∞). Let us put := [t∗, ∞) × C and
:= {(t, y) : t p∗
, (t)>y> (t)}.
Definition 3. A system of initial functions SE, with respect to nonempty sets E and
where E ⊂ is defined as a continuous mapping : E → C such that (a) and (b) in the
following text hold:
(a) For each z = (t, y) ∈ E ∩ int and ϑ ∈ [−1, 0] : (t + ϑ, (z)(p(t, ϑ))) ∈ .
(b) For each z=(t, y) ∈ E∩j and ϑ ∈ [−1, 0) : (t +ϑ, (z)(p(t, ϑ))) ∈ and, moreover,
(t, (z)(p(t, 0))) = z.
We define as S1
E, a system of initial functions SE, if all functions (z), z = (t, y) ∈ E
are continuously differentiable on [−1, 0).
The next lemma deals with sufficient conditions for existence of solutions of system (1),
the graphs of which remain in the set . The proof of this lemma is based on the retract
method and the Lyapunoff method and can be found in [6, Theorem 1]. Since this result
will be used in the following, we modify slightly its original formulation underlying the
necessary (for our purposes) fact that every set of initial functions contains at least one
initial function generating solution with desired properties. This claim is a consequence of
the proof of cited result.
Lemma 1. Suppose f ∈ C( , Rn
) is locally Lipschitzian with respect to the second argument,
quasibounded and moreover:
(i) For any i = 1, . . . , p (with p ∈ {0, 1, . . . , n}), t t∗ and ∈ C([p(t, −1), t], Rn
) such
that ( , ( )) ∈ for all ∈ [p(t, −1), t), (t, (t)) ∈ j it follows:
i(t) < fi(t, t ) when i(t) = i(t) (4)
and
i(t) > fi(t, t ) when i(t) = i(t). (5)
(ii) For any i = p + 1, . . . , n, t t∗ and ∈ C([p(t, −1), t], Rn
) such that ( , ( )) ∈
for all ∈ [p(t, −1), t), (t, (t)) ∈ j it follows:
i(t) > fi(t, t ) when i(t) = i(t) (6)
J. Diblík, Z. Svoboda / Nonlinear Analysis 64 (2006) 1831–1848 1835
and
i(t) < fi(t, t ) when i(t) = i(t). (7)
Then at every set of initial functions SE, with
E := {(t, y) : t = t∗
, (t) y (t)}
There exist at least one = ∗ ∈ SE, defined by a z∗ = (t∗, y∗) ∈ E ∩ int such that for
corresponding solution y(t∗, ∗(z∗)) we have
(t, y(t∗
, ∗
(z∗
))(t)) ∈ (8)
for every t p∗.
3. Nonlinear results
Definition 4. We say that the functional g ∈ C( , Rn
) is i-strongly decreasing (or
i-strongly increasing), i ∈ {1, 2, . . . , n} if for each (t, ) ∈ and (t, ) ∈ such that
(p(t, ϑ))> (p(t, ϑ)) where ϑ ∈ [−1, 0) and i(p(t, 0)) = i(p(t, 0))
the inequality
gi(t, ) > gi(t, ) (or gi(t, ) < gi(t, ))
holds.
The following lemma state a necessary for consequent criterion (Theorem 1) fact, that
if a positive solution of (1) exists then there exists a positive solution on the same interval
through a function ∈ C under an additional condition that is continuously differentiable.
Lemma 2. Suppose f ∈ C( , Rn
) is locally Lipschitzian with respect to the second argument,
quasibounded and, moreover:
(i) f is i-strongly decreasing if i = 1, . . . , p and i-strongly increasing if i = p + 1, . . . , n.
(ii) fi(t, 0) 0 for i = 1, . . . , p and fi(t, 0) 0 for i = p + 1, . . . , n if (t, 0) ∈ .
Ifthesystemp-RFDEs(1)hasapositivesolutiony=y(t)on[ p∗, ∞)(wherep∗=p(t∗, −1))
then it has a positive solution y = Y(t) on[ p∗, ∞) which is continuously differentiable on
[ p∗, t∗), too.
Proof. As it follows from the definition of a solution y of (1) (Definition 2) it include the
properties (with respect to the case considered): y ∈ C([ p∗, ∞), Rn
) ∩ C1([t∗, ∞), Rn
).
In our case is necessary to prove: Y ∈ C([ p∗, ∞), Rn
) ∩ C1([ p∗, t∗) ∪ [t∗, ∞), Rn
), i.e.,
for the proof only suffice to show that a solution Y is differentiable on [ p∗, t∗).
Let us employ Lemma 1. Define (t) := 0 and (t) := y(t). In this case inequality
(4) holds since, for i = 1, . . . , p and i(t) = i(t) = yi(t) (in view of condition (i) of
1836 J. Diblík, Z. Svoboda / Nonlinear Analysis 64 (2006) 1831–1848
Lemma 2), we get for t t∗
fi(t, t ) − i(t) = fi(t, t ) − yi(t)
= fi(t, t ) − fi(t, yt ) > fi(t, t ) − fi(t, t ) = 0.
Inequality (5) holds too since, for i = 1, . . . , p; i(t) = i(t) = 0 in view of conditions (i),
(ii) of Lemma 2, we get for t t∗
i(t) − fi(t, t ) = −fi(t, t ) > − fi(t, 0) 0.
Inequalities (6) and (7) can be verified in a similar manner. Inequality (6) holds since, for
i = p + 1, . . . , n and i(t) = i(t) = yi(t) (in view of condition (ii) of Lemma 2), we get
for t t∗
i(t) − fi(t, t ) = yi(t) − fi(t, t )
= fi(t, yt ) − fi(t, t ) > fi(t, t ) − fi(t, t ) = 0.
Inequality (6) holds too since, for i = p + 1, . . . , n; i(t) = i(t) = 0 in view of conditions
(i), (ii) of Lemma 2, we get for t t∗
i(t) − fi(t, t ) = −fi(t, t ) < − fi(t, 0) 0.
All conditions of Lemma 1 are valid. Then at every set of initial functions SE, with
E := {(t, y) : t = t∗
, 0 y y(t)}
there exist at least one = ∗ ∈ SE, defined by a z∗ = (t∗, y∗) ∈ E ∩ int such that for
corresponding solution y∗ = y∗(t∗, ∗(z∗)) we have
0 < (y∗
(t∗
, ∗
(z∗
))(t)) < y(t)
for every t p∗. Since the set of initial functions SE, can be taken arbitrarily we can
suppose that all initial functions are continuously differentiable, i.e., we put SE, ≡ S1
E, .
Suppose this situation from beginning of the proof. Then the choice Y := y∗(t∗, ∗(z∗))
ends it.
3.1. Sufficient and necessary conditions
Let k = (k1, . . . , kn)?0 be a constant vector. Let (t) = ( 1(t), . . . , n(t)) denote a
vector, defined and locally integrable on [ p∗, ∞). Define an auxiliary operator
T (k, )(t) := ke
t
p∗ (s) ds
= (k1e
t
p∗ 1(s) ds
, . . . , kne
t
p∗ n(s) ds
). (9)
J. Diblík, Z. Svoboda / Nonlinear Analysis 64 (2006) 1831–1848 1837
Theorem 1. Suppose f ∈ C( , Rn
) is locally Lipschitzian with respect to the second
argument, quasibounded and, moreover:
(i) f is i-strongly decreasing if i = 1, . . . , p and i-strongly increasing if i = p + 1, . . . , n.
(ii) fi(t, 0) 0 for i = 1, . . . , p and fi(t, 0) 0 for i = p + 1, . . . , n if (t, 0) ∈ .
Then for existence of a positive solution y = y(t) on [ p∗, ∞) of the system p-RFDEs (1)
(where p∗ = p(t∗, −1)) is necessary and sufficient existence of a positive constant vector
k and a locally integrable vector : [ p∗, ∞) → Rn
continuous on [ p∗, t∗) ∪ [t∗, ∞)
satisfying the system of integral inequalities
i i(t) i
ki
e−
t
p∗ i(s) ds
· fi(t, T (k, )t ), i = 1, . . . , n (10)
for t t∗ with i = −1 for i = 1, . . . , p and i = 1 for i = p + 1, . . . , n.
Proof. Necessity. Let y be a positive solution of system (1) on [t∗, ∞), i.e., in view of
Definition 2 (and taking into account the case considered) y(t) ∈ C([ p∗, ∞), Rn
) ∩
C1([ t∗, ∞), Rn
). Since all suppositions of Lemma 2 hold, there exists a positive solution
y = Y(t) on [ p∗, ∞) which is continuously differentiable on [ p∗, t∗), too, i.e., Y ∈
C([ p∗, ∞), Rn
) ∩ C1([ p∗, t∗) ∪ [t∗, ∞), Rn
). Define
i(t) :=
⎧
⎪⎨
⎪⎩
Yi (t)
Yi(t)
if t ∈ [ p∗, t∗) ∪ (t∗, ∞),
Yi (t∗ + 0)
Yi(t∗)
if t = t∗.
Then the vector is well defined and locally integrable on [ p∗, ∞) and is continuous on
[ p∗, t∗) ∪ [t∗, ∞). Now it is easy to verify
Y(t) ≡ T (k, )(t) = k e
t
p∗ (s) ds
= (k1 e
t
p∗ 1(s) ds
, . . . , kn e
t
p∗ n(s) ds
), t ∈ [ p∗
, ∞)
with k = (Y1(p∗), . . . , Yn(p∗))?0. Since Yi (t) ≡ fi(t, Yt ) on [t∗, ∞), we get
ki i(t)e
t
p∗ i(s) ds
≡ fi(t, T (k, )t ), i = 1, . . . , n, t ∈ [t∗
, ∞)
or, equivalently,
i i(t) ≡ i
ki
e−
t
p∗ i(s) ds
· fi(t, T (k, )t ), i = 1, . . . , n, t ∈ [t∗
, ∞),
where all operations are well defined. The last identity ends the proof of necessity since
inequalities (10) hold on [t∗, ∞).
Sufficiency. This part of the proof uses Lemma 1. Put
(t) := 0, (t) := T (k, )(t).
1838 J. Diblík, Z. Svoboda / Nonlinear Analysis 64 (2006) 1831–1848
In this case inequality (4) holds since, for i = 1, . . . , p and i(t) = i(t) (in view of (10)
and condition (i) of Theorem 1), we get for t t∗ and i = −1:
fi(t, t ) − i(t) = i( i(t) − fi(t, t )) = i(ki i(t)e
t
p∗ i(s) ds
− fi(t, t ))
= kie
t
p∗ i(s) ds
i i(t) − i
ki
e−
t
p∗ i(s) ds
fi(t, t )
[in view of (10)] i(fi(t, T (k, )t ) − fi(t, t ))
= fi(t, t ) − fi(t, T (k, )t )
> [in view of (i) since T (k, )t (ϑ)
> t (ϑ) for ϑ ∈ [−1, 0) and Ti(k, )(0) = i(0)]
> fi(t, T (k, )t ) − fi(t, T (k, )t ) = 0.
Inequality (5) holds too since, for i = 1, . . . , p; i(t) = i(t) = 0 in view of conditions (i),
(ii) of Theorem 1, we get for t t∗
i(t) − fi(t, t ) = −fi(t, t ) > − fi(t, 0) 0.
Inequalities (6) and (7) will be verified in a similar manner. Inequality (6) holds since, for
i = p + 1, . . . , n and i(t) = i(t) (in view of (10) and condition (ii) of Theorem 1), we
get for t t∗ and i = 1:
i(t) − fi(t, t ) = i( i(t) − fi(t, t )) = i(ki i(t)e
t
p∗ i(s) ds
− fi(t, t ))
= kie
t
p∗ i(s) ds
i i(t) − i
ki
e−
t
p∗ i(s) ds
fi(t, t )
[in view of (10)] i(fi(t, T (k, )t ) − fi(t, t ))
= fi(t, T (k, )t ) − fi(t, t )
> [in view of (ii) since T (k, )t (ϑ)
> t (ϑ) for ϑ ∈ [−1, 0)and Ti(k, )(0) = i(0)]
> fi(t, T (k, )t ) − fi(t, T (k, )t ) = 0.
Inequality (7) holds too since, for i = p + 1, . . . , n; i(t) = i(t) = 0 in view of conditions
(i), (ii) of Theorem 1, we get for t t∗
i(t) − fi(t, t ) = −fi(t, t ) < − fi(t, 0) 0.
All conditions of Lemma 1 are satisfied. Its conclusion ends the proof of this part. Theorem
1 is proved.
3.2. Sufficient conditions
Let a constant vector k?0 and a vector (t) defined and locally integrable on [ p∗, ∞)
be given. Then the operator T is well defined by (9). Define for every i ∈ {1, 2, . . . , n} two
J. Diblík, Z. Svoboda / Nonlinear Analysis 64 (2006) 1831–1848 1839
types of subsets of the set C:
Ti
:= 
 ∈ C : 0>
(ϑ)>T (k, )t (ϑ), ϑ ∈ [−1, 0] except for 
i(0)
= kie
t
p∗ i(s) ds
and
Ti := {
 ∈ C : 0>
(ϑ)>T (k, )t (ϑ), ϑ ∈ [−1, 0] except for 
i(0) = 0}.
Theorem 2. Suppose f ∈ C( , Rn
) is locally Lipschitzian with respect to the second
argument and quasibounded. Let a constant vector k?0 and a vector (t) defined and
locally integrable on [ p∗, ∞) are given. If, moreover, inequalities
i i(t) > i
ki
e−
t
p∗ i(s) ds
· fi(t, 
) (11)
hold for every i ∈ {1, 2, . . . , n}, (t, 
) ∈ [t∗, ∞) × Ti
and inequalities
ifi(t, 
) > 0 (12)
hold for every i ∈ {1, 2, . . . , n}, (t, 
) ∈ [t∗, ∞) × Ti, where i = −1 for i = 1, . . . , p
and i = 1 for i = p + 1, . . . , n, then there exists a positive solution y = y(t) on [ p∗, ∞)
of the system p-RFDEs (1).
Proof. We will employ Lemma 1 again. Put (t) := 0, (t) := T (k, )(t). Let us suppose
i ∈ {1, . . . , p}. It is easy to conclude that inequality (4) is equivalent to
i(t) < fi(t, 
) when 
 ∈ Ti
(13)
if the function is changed by the function 
 ∈ Ti
and inequality (5) is equivalent to
i(t) > fi(t, 
) when 
 ∈ Ti (14)
if the function is changed by the function 
 ∈ Ti. Similarly, for i ∈ {p + 1, . . . , n} we
conclude that inequality (7) is equivalent to
i(t) > fi(t, 
) when 
 ∈ Ti
(15)
if the function is changed by the function 
 ∈ Ti
and inequality (6) is equivalent to
i(t) < fi(t, 
) when 
 ∈ Ti (16)
1840 J. Diblík, Z. Svoboda / Nonlinear Analysis 64 (2006) 1831–1848
if the function is changed by the function 
 ∈ Ti. Let us verify that above inequalities
are valid. For t t∗ we get
fi(t, 
) − i(t) if i ∈ {1, . . . , p}, i = −1
i(t) − fi(t, 
) if i ∈ {p + 1, . . . , n}, i = 1
= i( i(t) − fi(t, 
))
= i(ki i(t)e
t
p∗ i(s) ds
− fi(t, 
))
= kie
t
p∗ i(s) ds
i i(t) − i
ki
e−
t
p∗ i(s) ds
fi(t, 
)
> [in view of (11)] > kie
t
p∗ i(s) ds
( i i(t) − i i(t)) = 0.
Therefore inequalities (13), (15) hold. Inequalities (14), (16) are valid, too since, due to (12)
i(t) − fi(t, 
) if i ∈ {1, . . . , p}, i = −1
fi(t, 
) − i(t) if i ∈ {p + 1, . . . , n}, i = 1
= ifi(t, 
) > 0.
All conditions of Lemma 1 are satisfied. From its conclusion we immediately get the desired
statement. Theorem 2 is proved.
Remark 1. Let us underline that if Lemma 2 or sufficiency part of Theorems 1 or 2 hold,
then indicated positive solution y = y(t) satisfies on [p∗, ∞] inequalities
0>y(t)> (t)
with corresponding given .
3.3. A nonlinear example
The following example demonstrates that results can be successfully applied to nonlinear
systems. Let us show that the system
y1(t) = −1
2 [y4
1(t1/2
) + y2
1(t) · y2(t)],
y2(t) = y2(t) − y1(t) · y2(t1/2
) · y3(t),
y3(t) = y2
1(t1/2
) · y2
3(t1/2
) (17)
has a positive solution on interval [2, ∞). Define p(t, ϑ) := t + (t −
√
t)ϑ, ϑ ∈ [−1, 0].
Then system (17) can be rewritten as
y1(t) = f1(t, yt ) := −1
2 [y4
1(p(t, −1)) + y2
1(p(t, 0)) · y2(p(t, 0))],
y2(t) = f2(t, yt ) := y2(p(t, 0)) − y1(p(t, 0)) · y2(p(t, −1)) · y3(p(t, 0)),
y3(t) = f3(t, yt ) := y2
1(p(t, −1)) · y2
3(p(t, −1)).
Let us verify that Theorem 2 can be used. For it we put: p∗ = 2 = p(t∗, −1), t∗ = 4,
k = (k1, k2, k3) = (1
4 , 1, 1
2 ), = ( 1, 2, 3) = (−1/t, 0, 1/t), 1 = 2 = −1 and 3 = 1.
J. Diblík, Z. Svoboda / Nonlinear Analysis 64 (2006) 1831–1848 1841
Then
T (k, )(t) := ke
t
2 (s) ds
= (e−
t
2 ds/s
/4, 1, e
t
2 ds/s
/2)
= (1/(2t), 1, t/4).
Let us verify inequalities (11) and (12). If i = 1 and 
 ∈ T1
then
1
k1
e−
t
p∗ 1(s) ds
· f1(t, 
) = − 2t · f1(t, 
) < t ·
1
2
√
t
4
+
1
2t
2
=
3
8t
<
1
t
= 1 1(t),
if i = 2 and 
 ∈ T2
then
2
k2
e−
t
p∗ 2(s) ds
· f2(t, 
) = −
2
t
· f2(t, 
)
=
2
t
· 1 − 
2(−1) ·
1
2t
·
t
4
<
2
t
· −1 +
1
8
= −
7
4t
< 0 = 2 2(t)
and if i = 3 and 
 ∈ T3
then
3
k3
e−
t
p∗ 3(s) ds
· f3(t, 
) =
4
t
· f3(t, 
) <
4
t
·
1
2
√
t
2
·
√
t
4
2
=
1
16t
<
1
t
= 3 3(t)
and inequalities (11) on interval [4, ∞) hold.
Inequalities (12) hold on interval [4, ∞) since if i = 1 and 
 ∈ T1 then
1
k1
· f1(t, 
) = −4f1(t, 
) = 2[
4
1(−1) + 
2
1(0) · 
2(0)] > 0,
if i = 2 and 
 ∈ T2 then
2
k2
· f2(t, 
) = − f2(t, 
) = −[
2(0) − 
1(0) · 
2(−1) · 
3(0)]
= 
1(0) · 
2(−1) · 
3(0) > 0
and if i = 3 and 
 ∈ T3 then
3
k3
· f3(t, 
) =
1
2
f3(t, 
) =
1
2
[
2
1(−1) · 
2
3(−1)] > 0.
All conditions of Theorem 2 are valid. Therefore a positive solution
y = y(t) = (y1(t), y2(t), y3(t)),
1842 J. Diblík, Z. Svoboda / Nonlinear Analysis 64 (2006) 1831–1848
of system (17) exists on [2, ∞). Taking into account Remark 1 we conclude that on the
interval considered inequalities
0 < y1(t) < 1/(2t),
0 < y2(t) < 1,
0 < y3(t) < t/4
hold.
4. Linear case
The main results can be easily reformulated for the linear case. Let us consider the system
˙y(t) = L(t, yt ), (18)
where L ∈ C( × C, Rn
) is a linear functional and yt is defined in accordance with
Definition 1. Then corresponding linear analogies to Theorems 1, 2 are given in following
two theorems.
Theorem 3. Suppose L ∈ C( × C, Rn
) and, moreover:
(i) L is i-strongly decreasing if i = 1, . . . , p and i-strongly increasing if i = p + 1, . . . , n.
(ii) Li(t, 0) 0 for i = 1, . . . , p and Li(t, 0) 0 for i = p + 1, . . . , n if (t, 0) ∈ .
Then for existence of a positive solution y = y(t) on [ p∗, ∞) of the system p-RFDEs (18)
(where p∗ = p(t∗, −1)) is necessary and sufficient existence of a positive constant vector
k and a locally integrable vector : [ p∗, ∞) → Rn
continuous on [ p∗, t∗) ∪ [t∗, ∞)
satisfying the system of integral inequalities
i i(t) i
ki
· Li(t, e−
t
p∗ i(s) ds
· T (k, )t ), i = 1, . . . , n (19)
for t t∗ with i = −1 for i = 1, . . . , p and i = 1 for i = p + 1, . . . , n.
Theorem 4. Suppose L ∈ C( , Rn
). Let a constant vector k?0 and a vector (t) defined
and locally integrable on [ p∗, ∞) are given. If, moreover, inequalities
i i(t) > i
ki
· Li(t, e−
t
p∗ i(s) ds
· 
) (20)
hold for every i ∈ {1, 2, . . . , n}, (t, 
) ∈ [t∗, ∞) × Ti
and inequalities
iLi(t, 
) > 0 (21)
hold for every i ∈ {1, 2, . . . , n}, (t, 
) ∈ [t∗, ∞) × Ti, where i = −1 for i = 1, . . . , p
and i = 1 for i = p + 1, . . . , n, then there exists a positive solution y = y(t) on [ p∗, ∞)
of the system p-RFDEs (18).
Now, let us give several linear applications.
J. Diblík, Z. Svoboda / Nonlinear Analysis 64 (2006) 1831–1848 1843
4.1. A scalar equation with discrete delays
Let us study conditions for existence of a positive solution of a scalar equation with
discrete delays
˙y(t) = −
m
q=1
cq(t)y(p(t, ϑq)) (22)
with −1 = ϑ1 < ϑ2 < · · · < ϑm = 0, continuous on [t∗, ∞) functions cq, q = 1, 2, . . . , m,
which are nonnegative if q = 1, 2, . . . , m − 1 and satisfy inequality m−1
q=1 cq(t) > 0 for
t ∈ [t∗, ∞).
Theorem 5. Forexistenceofapositivesolutiony=y(t)on[ p∗, ∞)(wherep∗=p(t∗, −1))
of Eq. (22) is necessary and sufficient existence of a locally integrable function ∗
:
[ p∗, ∞) → R continuous on [ p∗, t∗) ∪ [t∗, ∞) and satisfying the integral inequality
∗
(t)
m
q=1
cq(t)e
t
p(t,ϑq )
∗
(s) ds
(23)
for t t∗.
Proof. The proof usesTheorem 3. Let us put n=p=1 and define functional L corresponding
to the right-hand side of (23):
L(t, ) := −
m
q=1
cq(t) (ϑq),
where (t, ) ∈ ×C. Then conditions (i), (ii) of Theorem 3 are satisfied. Note that the sign
constancy of the function cm is not necessary for verifying that L is a 1-strongly decreasing
functional. Conclusion of Theorem 5 is now a consequence of scalar inequality (19) if
:= − ∗
.
Remark 2. The result mentioned in Section 1.3 can be considered as a partial case of
Theorem 5 if m = 1. Let us underline that a condition equivalent to limt→∞ (t − (t)) = ∞
is not involved in Theorem 5.
Theorem 5 can serve as a source of various sufficient conditions including well known
sufficient conditions given e.g., in [10,11]. Let us give several concrete consequences of
Theorem 5 concerning the equation
˙y(t) = −c(t)y(p(t, −1)) (24)
with a positive continuous function c. Obviously, Eq. (24) is a partial case of (22) if m = 1.
1844 J. Diblík, Z. Svoboda / Nonlinear Analysis 64 (2006) 1831–1848
Theorem 6. Let c be a positive continuous function on [ p∗, ∞) and inequality
e ·
t
p(t,−1)
c(s) ds 1 (25)
holds on [t∗, ∞) (with p∗ = p(t∗, −1)). Then (24) has a positive solution y = y(t) on
[ p∗, ∞).
Proof. We employ Theorem 5. Define ∗
(t) := c(t) · e. Then, due to (25), inequality (23)
holds on [t∗, ∞) since it turns into
e exp e ·
t
p(t,−1)
c(s) ds .
Directly from (25) follows the following corollary.
Corollary 1. Let all conditions of Theorem 6 be valid and there exists a nondecreasing
function b(t), t ∈ [p∗, ∞) such that c(t) b(t) holds on [p∗, ∞) and
b(t)
1
e · [t − p(t, −1)]
(26)
holds on [t∗, ∞). Then (24) has a positive solution y = y(t) on [ p∗, ∞).
Theorem 7. Let c be a positive continuous function on [t∗, ∞) and there exists a positive
constant K such that
c(t) Ke−K(t−p(t,−1))
(27)
on [t∗, ∞). Then (24) has a positive solution y = y(t) on[ p∗, ∞) (with p∗ = p(t∗, −1)).
Proof. We employ Theorem 5. Define ∗
(t) := K. Then, due to (27), inequality (23) holds
on [t∗, ∞) since it can be replaced by
K c(t) exp [K(t − p(t, −1))] .
Remark 3. Presented results are sharp. We can demonstrate it, e.g., on the last result. If
p(t, −1) := t − with a positive constant , c(t) ≡ c = const and if K := 1/ , then (27)
yields a classical result [11, Theorem 2.2.3] ensuring existence of a positive solution:
c e 1.
4.2. A scalar equation with distributed delay
Consider existence of a positive solution of a scalar equation having distributed delay
˙y(t) = −
ϑ∗
−1
c(t, ϑ)y(p(t, ϑ)) dϑ (28)
with ϑ∗ ∈ (−1, 0], and continuous c : [t∗, ∞) × [−1, ϑ∗] → (0, ∞).
J. Diblík, Z. Svoboda / Nonlinear Analysis 64 (2006) 1831–1848 1845
Theorem 8. Forexistenceofapositivesolutiony=y(t)on[ p∗, ∞)(wherep∗=p(t∗, −1))
of Eq. (28) is necessary and sufficient existence of a locally integrable function ∗
:
[ p∗, ∞) → R continuous on [ p∗, t∗) ∪ [t∗, ∞) and satisfying the integral inequality
∗
(t)
ϑ∗
−1
c(t, ϑ)e
t
p(t,ϑ)
∗
(q)dq
dϑ (29)
for t t∗.
Proof. The proof uses Theorem 3 again. Let us put n = p = 1 and define functional L
corresponding to the right-hand side of (28):
L(t, ) := −
ϑ∗
−1
c(t, ϑ) (ϑ) dϑ,
where (t, ) ∈ × C. Then conditions (i), (ii) of Theorem 3 are satisfied. Conclusion of
Theorem 8 is now a consequence of scalar inequality (19) if := − ∗
.
The following results are consequences of Theorem 8.
Theorem 9. Let there exists a positive constant K such that inequality
ϑ∗
−1
c(t, ϑ) dϑ Ke−K·[t−p(t,−1)]
(30)
holds on [t∗, ∞). Then Eq. (28) with a positive continuous function c on [t∗, ∞)×[−1, ϑ∗]
has a positive solution y = y(t) on [ p∗, ∞) (where p∗ = p(t∗, −1)).
Proof. Put ∗
(t) := K in Theorem 8. Then inequality (29) holds (due to (30)) on [t∗, ∞)
since its right-hand side equals
ϑ∗
−1
c(t, ϑ)e
t
p(t,ϑ)
∗
(q) dq
dϑ =
ϑ∗
−1
c(t, ϑ)e
t
p(t,ϑ) Kdq
dϑ
= eK·[t−p(t,−1)]
·
ϑ∗
−1
c(t, ϑ) dϑ.
Theorem 10. Let the difference t −p(t, −1) be a nonincreasing on [t∗, ∞) function. Then
Eq. (28) with a positive continuous function c on [t∗, ∞)×[−1, ϑ∗] has a positive solution
y = y(t) on [ p∗, ∞) (where p∗ = p(t∗, −1)) if the inequality
ϑ∗
−1
c(t, ϑ) dϑ
1
e · [t − p(t, −1)]
(31)
holds on [t∗, ∞).
Proof. We employ Theorem 8. Define ∗
as a nondecreasing function ∗
(t) := 1/(t −
p(t, −1)). Then inequality (29) holds (due to (31)) on [t∗, ∞) since its right-hand side can
1846 J. Diblík, Z. Svoboda / Nonlinear Analysis 64 (2006) 1831–1848
be estimated as
ϑ∗
−1
c(t, ϑ)e
t
p(t,ϑ)
∗
(q) dq
dϑ
ϑ∗
−1
c(t, ϑ)e
t
p(t,ϑ)
∗
(t) dq
dϑ
=
ϑ∗
−1
c(t, ϑ) exp
t − p(t, ϑ)
t − p(t, −1)
dϑ
< e
ϑ∗
−1
c(t, ϑ) dϑ.
A straightforward consequence of inequality (31) is the following corollary.
Corollary 2. Let all conditions of Theorem 10 be valid and there exists a function b :
[t∗, ∞) × [−1, ϑ∗] → R, nondecreasing in ϑ on [−1, ϑ∗] for each t ∈ [t∗, ∞), such that
c(t, ϑ) b(t, ϑ) on [t∗, ∞) × [−1, ϑ∗]. If, moreover,
b(t, ϑ∗)
1
e · [t − p(t, −1)](1 + ϑ∗)
(32)
holds on [t∗, ∞) then (28) has a positive solution y = y(t) on [ p∗, ∞).
4.3. Positive solutions of a linear system
Let us establish sufficient conditions for existence of positive solutions of the following
linear system:
y (t) = −A(t)y(p(t, −1)), (33)
where A = {aij } is n × n matrix with continuous on [t∗, ∞) entries satisfying aij (t) 0,
i, j = 1, 2, . . . , n and n
j=1 aij (t) > 0 for every i = 1, 2, . . . , n,.
Theorem 11. For existence of a positive solution y=y(t) on [ p∗, ∞) (with p∗=p(t∗, −1))
of linear system (33)) is sufficient condition the existence of a positive constant vector k
and a locally integrable function ∗
: [ p∗, ∞) → R continuous on [ p∗, t∗) ∪ [t∗, ∞) and
satisfying the integral inequality
∗
(t)e−
t
p(t,−1)
∗
(q) dq
max
i=1,2,...,n
⎧
⎨
⎩
1
ki
n
j=1
kj aij (t)
⎫
⎬
⎭
(34)
for t t∗.
Proof. Functional L ∈ C( × C, Rn
), corresponding to system (33) has the form
L(t, ) := −A(t) (−1)
and is i-strongly decreasing if i=1, 2, . . . , n, and L(t, 0)=0 if (t, 0) ∈ .Then, as it follows
from Theorem 3, for existence of a positive solution on [ p∗, ∞) is sufficient if inequalities
J. Diblík, Z. Svoboda / Nonlinear Analysis 64 (2006) 1831–1848 1847
(19) with i =−1, i=1, 2, . . . , n hold for t t∗. Let us suppose 1 ≡ 2 ≡ · · · ≡ n ≡ − ∗
.
Then inequalities (19) turn into
∗
(t)
1
ki
· e
t
p(t,−1)
∗
(q) dq
·
n
j=1
kj aij (t),
where i = 1, 2, . . . , n, and hold on [t∗, ∞) if inequality (34) is valid.
Inequality (34) gives a lot of possibilities to develop concrete sufficient conditions. We
consider two of them.
Theorem 12. Suppose that a continuous nondecreasing function ∗
: [ p∗, ∞) → R
satisfies the inequality
∗
(t)e− ∗
(t)·[t−p(t,−1)]
max
i=1,2,...,n
⎧
⎨
⎩
1
ki
n
j=1
kj aij (t)
⎫
⎬
⎭
(35)
for t t∗, where k = (k1, k2, . . . , kn) is a suitable positive constant vector. Then linear
system (33) has a positive solution y = y(t) on [p∗, ∞) (with p∗ = p(t∗, −1)).
Proof. Presented result is a straightforward consequence of Theorem 11 since obviously
∗
(t)e−
t
p(t,−1)
∗
(q) dq ∗
(t)e− ∗
(t)·[t−p(t,−1)]
.
Then inequality (34) is a consequence of inequality (35).
Theorem 13. Let matrix A be an indecomposable constant matrix. Then for existence of a
positive solution y=y(t) on [p∗, ∞) (with p∗=p(t∗, −1)) of linear system (33) is sufficient
ifalocallyintegrablefunction ∗
: [p∗, ∞) → R,continuouson [ p∗, t∗)∪[t∗, ∞),satisfies
the inequality
∗
(t)e−
t
p(t,−1)
∗
(q) dq
(A) (36)
for t t∗, where (A) is the spectral radius of the matrix A.
Proof. Let us estimate the right-hand side of inequality (34) if the matrix A is a constant
matrix. Suppose that for every i ∈ {1, 2, . . . , n}
aii +
1
ki
n
j=1, j=i
kj aij = , (37)
where k = (k1, k2, . . . , kn) is a constant positive vector. This vector and constant satisfy
the system
(A − E)kT
= 0 (38)
with n × n unit matrix E. As it follows from Frobenius theorem, the nonnegative and
indecomposable matrix A always has a positive eigenvalue, and a positive eigenvector m =
1848 J. Diblík, Z. Svoboda / Nonlinear Analysis 64 (2006) 1831–1848
(m1, m2, . . . , mn) always corresponds to its maximal positive eigenvalue max. Obviously
max = (A). Let us put = max = (A) and k = m. Then, in view of (37) and (36),
inequality (34) holds. Consequently, conclusion of Theorem 13 is now a consequence of
Theorem 11.
Remark 4. If Theorem 13 holds and p(t, −1) := t − with a positive constant , then for
the choice ∗
(t) := 1/ inequality (36) turns into
(A) e 1.
This is the result of Theorem 6 in [4], i.e., it is a partial case of Theorem 13.
Acknowledgements
This research was supported by the Grant A 1163401 of Grant Agency of the ASCR and
by the Council of Czech Government MSM 2622000 13 and MSM 0021630503.
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Author's personal copy
Computers and Mathematics with Applications 56 (2008) 556–564
www.elsevier.com/locate/camwa
Explicit criteria for the existence of positive solutions for a scalar
differential equation with variable delay in the critical case
Josef Dibl´ık∗, Zdenˇek Svoboda, Zdenˇek ˇSmarda
Brno University of Technology, Brno, Czech Republic
Received 28 February 2007; received in revised form 28 December 2007; accepted 4 January 2008
Abstract
A scalar linear differential equation with time-dependent delay ˙x(t) = −a(t)x(t − τ(t)) is considered, where t ∈ I := [t0, ∞),
t0 ∈ R, a: I → R+ := (0, ∞) is a continuous function and τ: I → R+ is a continuous function such that t − τ(t) > t0 − τ(t0) if
t > t0. The goal of our investigation is to give sufficient conditions for the existence of positive solutions as t → ∞ in the critical
case in terms of inequalities on a and τ. A generalization of one known final (in a certain sense) result is given for the case of τ
being not a constant. Analysing this generalization, we show, e.g., that it differs from the original statement with a constant delay
since it does not give the best possible result. This is demonstrated on a suitable example.
c 2008 Elsevier Ltd. All rights reserved.
Keywords: Positive solution; Delayed equation; Critical case; Infinite delay; p-function
1. Preliminaries
In this paper we consider a scalar linear differential equation with time-dependent delay
˙x(t) = −a(t)x(t − τ(t)), (1)
where t ∈ I := [t0, ∞), t0 ∈ R, a: I → R+ := (0, ∞) is a continuous function and τ: I → R+ is a continuous
function such that t − τ(t) > t0 − τ(t0) if t > t0. The goal of our investigation is to give sufficient conditions for the
existence of positive solutions of (1) as t → ∞ in terms of inequalities on a and τ. In the literature, several results
have been derived with the aid of a suitable estimation of function a. A final result (in a certain sense) in one of the
directions pursued is given in [1] for the case of a constant delay. Namely, it holds.
Theorem 1. (I) Let us assume that a(t) ≤ ak(t) with
ak(t) :=
1
eτ
+
τ
8et2
+
τ
8e(t ln t)2
+
τ
8e(t ln t ln2 t)2
+ · · · +
τ
8e(t ln t ln2 t . . . lnk t)2
(2)
∗ Corresponding address: Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology,
Technick´a 8, 616 00 Brno, Czech Republic. Tel.: + 420 541143155; fax: +420 541143392.
E-mail addresses: diblik.j@fce.vutbr.cz, diblik@feec.vutbr.cz (J. Dibl´ık), svobodaz@feec.vutbr.cz (Z. Svoboda), smarda@feec.vutbr.cz
(Z. ˇSmarda).
0898-1221/$ - see front matter c 2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.camwa.2008.01.015
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if t → ∞ and an integer k ≥ 0. Then there exists a positive solution x = x(t) of (1) with τ(t) ≡ τ = const.
Moreover,
x(t) < νk(t) := e−t/τ
t ln t ln2 t · · · lnk t
as t → ∞.
(II) Let us assume that
a(t) > ak−2(t) +
θτ
8e(t ln t ln2 t · · · lnk−1 t)2
(3)
if t → ∞, an integer k ≥ 2 and a constant θ > 1. Then all the solutions of (1) with τ(t) ≡ τ = const oscillate.
In this theorem, lnk t := ln(lnk−1 t), k ≥ 1, ln0 t := t and it is assumed that t > expk−2 1 where expk t :=
exp(expk−1 t), k ≥ 1, exp0 t := t, and exp−1 t := 0.
Theorem 1 can be applied to what is called the critical case since inequalities (2) and (3) are almost opposite. With
respect to the critical case, we refer (in addition to the paper, mentioned above) to the papers [2–5] and the book [6].
We give a generalization of the first part of Theorem 1 for the case of τ being not a constant. As a tool for this
generalization, we use the results on the existence of positive solutions for retarded functional differential equations
with unbounded delay and finite memory. The necessary relevant information is given in Section 2. The generalization
of Theorem 1 is given in Section 3. Analysing this generalization, we conclude that it differs from the original
statement with a constant delay since it does not give the best possible result. To show this, in Section 3 we formulate
another sufficient condition of positivity and in Section 4 we show that, for a class of delays, it yields a better result.
Finally, in Section 5 we explain why a generalization of Theorem 1 (i.e., generalization in both its parts) for the case of
τ being not a constant is not possible. Other results concerning the existence of positive solutions, may, for example,
be found in [7–20].
2. Positive solutions of equations with p-functions
A continuous function p: R × [−1, 0] → R is called a p-function if it has the following properties [21, p. 8]:
p(t, 0) = t, p(t, −1) is a nondecreasing function of t, and there exists a σ ≥ −∞ such that p(t, ϑ) is an increasing
function for ϑ for each t ∈ (σ, ∞). Throughout the following text, we assume σ = t0. We define p0 := p(t0, −1).
We consider a differential equation with p -functions
˙x(t) = −
m
q=1
cq(t)x(p(t, ϑq)), (4)
where ϑq = const, q = 1, . . . , m, −1 = ϑ1 < ϑ2 < · · · < ϑm = 0, functions cq: [t0, ∞) → R+ := [0, ∞) are
continuous and m−1
q=1 cq(t) > 0 for t ∈ [t0, ∞). We will use one result derived in [14] concerning necessary and
sufficient conditions of the existence of positive solutions for equations with p -functions:
Theorem 2. A positive solution x = x(t) on [ p0, ∞) of (4) exists if and only if a locally integrable function
λ: [ p0, ∞) → R exists continuous on [ p0, t0) ∪ [t0, ∞) and satisfying the integral inequality
λ(t) ≥
m
q=1
cq(t)e
t
p(t,ϑq ) λ(s)ds
for t ≥ t0. Moreover, x(t) < exp −
t
p0
λ(s)ds .
Eq. (1) is a particular case of (4). This becomes clear if we define
p(t, ϑ) :=



t + 2ϑτ(t) if − 1/2 ≤ ϑ ≤ 0,
t0 − τ(t0) + 2(1 + ϑ) (t − τ(t) − (t0 − τ(t0)))
if − 1 ≤ ϑ ≤ −1/2,
m = 3, ϑ1 = −1, ϑ2 = −1/2, ϑ3 = 0, c1(t) = 0, c2(t) = a(t) and c3(t) = 0. Then p0 = t0 − τ(t0) and Theorem 2
reduces to:
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Theorem 3. A positive solution x = x(t) on [t0 − τ(t0), ∞) of the Eq. (1) exists if and only if a locally integrable
function λ: [t0 − τ(t0), ∞) → R exists continuous on [t0 − τ(t0), t0) ∪ [t0, ∞) and satisfying the integral inequality
λ(t) ≥ a(t)e
t
t−τ(t) λ(s)ds
(5)
for t ≥ t0. Moreover, x(t) < exp −
t
t0−τ(t0) λ(s)ds .
This theorem will be used for finding two explicit criteria for the existence of positive solutions of (1).
Remark 1. The above specification of p-function is obviously not unique. One can put e.g. p(t, ϑ) := t + ϑτ(t),
m = 2, ϑ1 = −1, ϑ2 = 0, c1(t) = a(t) and c2(t) = 0 and assume that t − τ(t) is a nondecreasing function of t on
I rather than assuming t − τ(t) > t0 − τ(t0) if t > t0 as above. Then, p0 = t0 − τ(t0) and Theorem 2 reduces to
Theorem 3 again.
3. Criteria of existence of positive solutions
3.1. First criterion — a generalization of Theorem 1, part I
Now we give a generalization of Theorem 1 with the aid of a suitable auxiliary function more general than the
function ak(t) given by (2). The form of this new function formally copies the old one, but now delay τ will be a
function. The proof needs some auxiliary results. Below, symbols O and o mean the Landau order symbols. If real
functions f1, f2, f3 are defined as t → ∞, then the relation f1(t) = f2(t) + O( f3(t)) means that there exists a
positive constant M such that
| f1(t) − f2(t)| ≤ M| f3(t)|
as t → ∞, and the relation f1(t) = f2(t) + o( f3(t)) is equivalent with
lim
t→∞
f1(t) − f2(t)
f3(t)
= 0
if f3(t) = 0.
Lemma 1. Let τ(t) = o(t) as t → ∞. Then
(t − τ(t))σ
= tσ
1 −
στ(t)
t
+
σ(σ − 1)τ2(t)
2t2
−
σ(σ − 1)(σ − 2)τ3(t)
6t3
+ O
τ4(t)
t4
(6)
for t → ∞ and any fixed σ ∈ R.
Proof. This can be verified easily using the binomial formula.
Lemma 2. Let τ(t) ln t = o(t) as t → ∞. Then
[ln(t − τ(t))]
1
2 = (ln t)
1
2 1 −
τ(t)
2t ln t
−
τ2(t)
4t2 ln t
1 +
1
2 ln t
+ O
τ3(t)
t3 ln t
as t → ∞.
Proof. For t → ∞ we have
[ln(t − τ(t))]
1
2 = (ln t)
1
2 1 +
1
ln t
ln 1 −
τ(t)
t
1
2
= (ln t)
1
2 1 −
1
ln t
τ(t)
t
+
τ2(t)
2t2
+ O
τ3(t)
t3
1
2
.
The proof can be finished by expanding the expression in square brackets using the binomial formula.
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Lemma 3. Let τ(t) ln t ln2 t . . . lnk t = o(t) as t → ∞. Then
[lnk(t − τ(t))]
1
2 = (lnk t)
1
2 1 −
τ(t)
2t ln t ln2 t · · · lnk−1 t lnk t
−
τ2(t)
4t2 ln t ln2 t · · · lnk−1 t lnk t
1 +
1
ln t
+ · · · +
1
ln t ln2 t · · · lnk−1 t
+
1
2 ln t ln2 t · · · lnk−1 t lnk t
+ O
τ3(t)
t3 ln t ln2 t · · · lnk t
(7)
for t → ∞ and any fixed k ≥ 1.
Proof. For k = 1, the proof follows from Lemma 2. Suppose that (7) holds with for k−1 (instead of k) and (k−1) ≥ 1.
We use it for the representation of lnk−1(t − τ(t)) in the relation
[lnk(t − τ(t))]
1
2 = (lnk t)
1
2 1 +
1
lnk t
ln
lnk−1(t − τ(t))
lnk−1 t
1
2
.
We get
[lnk(t − τ(t))]
1
2 = (lnk t)
1
2 1 +
2
lnk t
ln 1 −
τ(t)
2t ln t ln2 t · · · lnk−2 t lnk−1 t
−
τ2(t)
4t2 ln t ln2 t · · · lnk−2 t lnk−1 t
1 +
1
ln t
+ · · · +
1
ln t ln2 t · · · lnk−2 t
+
1
2 ln t ln2 t · · · lnk−2 t lnk−1 t
+ O
τ3(t)
t3 ln t ln2 t . . . lnk−1 t
1
2
.
After decomposing logarithm ln( 1 − · · ·) into its Taylor’s polynomial, we expand the expression in square brackets
by the binomial formula. Then, using only the necessary terms, we get the representation (7).
Let us consider now a linear equation
˙x(t) = −A(t)x(t − τ(t)) (8)
with A: I → R.
Lemma 4 ([1]). Let a(t) ≤ A(t) on I and (8) have a positive solution x = µ(t) on [t0 − τ(t0), ∞). Then (1) has
a positive solution x = x(t) on [t0 − τ(t0), ∞) and, moreover, x(t) < µ(t) holds.
Now we define a new auxiliary function
akτ (t) :=
1
eτ(t)
+
τ(t)
8et2
+
τ(t)
8e(t ln t)2
+
τ(t)
8e(t ln t ln2 t)2
+ · · · +
τ(t)
8e(t ln t ln2 t . . . lnk t)2
(9)
for t → ∞ and an integer k ≥ 0.
Theorem 4. Let us assume that a(t) ≤ akτ (t) and
t
t−τ(t) ds/τ(s) ≤ 1 if t → ∞ and an integer k ≥ 0. Let moreover
τ(t) ln t ln2 t . . . lnk t = o(t) as t → ∞. Then there exists a positive solution x = x(t) of (1) satisfying
x(t) < t ln t ln2 t . . . lnk t · exp
t
t0−τ(t0)
−1
τ(s)
ds (10)
as t → ∞.
Proof. Let us consider an auxiliary equation
˙x(t) = −akτ (t)x(t − τ(t)) (11)
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and let us prove the existence of a positive solution. We verify (5) for a(t) := akτ (t) and
λ(t) := λk(t) =
1
τ(t)
−
1
2t
−
1
2t ln t
−
1
2t ln t ln2 t
− · · · −
1
2t ln t ln2 t . . . lnk t
.
With the aid of Lemma 1 (substituting σ = 1/2 in (6)), Lemmas 2 and 3 we estimate the exponential term on the
right-hand side of (5). We obtain
exp
t
t−τ(t)
λk(s)ds = exp
t
t−τ(t)
ds
τ(s)
·
t − τ(t)
t
·
ln(t − τ(t))
ln t
· · ·
lnk(t − τ(t))
lnk t
≤ E := e 1 −
τ(t)
2t
−
τ2(t)
8t2
−
τ3(t)
16t3
+ O
τ4(t)
t4
× 1 −
τ(t)
2t ln t
−
τ2(t)
4t2 ln t
1 +
1
2 ln t
+ O
τ3(t)
t3 ln t
· · ·
× 1 −
τ(t)
2t ln t ln2 t . . . lnk−1 t lnk t
−
τ2(t)
4t2 ln t ln2 t . . . lnk−1 t lnk t
× 1 +
1
ln t
+ · · · +
1
ln t ln2 t . . . lnk−1 t
+
1
2 ln t ln2 t . . . lnk−1 t lnk t
+ O
τ3(t)
t3 ln t ln2 t . . . lnk−1 t lnk t
.
After some simplification, we get
exp
t
t−τ(t)
λk(s)ds ≤ E = e 1 −
τ(t)
2t
1 +
1
ln t
+ · · · +
1
ln t . . . lnk t
−
τ2(t)
8t2
1 +
1
(ln t)2
+ · · · +
1
(ln t . . . lnk t)2
−
τ3(t)
16t3
+ O
τ3(t)
t3 ln t
.
Now we have, for the right-hand side R of (5),
R ≤
1
τ(t)
+
τ(t)
8t2
+
τ(t)
8(t ln t)2
+
τ(t)
8(t ln t ln2 t)2
+ · · · +
τ(t)
8(t ln t ln2 t . . . lnk t)2
e−1
· exp
t
t−τ(t)
λk(s)ds
≤
1
τ(t)
−
1
2t
−
1
2t ln t
−
1
2t ln t ln2 t
− · · · −
1
2t ln t ln2 t . . . lnk t
−
τ2(t)
8t3
+ O
τ2(t)
t3 ln t
.
Comparing the left-hand side L of (5) and the right-hand side R of (5) we conclude that for L ≥ R
0 ≥ −
τ2(t)
8t3
+ O
τ2(t)
t3 ln t
is sufficient. This inequality obviously holds as t → ∞. Therefore, (5) is valid and (11) has a positive solution
x = µk(t). Now it remains to apply Lemma 4 with A(t) := akτ (t). Consequently, (1) has a positive solution x = x(t)
that satisfies the inequality x(t) < µk(t) as t → ∞. For µk(t), we have an estimate
µk(t) < exp −
t
t0−τ(t0)
λk(s)ds
=
t ln t . . . lnk t
(t0 − τ(t0)) ln(t0 − τ(t0)) . . . lnk(t0 − τ(t0))
1
2
exp −
t
t0−τ(t0)
1
τ(s)
ds .
From the linearity of (1), it follows that there exists a positive solution satisfying (10).
3.2. Second criterion
The second sufficient condition for the existence of a positive solution can be derived from inequality (5).
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Theorem 5. Let us assume that
a(t) ≤
1
τ(t)
· exp −
t
t−τ(t)
ds
τ(s)
(12)
as t → ∞. Then there exists a positive solution x = x(t) of (1). Moreover,
x(t) < exp −
t
t0−τ(t0)
ds
τ(s)
.
Since the statement of Theorem 5 is a straightforward consequence of (5) with λ(t) := 1/τ(t), no proof is necessary.
We remark only that, for τ(t) = τ = const, inequality (12) gives a classical sufficient condition for the existence of
positive solutions, namely, the condition a(t) ≤ 1/(τe).
4. Analysis of both criteria
To compare Theorem 4 with Theorem 5, we will investigate equation (1), where
τ(t) := c + d/t (13)
and c, d are positive constants, i.e., we consider an equation
˙x(t) = −a(t)x(t − c − d/t). (14)
4.1. Application of the first criterion
The delay (13) is decreasing, tends to c as t → ∞ and satisfies the inequality
t
t−τ(t)
ds
τ(s)
< 1.
If
a(t) ≤ akτ (t) (15)
for an integer k ≥ 0 as t → ∞ then, by Theorem 4, Eq. (14) has a positive solution. We will first develop several
terms of the asymptotic decomposition of akτ (t) with τ(t) given by (13) if t → ∞ and rewrite condition (15). We get
sufficient condition for the existence of a positive solution of (14) in the form
a(t) ≤ akτ (t) =
1
ec
−
d
ec2
·
1
t
+
1
e
·
d2
c3
+
c
8
·
1
t2
+ o
1
t2
. (16)
Remark 2. The right-hand side of (16) was obtained only with the aid of two terms of expression (9) and does
not explicitly contain index k. In other words, we used only the necessary (for our following analysis) part of the
expression (9). Therefore, our decomposition and, consequently, inequality (16) holds for every k ≥ 0.
4.2. Application of the second criterion
We compute
t
t−τ(t)
ds
τ(s)
=
t
t−c−d/t
ds
c + d/s
=
s
c
−
d
c2
ln(cs + d)
t
t−c−d/t
= 1 +
d
ct
−
d
c2
ln
t
t − c
.
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562 J. Dibl´ık et al. / Computers and Mathematics with Applications 56 (2008) 556–564
Now we are able to asymptotically decompose the right-hand side of inequality (12) as t → ∞. We get
1
τ(t)
· exp −
t
t−τ(t)
ds
τ(s)
=
1
c + d/t
· exp −1 −
d
ct
+
d
c2
ln
t
t − c
=
1
ec
·
1
1 + d/(ct)
·
t − c
t
−d/c2
· e−d/(ct)
= [to decompose the third term, we use Lemma 1 withσ = −d/c2
and τ(t) ≡ c in (6)]
=
1
ec
· 1 −
d
ct
+
d2
c2t2
+ o
1
t2
· 1 +
d
ct
+
d(d + c2)
2c2t2
+ o
1
t2
× 1 −
d
ct
+
d2
2c2t2
+ o
1
t2
=
1
ec
−
d
ec2
·
1
t
+
1
e
·
d2
c3
+
d
2c
·
1
t2
+ o
1
t2
.
Finally, by the second criterion, the sufficient condition for the existence of a positive solution of (14) is
a(t) ≤
1
ec
−
d
ec2
·
1
t
+
1
e
·
d2
c3
+
d
2c
·
1
t2
+ o
1
t2
. (17)
4.3. Final comparison
Comparing the right-hand sides of expressions (16) and (17), we see that the first two terms of both decompositions
coincide. The quality of every criterion is expressed by the coefficients of the term 1/t2, i.e., by the coefficient
CI
2 =
1
e
·
d2
c3
+
c
8
in the case of expression (16) and by the coefficient
CII
2 =
1
e
·
d2
c3
+
d
2c
in the case of expression (17). We conclude CI
2 < CII
2 if c2 < 4d and CI
2 > CII
2 if c2 > 4d. Thus, we have
Theorem 6. The first criterion is more general in the case of c2 > 4d; the second criterion is more general if
c2 < 4d.
5. Theorem 1 cannot be generalized for variable delay
Let us formulate the following natural conjecture which is a generalization of Theorem 1 for variable delay (we
omit the inequality for a positive solution):
Conjecture 1. Let us assume
t
t−τ(t) ds/τ(s) ≤ 1 as t → ∞.
(a) If
a(t) ≤ akτ (t)
with akτ (t) defined by formula (9) for t → ∞ and an integer k ≥ 0, then there exists a positive solution x = x(t)
of (1).
(b) If
a(t) > ak−2,τ (t) +
θτ(t)
8e(t ln t ln2 t . . . lnk−1 t)2
(18)
for t → ∞, an integer k ≥ 2 and a constant θ > 1, then all the solutions of (1) oscillate.
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Comparing the results in Section 4, we can conclude that the Conjecture 1 does not hold. This can be proved by
showing that Conjecture 1 is false for at least one variable delay. We will show that it does not hold for an equation of
the type (14) with variable delay (13). We set c = d = 1, τ(t) = 1 + 1/t, k = 2,
a(t) :=
1
e
1 −
1
t
+
4
3
·
1
t2
and consider an equation of the type (14), i.e.,
˙x(t) = −
1
e
1 −
1
t
+
4
3
·
1
t2
x t − 1 −
1
t
. (19)
We will verify inequality (18). Due to Remark 2 and the decomposition (16), we have
a0τ (t) +
θτ(t)
8e(t ln t)2
=
1
e
1 −
1
t
+
9
8
·
1
t2
+ o
1
t2
as t → ∞. Inequality (18) holds since
a(t) =
1
e
1 −
1
t
+
4
3
·
1
t2
> a0τ (t) +
θτ(t)
8e(t ln t)2
=
1
e
1 −
1
t
+
9
8
·
1
t2
+ o
1
t2
as t → ∞. Then all the solutions of (19) should oscillate by Conjecture 1, part (b). In our case, however,
a(t) =
1
e
1 −
1
t
+
4
3
·
1
t2
<
1
e
1 −
1
t
+
3
2
·
1
t2
+ o
1
t2
as t → ∞ and inequality (17) holds. Then, by Theorem 5, Eq. (19) has a positive solution as t → ∞. This is a
contradiction with Conjecture 1.
Acknowledgment
This research was supported by the Council of Czech Government MSM 0021630503, MSM 0021630519 and
MSM 0021630529, and by the Grant 201/08/0469 of Czech Grant Agency.
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Nonlinear Analysis 71 (2009) e1393–e1400
Contents lists available at ScienceDirect
Nonlinear Analysis
journal homepage: www.elsevier.com/locate/na
Retract principle for neutral functional differential equations
Josef Diblík∗
, Zdeněk Svoboda, Zdeněk Šmarda
Brno University of Technology, Brno, Czech Republic
a r t i c l e i n f o
MSC:
34K15
34K25
Keywords:
Retract principle
Neutral functional differential equations
Positive solution
a b s t r a c t
The investigation of asymptotic behaviour of solutions of ordinary differential equations
is often based on the application of the retract principle. Initially developed for ordinary
differential equations, this technique was extended to other classes of equations. Not
answered remains a problem concerning the possibility of extending this principle to
neutral differential equations. The goal of the present paper is to partially fill this gap and
develop a corresponding technique for the application of this principle. The applicability of
the main result is illustrated on a nonlinear equation and sufficient conditions for existence
of a positive solution are derived.
© 2009 Elsevier Ltd. All rights reserved.
1. Introduction and preliminaries
The investigation of the asymptotic behavior of solutions of ordinary differential equations is often based on the
application of a retract (or Ważewski’s) principle. This is a method of proving the existence of solutions which remain in
a given set. For sources we refer, e.g., to [1] — one of Ważewski’s original papers or to comprehensive explanations [2,3].
Initially developed for ordinary differential equations, this technique was extended to other classes of equations, e.g., to
partial differential equations in [4,5], discrete equations in [6], to the investigation of retarded functional differential
equations with bounded retardation in [7,8] or to retarded functional differential equations with unbounded delay but with
finite memory developed in [9–11].
Not answered remains a problem concerning the possibility of extending this principle to neutral differential equations.
The goal of the present paper is to partially fill this gap and develop a corresponding technique for the application of this
principle.
The paper is structured as follows: Section 1.1 is devoted to basic theoretical results concerning the existence of a solution
of the initial problem, its uniqueness, continuability and continuous dependence on the initial input values. A formulation
of the retract principle for a system of curves is given in Section 1.2. As a tool for applications of this principle we build a
notion of a regular polyfacial set in Section 2 and the main result is formulated in Section 3. Section 4 shows the applicability
of the main result using an illustrative nonlinear example where sufficient conditions for the existence of a positive solution
are derived. In the last Section 5 we give recommendations regarding further investigation.
1.1. Neutral functional differential equations
We consider a neutral functional differential system of the form
˙y(t) = f (t, yt , ˙yt ) (1)
∗ Corresponding address: Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická
8, 616 00 Brno, Czech Republic. Tel.: +420 541143155; fax: +420 541143392.
E-mail addresses: diblik@feec.vutbr.cz, diblik.j@fce.vutbr.cz (J. Diblík), svobodaz@feec.vutbr.cz (Z. Svoboda), smarda@feec.vutbr.cz (Z. Šmarda).
0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.na.2009.01.164
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e1394 J. Diblík et al. / Nonlinear Analysis 71 (2009) e1393–e1400
where the symbol ˙y (sometimes we use y ) stands for the derivative (considered, if necessary, as one-sided). First we give
the necessary auxiliary background regarding this equation.
Let C be the set of all continuous functions ϕ: [−h, 0] → Rn
and C1
be the set of all continuously differentiable functions
ϕ: [−h, 0] → Rn
.
We assume t ≥ t0, yt (θ) = y(t + θ), θ ∈ [−h, 0] where h > 0 is a constant and f : Eh → Rn
with Eh := [t0, ∞) × C × C.
We pose an initial problem for (1):
yt0
= ϕ, ˙yt0
= ˙ϕ (2)
where ϕ ∈ C1
. The norm of ϕ ∈ C is defined as ϕ h := maxθ∈[−h,0] ϕ(θ) and, if ϕ ∈ C1
, then
ϕ h := max
θ∈[−h,0]
ϕ(θ) + max
θ∈[−h,0]
ϕ (θ) .
A function y: [t0 −h, tϕ) → Rn
, tϕ ∈ (t0, ∞], is a solution of (1), (2) if yt0
= ϕ, ˙yt0
= ˙ϕ and (1) is satisfied for any t ∈ [t0, tϕ).
The following result is taken from the known book by Kolmanovskii and Myshkis [12, p. 107].
Theorem 1. Let f : Eh → Rn
be a continuous functional satisfying, in some neighborhood of any point of Eh, the condition
f (t, ψ1, χ1) − f (t, ψ2, χ2) ≤ L ψ1 − ψ2 h + χ1 − χ2 h (3)
with constants L ∈ [0, ∞), ∈ [0, 1). Assume also ϕ ∈ C1
and the sewing condition
˙ϕ(0) = f (t0, ϕ, ˙ϕ) (4)
being fulfilled. Then there exists a tϕ ∈ (t0, ∞] such that:
(a) There exists a solution y of (1), (2) on [t0 − h, tϕ).
(b) On any interval [t0 − h, t1] ⊂ [t0 − h, tϕ), t1 > t0 this solution is unique.
(c) If tϕ < ∞, then ˙x(t) has not a finite limit as t → t−
ϕ .
(d) The solution y and ˙y depend continuously on f , ϕ.
For a particular case of system (1) given by
f (t, yt , ˙yt ) := f (t, y(t − h1(t)), . . . , y(t − ho(t)), ˙y(t − g1(t)), . . . , ˙y(t − g (t)))
where indices o and are non-negative, i.e.,
˙y(t) = f (t, y(t − h1(t)), . . . , y(t − ho(t)), ˙y(t − g1(t)), . . . , ˙y(t − g (t))), (5)
a more general result can be proved easily by the method of steps (compare [12, pp. 111, 96 and 15]).
Theorem 2. Let f : [t0, ∞) × Ro+
→ Rn
, hi: [t0, ∞) → (0, h], i = 1, . . . , o and gj: [t0, ∞) → (0, h], j = 1, . . . , be
continuous functions. Assume also ϕ ∈ C1
and the sewing condition (4), in the case considered having the form
˙ϕ(0) = f (t0, ϕ(−h1(t0)), . . . , ϕ(−ho(t0)), ˙ϕ(−g1(t0)), . . . , ˙ϕ(−g (t0))) (6)
being fulfilled. Then:
(a) There exists a solution y of (1), (2) on [t0 − h, ∞).
(b) On any interval [t0 − h, t1] ⊂ [t0 − h, ∞), t1 > t0 this solution is unique.
(c) The solution y and ˙y depend continuously on f , ϕ.
1.2. System of curves and retract principle
In this part, we formulate the retract principle for a system of curves. This principle gives (roughly speaking) the necessary
conditions for the existence of at least one curve (within a given family of curves), with its graph lying in a prescribed set.
Definition 1 and Lemma 1 below are modifications of the corresponding Definition 2.2 and Theorem 2.1 by Rybakowski [8]
(see also [9]). Therefore, we omit the proof.
If a set A ⊂ R × Rn
is given, then int A, A and ∂A denote, as usual, the interior, the closure, and the boundary of A,
respectively.
Definition 1 (System of Curves). Let Λ be a topological space, let a subset ˜Ω ⊂ R × Λ be open in R × Λ, and let x be
a mapping, associating with every (δ, λ) ∈ ˜Ω a function x(δ, λ) : Dδ,λ → Rn
where Dδ,λ is an interval in R. Assume (1)–(3):
(1) δ ∈ Dδ,λ.
(2) If t ∈ int Dδ,λ, then there is an open neighbourhood O(δ, λ) of (δ, λ) in ˜Ω such that t ∈ Dδ ,λ holds for all (δ , λ ) ∈
O(δ, λ).
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(3) If (δ , λ ), (δ, λ) ∈ ˜Ω, and t ∈ Dδ ,λ , t ∈ Dδ,λ, then
lim
(δ ,λ ,t )→(δ,λ,t)
x(δ , λ )(t ) = x(δ, λ)(t).
If all these conditions are satisfied, then (Λ, ˜Ω, x) is called a system of curves in Rn
.
Definition 2 (Retract and Retraction). If A ⊂ A∗
are any two sets of a topological space and π: A∗
→ A is a continuous
mapping from A∗
onto A such that π(p) = p for every p ∈ A, then π is said to be a retraction of A∗
onto A. If there exists
a retraction of A∗
onto A, A is called a retract of A∗
.
Lemma 1 (Retract Principle). Let (Λ, ˜Ω, x) be a system of curves in Rn
. Let ˜ω, W, Z be sets. Assume that conditions (1)–(4) below
hold:
(1) (a) ˜ω ⊂ [t0 −h, t∗)×Rn
, t∗ > t0 and the cross-section {(˜t, y) ∈ ˜ω} is an open simply connected set for every ˜t ∈ [t0 −h, t∗),
W ⊂ ∂ ˜ω,
(b) Z ⊂ ˜ω ∪ W, Z ∩ W is a retract of W, but not a retract of Z.
(2) There is a continuous map q: B → Λ where B = Z ∩ (Z ∪ W) such that, for any z = (δ, y) ∈ B, (δ, q(z)) ∈ ˜Ω, and, if also
z ∈ W, then x(δ, q(z))(δ) = y.
(3) Let A be the set of all z = (δ, y) ∈ Z ∩ ˜ω such that, for fixed (δ, y) ∈ A, there is a t > δ, t ∈ Dδ,q(z) and (t, x(δ, q(z))(t)) ∈ ˜ω.
Assume that, for every z = (δ, y) ∈ A, there is a t(z), t(z) > δ, such that:
(a) t(z) ∈ Dδ,q(z) and, for all t, δ ≤ t < t(z), (t, x(δ, q(z))(t)) ∈ ˜ω,
(b) (t(z), x(δ, q(z))(t(z))) ∈ W,
(c) For any σ > 0, there is a t = t(σ, z), t(z) < t ≤ t(z) + σ such that t ∈ Dδ,q(z) and (t, x(δ, q(z))(t)) ∈ ˜ω.
(4) For any z = (δ, y) ∈ W ∩ B and all σ > 0, there is a t = t(σ, z), δ < t ≤ δ + σ such that t ∈ Dδ,q(z) and
(t, x(δ, q(z))(t)) ∈ ˜ω.
Then there is a z0 = (δ0, y0) ∈ Z ∩ ˜ω such that, for every t ∈ Dδ0,q(z0),
(t, x(δ0, q(z0))(t)) ∈ ˜ω.
Remark 1. Let Λ = C1
, ˜Ω ⊂ {(t, λ) ∈ [t0, ∞)×C1
such that ˙λ(0) = f (t0, λ, ˙λ)}and function f satisfies all the assumptions
of Theorem 1. In this case, through each (t0, λ) ∈ ˜Ω, there exists a unique solution y(t0, λ) of (1) defined on the maximal
interval [t0 − h, aλ). Let Dt0,λ = [t0 − h, aλ) where aλ > t0. Then (Λ, ˜Ω, y) is a system of curves in Rn
in the sense of
Definition 1. A similar remark holds when all the assumptions of Theorem 2 are satisfied.
2. Polyfacial set and regular polyfacial set
To extend the retract principle for systems of curves generated by neutral differential equations, a suitable tool is
necessary. Very often constructions are used based on applying polyfacial sets and regular polyfacial sets (e.g., by Hartman [2]
in the theory of ordinary differential equations or by Rybakowski [7,8] for retarded differential equations). We shall give
the definition of a polyfacial set and a modification of the notion of a regular polyfacial set suitable for neutral differential
equations.
Definition 3 (Polyfacial Set). Let li, mj, i = 1, . . . , p, j = 1, . . . , s, p +s > 0 be real-valued C1
-functions defined on R ×Rn
and t∗ > t0. The set
ω = {(t, y) ∈ [t0 − h, t∗) × Rn
, li(t, y) < 0, mj(t, y) < 0, for all i, j}
will be called a polyfacial set provided that, for every fixed t∗
∈ [t0 − h, t∗), the cross-section ω ∩ {(t, y): t = t∗
, y ∈ Rn
} is
an open and simply connected set.
When we investigate solutions of ordinary differential equations with graphs remaining in a polyfacial set, we often
compute the full derivative of a Liapunov–type function on trajectories of a given system at boundary points of this set and
investigate the sign of this derivative. In the case of retarded functional differential equations, it is sufficient, in a similar
computation, in accordance with an ingenious idea of Razumikhin’s (see, e.g., [13]) applied in the stability theory, to take
into account only the corresponding ‘‘time-history’’ of the solution (which usually coincides with the length of delay). In
our case, this means that it is enough in estimating the derivatives of Liapunov-type functions to use only those solutions
with the ‘‘time-history’’ satisfying prescribed conditions. An additional complication, when considering neutral differential
equations, arises due to the derivatives depending also on the derivatives of the ‘‘time-history’’ of solutions. This is a problem
of estimating the derivatives of the Liapunov-type functions containing retarded derivatives of solutions. In some cases, it
is possible to estimate them using the properties of the polyfacial set and the prescribed properties of the sets of initial
functions used. Below, such properties are expressed in the form of subsidiary inequalities. This is a novelty in our approach.
Extending Razumikhin’s idea to the derivatives of solutions, taking into account the relevant ‘‘time-history’’ of the derivatives
of solutions together with the ‘‘time-history’’ of the solutions themselves, we are able to overcome the difficulties described
above.
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Definition 4 (Regular Polyfacial Set). Let ck: [t0 − h, tϕ) × Rn
× Rn
→ R, k = 1, . . . , q, be continuous functions. A polyfacial
set ω is called regular with respect to Eq. (1) and subsidiary inequalities
ck(t, y, x) ≤ 0, k = 1, . . . , q (7)
if (α)–(δ) below hold:
(α) If (t, ϕ) ∈ R × C1
and (t + θ, ϕ(θ)) ∈ ω for θ ∈ [−h, 0), then (t, ϕ) ∈ ˜Ω.
(β) If (t, ϕ) ∈ R × C1
, (t + θ, ϕ(θ)) ∈ ω for θ ∈ [−h, 0) and, moreover,
ck(t + θ, ϕ(θ), ˙ϕ(θ)) ≤ 0, θ ∈ [−h, 0), k = 1, . . . , q (8)
then
ck(t, ϕ(θ), f (t, ϕ, ˙ϕ)) ≤ 0, k = 1, . . . , q. (9)
(γ ) For all i = 1, . . . , p, all (t, y) ∈ ∂ω for which li(t, y) = 0 and for all ϕ ∈ C1
for which ϕ(0) = y,
(t + θ, ϕ(θ)) ∈ ω, θ ∈ [−h, 0)
and
ck(t + θ, ϕ(θ), ˙ϕ(θ)) ≤ 0, θ ∈ [−h, 0), k = 1, . . . , q (10)
it follows that:
Dli(t, y) ≡
∂li
∂t
(t, y) +
n
r=1
∂li
∂yr
(t, y) · fr (t, ϕ, ˙ϕ) > 0. (11)
(δ) For all j = 1, . . . , s, all (t, y) ∈ ∂ω for which mj(t, y) = 0 and for all ϕ ∈ C1
for which ϕ(0) = y,
(t + θ, ϕ(θ)) ∈ ω, θ ∈ [−h, 0)
and
ck(t + θ, ϕ(θ), ˙ϕ(θ)) ≤ 0, θ ∈ [−h, 0), k = 1, . . . , q (12)
for all θ ∈ [−1, 0), it follows that:
Dmj(t, y) ≡
∂mj
∂t
(t, y) +
n
r=1
∂mj
∂yr
(t, y) · fr (t, ϕ, ˙ϕ) < 0.
Note that functions ck, k = 1, . . . , q can be undefined and, in such a case, subsidiary inequalities (7) are not prescribed. Then
the regular polyfacial set with respect to the neutral system (1) turns (after omitting (7), assumption (β), (10) and (12))
into a usual regular polyfacial set for delayed functional differential equations (compare, e.g., [8]). In addition, we focus our
attention to the fact that computations in (β)–(δ) assume that t ≥ t0. This is a consequence of the inclusion (t+θ, ϕ(θ)) ∈ ω
for θ ∈ [−h, 0) or for θ ∈ [−h, 0].
3. Main result
In Section 1.1, two theorems (Theorems 1 and 2) were formulated in order to define a set of the assumptions for the
existence of a solution of problem (1), (2) satisfying the properties indicated. In the formulation of the main result (Theorem 3
below), we will assume that the solution of problem (1), (2) exists and satisfies the required properties irrespective of the
assumptions of these theorems. We collect the necessary requirements as
Hypothesis A. Assume ϕ ∈ C1
and the sewing condition (4) being fulfilled. Let a tϕ ∈ (t0, ∞] exists such that:
(a) There exists a solution y of (1), (2) on [t0 − h, tϕ).
(b) On any interval [t0 − h, t1] ⊂ [t0 − h, tϕ), t1 > t0 this solution is unique.
(c) If tϕ < ∞, then ˙x(t) has not a finite limit as t → t−
ϕ .
(d) The solution y and ˙y depend continuously on f , ϕ.
Now, according with the main goal of the retract principle, we are going to prove the existence of a solution y = y(t)
of (1) defined by the initial data (2) such that its graph lies in a given set. We assume that such a set can be expressed as a
polyfacial set ω, i.e., we prove that, under certain assumptions, (t, y(t)) ∈ ω, t ∈ [t0 − h, min{tϕ, t∗}).
Let ω be a polyfacial set. Define
W := {(t, y) ∈ ∂ω : mj(t, y) < 0, j = 1, . . . , s}.
Let Z be a subset of ω∪W and let the mapping q: B → C1
, B := Z ∩(Z ∪W) be continuous. We assume that, if z = (δ, y) ∈ B,
then (δ, q(z)) ∈ ˜Ω, and:
(1) If z ∈ Z ∩ ω, then (δ + θ, q(z)(θ)) ∈ ω for θ ∈ [−h, 0].
(2) If z ∈ W ∩ B, then (δ, q(z)(δ)) = z and (δ + θ, q(z)(θ)) ∈ ω for θ ∈ [−h, 0).
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Theorem 3 (Main Result). Let ω be a nonempty polyfacial set, regular with respect to (1) and inequalities (7). We assume that
Hypothesis A holds and that the derivative of every solution y(δ, q(z))(t) of (1) defined by any z = (δ, y) ∈ B has a finite left
limit at every point t provided (t, y(δ, q(z))(t)) ∈ ω. Let, moreover, Z ∩ W be a retract of W, but not a retract of Z. Then there
exists at least one point z0 = (δ0, y0) ∈ Z ∩ ω such that a solution y(δ0, q(z0))(t) exists on [t0 − h, t∗) and
(t, y(δ0, q(z0))(t)) ∈ ω
holds for all t ∈ [t0 − h, t∗).
Proof. In the proof we use Lemma 1 defining ˜ω := ω, Λ := C1
, ˜Ω := [t0, t∗)×C1
. Note that such a definition of ˜Ω is correct
because all points in Definition 1 using open intervals can be restricted, without loss of generality, to half-open intervals
open at the right.
Condition (1) of Lemma 1 is satisfied evidently due to the definition of the polyfacial set ω. In much the same way,
condition (2) of Lemma 1 is satisfied. Let us verify conditions (3) and (4).
Verifying condition (3): Let z = (δ, y) ∈ A, and let t(z) be the smallest of all t ≥ δ such that t ∈ Dδ,q(z) and
(t, y(δ, q(z))(t)) ∈ ω. Since
(δ, y(δ, q(z))(δ)) = (δ, q(z)(0)) ∈ ω,
it follows that δ < t(z) < ∞. Obviously,
(t(z), y(δ, q(z))(t(z))) ∈ ∂ω
and moreover, for δ ≤ t < t(z), it holds: (t, y(δ, q(z))(t)) ∈ ω, hence (3) (a) is satisfied.
Let ϕ ≡ yt(z)(δ, q(z)). Then ϕ ∈ C1
and (t(z), ϕ) ∈ ˜Ω. Moreover,
(t(z), ϕ(0)) = (t(z), y(δ, q(z))(t(z))) ∈ ∂ω
and
(t(z) + θ, ϕ(θ)) ∈ ω, for θ ∈ [−h, 0).
Now it becomes clear that we can use the regularity of the set ω. To prove condition (3)(b) suppose, on the contrary, that
(t(z), ϕ(0)) ∈ W.
Since (t(z), ϕ(0)) ∈ ∂ω, it follows
mj0
(t(z), ϕ(0)) = 0 for some j0 ∈ {1, . . . , s}.
Hence the inequality (δ) in Definition 4 is satisfied. Since y(δ, q(z))(t) is differentiable in t for t > δ, this inequality becomes
Dmj0
(t, y(δ, q(z))(t))|t=t(z) < 0,
i.e., for some σ > 0 and all 0 < ε < σ,
mj0
(t(z) − ε, y(δ, q(z))(t(z) − ε)) > mj0
(t(z), y(δ, q(z))(t(z)))
= mj0
(t(z), ϕ(0)) = 0.
Hence
(t(z) − ε, y(δ, q(z))(t(z) − ε)) ∈ ω.
This contradicts (3)(a). Then (t(z), ϕ(0)) ∈ W and, therefore, (3)(b) is satisfied.
It follows that li0
(t(z), ϕ(0)) = 0 for some i0 ∈ {1, . . . , p}. Applying (γ ) of Definition 4, we get
Dli0
(t, y(δ, q(z))(t))|t=t(z) > 0,
hence, for some σ > 0 and all 0 < ε < σ:
li0
(t(z) + ε, y(δ, q(z))(t(z) + ε)) > li0
(t(z), y(δ, q(z))(t(z)))
= li0
(t(z), ϕ(0)) = 0.
Hence
(t(z) + ε, y(δ, q(z))(t(z) + ε)) ∈ ω
and (3)(c) is satisfied.
Finally, we will verify condition (4): If z = (δ, y) ∈ W ∩ B, then there is a i0 ∈ {1, . . . , p} such that li0
(δ, y) = 0. We set
ϕ := q(z). Then
(δ + θ, ϕ(θ)) ∈ ω
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for all θ ∈ [−h, 0). Hence, the derivative from the right
Dli0
(t, y(δ, p(z))(t))|t=δ+0 > 0.
This implies the existence of a σ > 0 such that for all 0 < ε < σ:
li0
(δ + ε, y(δ, q(z))(δ + ε)) > li0
(δ, y(δ, q(z))(δ)) = li0
(δ, ϕ(0)) = 0,
i.e. (δ + ε, y(δ, q(z))(δ + ε)) ∈ ω for 0 < ε < σ. Thus, condition (4) of Lemma 1 holds. Lemma 1 is valid in the described
situation and from its conclusion, the conclusion of this theorem follows.
4. Example
Consider an equation
˙y(t) = −a(t)y(t − h)et2 ˙y(t−h)
, (13)
where h > 1 is a constant delay and a: [0, ∞) → (0, ∞). Set t0 = 0. Eq. (13) is a particular case of (5) with n = 1, o = = 1,
h1(t) ≡ h, g1(t) ≡ h and
f (t, y(t − h), ˙y(t − h)) := −a(t)y(t − h)et2 ˙y(t−h)
.
All the assumptions of Theorem 2 are fulfilled and, therefore, Hypothesis A holds with tϕ = ∞. Assume
a(t)eh
≤ 1 (14)
if t ∈ [0, ∞). We will prove that there exists a solution y = y(t) of Eq. (13) on [−h, ∞) such that
0 < y(t) < e−t
and − e−t
≤ ˙y(t) ≤ 0. (15)
Although in terms of the geometrical meaning a great deal of applicability moments of the retract principle are obvious, we
will make all the computations in detail.
Construction of a polyfacial set ω. We will construct a polyfacial set. Put n = 1, t∗ = ∞, p = 1, s = 0 and
l1(t, y) := y(y − e−t
) in Definition 3. Then
ω = {(t, y) ∈ [−h, ∞) × R, y · (y − e−t
) < 0}.
Regularity of ω. Now set q = 1 and define a function c1: [−h, ∞) × R × R → R as
c1(t, y, x) := x · (x + e−t
).
Let us verify that the polyfacial set ω is regular with respect to Eq. (13) and the inequality c1 ≤ 0. We will verify the conditions
(α) – (δ) of Definition 4.
Condition (α) is obviously satisfied for the choice ˜Ω := [0, ∞) × C1
.
Now we verify condition (β). We must show that inequality (9) holds if (8) is valid and the graphs of the functions used
lie in the regular set ω (in the computations below we assume t ≥ 0). More exactly, we must verify that
c1(t, ϕ(θ), f (t, ϕ(θ), ˙ϕ(θ))) = f (t, ϕ(θ), ˙ϕ(θ)) · f (t, ϕ(θ), ˙ϕ(θ)) + e−t
= −a(t)ϕ(θ)et2 ˙ϕ(θ)
−a(t)ϕ(θ)et2 ˙ϕ(θ)
+ e−t
≤ 0
if θ ∈ [−h, 0), 0 < ϕ(θ) < e−(t+θ)
and −e−(t+θ)
≤ ˙ϕ(θ) ≤ 0. This is obvious because
−a(t)ϕ(θ)et2 ˙ϕ(θ)
< 0
and (we use (14) as well)
−a(t)ϕ(θ)et2 ˙ϕ(θ)
+ e−t
> e−t
−a(t)eh
et2 ˙ϕ(θ)
+ 1 ≥ e−t
−a(t)eh
+ 1 ≥ 0.
Verification of condition (γ ): The boundary ∂ω is given as
∂ω = {(t, y) ∈ [−h, ∞) × R, y · (y − e−t
) = 0}
and can be split into two disjoint nonempty parts
∂ω1 = {(t, y) ∈ [−h, ∞) × R, y = 0}
and
∂ω2 = {(t, y) ∈ [−h, ∞) × R, y = e−t
}.
We must show that inequality (11) holds provided that inequality (10) is valid, graphs of the functions used (i.e. (t+θ, ϕ(θ)),
θ ∈ [−h, 0)) lie in ω and the point (t, ϕ(0)) ∈ ∂ω. Since
Dl1(t, y) = D y · (y − e−t
) = (y − e−t
) · −a(t)ϕ(−h)et2 ˙ϕ(−h)
+ y · −a(t)ϕ(−h)et2 ˙ϕ(−h)
+ e−t
Author's personal copy
J. Diblík et al. / Nonlinear Analysis 71 (2009) e1393–e1400 e1399
we consider sign Dl1(t, y) separately on ∂ω1 and ∂ω2. We get
sign Dl1(t, y)|(t,y)∈∂ω1
= sign(0 − e−t
) · −a(t)ϕ(−h)et2 ˙ϕ(−h)
= 1
and
sign Dl1(t, y)|(t,y)∈∂ω2
= sign e−t
· −a(t)ϕ(−h)et2 ˙ϕ(−h)
+ e−t
.
Since (we use properties of ϕ and inequality (14))
−a(t)ϕ(−h)et2 ˙ϕ(−h)
+ e−t
> e−t
−a(t)eh
et2 ˙ϕ(−h)
+ 1
≥ e−t
−a(t)eh
+ 1 ≥ 0,
sign Dl1(t, y)|(t,y)∈∂ω2
= 1. Finally, Dl1(t, y) > 0 and condition (γ ) is proved.
Condition (δ) is omitted because no function of the type mj is used. The set ω is regular.
Sets W, Z, B, mapping q and requirements (1), (2). In our case, since, as mentioned above, no function of the type mj is
used,
W = {(t, y) ∈ ∂ω} = ∂ω = {(t, y) ∈ [−h, ∞) × R, y · y − e−t
= 0}.
Let
Z := {(t, y) ∈ [−h, ∞) × R, t = 0, y · (y − 1) ≤ 0}.
Obviously, Z ∩ W is a retract of W, but not a retract of Z and the set B := Z ∩ (Z ∪ W) reduces to Z, i.e., B = Z.
We choose a suitable system of differentiable functions by defining the mapping q. For every point z = (0, y) ∈ Z, we
define q(z) as
q(z)(θ) := ye−λθ
, θ ∈ [−h, 0]
where λ is the unique solution of a transcendental equation
λ = a(0)eλh
(16)
satisfying inequality λh < 1. The existence of such root can be proved easily if the inequality a(0)he < 1, which is a
consequence of (14), holds (we refer, e.g, to [14]). Since h > 1, λ < 1. Mapping q is continuous if the point (0, y) varies
within Z. Such functions should satisfy requirements (1) and (2) formulated in Section 3. Now it is easy to see (δ = 0 in (1),
(2)) that, for
z ∈ Z ∩ ω = {(t, y) ∈ [−h, ∞) × R, t = 0, y · (y − 1) < 0},
we have
0 < q(z)(θ) = ye−λθ
< e−θ
, θ ∈ [−h, 0] (17)
since y ∈ (0, 1), θ ∈ [−h, 0] and λ ∈ (0, 1), and requirement (1) holds. If
z ∈ W ∩ B = {(t, y) ∈ [−h, ∞) × R, t = 0, y · (y − 1) = 0},
then, for θ ∈ [−h, 0), the previous inequalities hold,
(0, q(z)(0)) = (0, y) =
(0, 1) if z = (0, 1),
(0, 0) if z = (0, 0)
and requirement (2) is valid as well.
Sewing condition and subsidiary inequality for initial functions. It still remains to show the two properties of the set of
functions q(z)(θ) when z varies within Z. First we verify that, for such functions, the sewing condition (6), having the form
˙ϕ(0) = −a(0)ϕ(−h) (18)
in the case of Eq. (13), is satisfied. For ϕ(θ) := q(z)(θ) = ye−λθ
, θ ∈ [−h, 0], y ∈ [0, 1] we compute
ϕ(0) = q(z)(0) = y,
ϕ(−h) = q(z)(−h) = yeλh
,
˙ϕ(θ) = (q(z)(θ)) = −yλe−λθ
,
˙ϕ(0) = −yλ.
Then (18) turns into equality
−yλ = −a(0)yeλh
,
which is valid because λ is a root of the above transcendental equation (16).
Author's personal copy
e1400 J. Diblík et al. / Nonlinear Analysis 71 (2009) e1393–e1400
Next we must verify that the subsidiary inequality c1 ≤ 0 remains valid for initial functions as well, i.e. we must verify
that
−e−θ
≤ (q(z)(θ)) = −yλe−λθ
≤ 0
if θ ∈ [−h, 0]. The right-hand inequality is obvious. The left-hand inequality turns into e−θ
≥ yλe−λθ
and can be verified in
much the same way as (17).
All the assumptions of Theorem 3 are fulfilled and, therefore, (13) has at least one solution y = y(t) satisfying
inequalities (15) on [−h, ∞).
5. Concluding remarks
The retract method developed in the paper can be used, e.g., in asymptotic analysis of solutions of neutral differential
equations. Focus on the existence of positive solutions for neutral equations (the example considered was concerned with
this topic) seems to be of particular relevance. Using this new tool can add new important information to the existing results.
Note that results on the positivity and asymptotic behavior of solutions for neutral differential equations with delay and for
delayed differential equations can be found, e.g., in books [15–19,12] and papers [20–27]. The sewing conditions (4) and (6)
seem to be, in general, too restrictive and should be replaced with other assumptions when carrying on the investigation.
Acknowledgment
This research was supported by the Grant 201/08/0469 of Czech Grant Agency (Prague), by the Council of Czech
Government MSM 00216 30503, MSM 0021630519 and MSM 0021630529.
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[14] J. Diblík, M. Kúdelčíková, Nonoscillating solutions of the equation ˙x(t) = −(a + b/t)x(t − τ), Stud. Univ. Žilina Math. Ser. 15 (1) (2002) 11–24.
[15] R.P. Agarwal, M. Bohner, Wan-Tong Li, Nonoscillation and Oscillation: Theory for for Functional Differential Equations, Marcel Dekker, Inc., 2004.
[16] D.D. Bainov, D.P. Mishev, Oscillation Theory for Neutral Differential Equations with Delay, Adam Hilger, 1991.
[17] L.H. Erbe, Q. Kong, B.G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 1995.
[18] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, 1992.
[19] I. Györi, G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, Oxford, 1991.
[20] L. Berezansky, E. Braverman, On exponential stability of linear differential equations with several delays, J. Math. Anal. Appl. 324 (2006) 1336–1355.
[21] L. Berezansky, E. Braverman, Positive solutions for a scalar differential equation with several delays, Appl. Math. Lett. 21 (2008) 636–640.
[22] G.E. Chatzarakis, R. Koplatadze, I.P. Stavroulakis, Optimal oscillation criteria for first order difference equations with delay argument, Pacific J. Math.
235 (2008) 15–33.
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[24] Á. Elbert, I.P. Stavroulakis, Oscillation and non-oscillation criteria for delay differential equations, Proc. Amer. Math. Soc. 123 (1995) 1503–1510.
[25] L’ Hanuštiaková, R. Olach, Nonoscillatory bounded solutions of neutral differential systems, Nonlinear Anal. 68 (2008) 1816–1824.
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207–224.
DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2018008
DYNAMICAL SYSTEMS SERIES B
Volume 23, Number 1, January 2018 pp. 123–144
ASYMPTOTIC PROPERTIES OF DELAYED MATRIX
EXPONENTIAL FUNCTIONS VIA LAMBERT FUNCTION
Josef Dibl´ık
Brno University of Technology
CEITEC - Central European Institute of Technology
Purkyˇnova 656/123, 612 00 Brno, Czech Republic
Zdenˇek Svoboda∗
Brno University of Technology
CEITEC - Central European Institute of Technology
Purkyˇnova 656/123, 612 00 Brno, Czech Republic
Abstract. In the case of first-order linear systems with single constant delay
and with constant matrix, the application of the well-known “step by step”
method (when ordinary differential equations with delay are solved) has recently
been formalized using a special type matrix, called delayed matrix exponential.
This matrix function is defined on the intervals (k − 1)τ ≤ t < kτ,
k = 0, 1, . . . (where τ > 0 is a delay) as different matrix polynomials, and is
continuous at nodes t = kτ. In the paper, the asymptotic properties of delayed
matrix exponential are studied for k → ∞ and it is, e.g., proved that the
sequence of values of a delayed matrix exponential at nodes is approximately
represented by a geometric progression. A constant matrix has been found
such that its matrix exponential is the “quotient” factor that depends on the
principal branch of the Lambert function. Applications of the results obtained
are given as well.
1. Introduction. The well-known “step by step” method is one of the basic concepts
for the investigation of linear differential equations and systems with delay.
The application of this method to first-order linear systems with single constant
delay and with constant matrix of linear terms was formalized by using the notion
of delayed matrix exponential eBt
τ , where B is a square constant matrix and τ > 0
is a delay, in [5, 6]. A special delayed matrix function is defined on every interval
(k − 1)τ ≤ t < kτ, k = 0, 1, . . . (where τ > 0 is a delay) as a matrix polynomial
depending on B and is continuous at nodes t = kτ. Such a step by step definition
complicates its asymptotic analysis. The paper deals with the asymptotic properties
of delayed matrix exponential. Proofs of the results derived below make use of
the properties of the matrix Lambert function [8]. Therefore, some basic notations
and results related to this function are recalled in this part, too. Auxiliary results
overviewed in Part 1.1 can be found in [5, 6] and [2]. The results given in Part 1.2
are taken from [3] (see also the original source [8]). New auxiliary results are proved
in Part 1.3. In Part 2, auxiliary determinants are computed and the results applied
in Part 3 to prove the main result of the asymptotic behavior of a sequence of the
2010 Mathematics Subject Classification. Primary: 34K06; Secondary: 34K25.
Key words and phrases. Lambert function, delayed matrix exponential, asymptotic behavior,
principal part, instability.
∗ Corresponding author: Z. Svoboda.
123
124 JOSEF DIBL´IK AND ZDENˇEK SVOBODA
ratios of delayed matrix exponentials at adjacent nodes. The sequence of values
of delayed matrix exponential at nodes is approximately represented by a geometric
progression. A constant matrix is found such that its matrix exponential is
the “quotient” factor that depends on the principal branch of the Lambert function.
Moreover, some further results on asymptotic properties of delayed matrix
exponential are proved. Applications of the results derived are collected in Part 4.
1.1. First-order linear systems. Let B be an n×n constant matrix, Θ an n×n
null matrix, I an n×n unit matrix and let τ > 0 be a constant. The delayed matrix
exponential eBt
τ of the matrix B is an n × n matrix function mapping R to Rn×n
,
continuous on R \ {−τ} and defined as follows:
eBt
τ :=



k
j=0
Bj (t − (j − 1)τ)j
j!
, t ≥ −τ,
Θ, t < −τ
where k = t/τ is the ceiling function, i.e. the smallest integer greater than or equal
to t/τ. The main property of the delayed matrix exponential eBt
τ is the following:
eBt
τ = BeB(t−τ)
τ , t ∈ R \ {0}
and the matrix Y (t) = eBt
τ solves the initial problem for a matrix differential system
with a single delay
Y (t) = BY (t − τ), t ∈ [0, ∞),
Y (t) = I, t ∈ [−τ, 0].
If ϕ: [−τ, 0] → Rn
is a continuously differentiable vector-function, then the solution
of the initial-value problem
y (t) = By(t − τ), t ∈ [0, ∞), (1)
y(t) = ϕ(t), t ∈ [−τ, 0] (2)
can be represented in the form
y(t) = eBt
τ ϕ(−τ) +
0
−τ
eB(t−τ−s)
τ ϕ (s)ds, t ∈ [−τ, ∞). (3)
Let A be a regular n × n constant matrix and AB = BA. Then, the solution of the
initial-value problem
y (t) = Ay(t) + By(t − τ), t ∈ [0, ∞), (4)
y(t) = ϕ(t), t ∈ [−τ, 0] (5)
is given by the formula
y(t) = eA(t+τ)
eB1t
τ ϕ(−τ) +
0
−τ
eA(t−τ−s)
eB1(t−τ−s)
τ eAτ
[ϕ (s) − Aϕ(s)]ds (6)
where t ∈ [−τ, ∞) and B1 = e−Aτ
B.
ASYMPTOTIC PROPERTIES OF DELAYED MATRIX EXPONENTIAL FUNCTIONS 125
1.2. The Lambert W function. As can easily be seen from the definition, the
above delayed matrix function is defined on intervals (k−1)τ ≤ t < kτ, k = 0, 1, . . .
as different matrix polynomials. As mentioned in Introduction, this is the reason
why its asymptotic analysis is complicated.
Therefore, it seems to be important to study the sequence {eBkτ
τ }∞
k=0 of values
of the delayed exponential of a matrix B at nodes kτ, connecting two different
matrix polynomials, as k → ∞. Later, we will prove that, for the special matrix
considered, this sequence approximately equals a geometric progression and we
will find a constant n × n matrix C such that its ordinary exponential eCτ
is the
“quotient”, i.e., that
lim
k→∞
eB(k+1)τ
τ eBkτ
τ
−1
= eCτ
, (7)
where ( · )−1
denotes the inverse matrix, whose existence we assume.
This will be done using the so-called Lambert function (named after Johann
Heinrich Lambert, see [8]). Recall its definition and some basic results on the
Lambert function (published in [3]).
Lambert defined the function as the inverse to the function
f(w) = wew
.
This means that the Lambert function, usually denoted by W = W(z), is defined
implicitly by the equation
z = W(z)eW (z)
. (8)
Such a function is multi-valued (except for the point z = 0). For real arguments
z = x such that x > −1/e and real W(x) satisfying W(x) > −1, equation (8)
defines a single-valued function W = W0(x) called the principal branch of the
Lambert W(z) function, i.e.,
W0(x)eW0(x)
≡ x, x > −1/e. (9)
We prove that the matrix C in (7) is defined by the principal branch W0(z) of
the Lambert W(z) function (see Corollary 1 below).
The Maclaurin expansion of W0(x) can be found easily being given by the series
W0(x) =
∞
n=1
(−n)n−1
n!
xn
(10)
having the radius of convergence r = 1/e. The point x = 0 is a point of removable
singularity of the function W0(x)/x. It follows from (10) that the Maclaurin
expansion of the function
E(x) :=



W0(x)
x
, x = 0,
1, x = 0,
(11)
i.e.,
E(x) =
∞
n=1
(−n)n−1
n!
xn−1
(12)
has the same radius of convergence r = 1/e. The function E(x) is smooth and
infinitely many times differentiable. Moreover, applying the Lagrange inversion
126 JOSEF DIBL´IK AND ZDENˇEK SVOBODA
theorem (Lagrange-B¨urmann formula), we obtain
W0(x)
x
r
= exp(−rW0(x)) =
∞
n=0
r(n + r)n−1
n!
(−x)n
. (13)
Differentiating the defining equation (8), we conclude that all branches of W(z)
satisfy the differential equation
z(1 + W)
dW
dz
= W, z = 0. (14)
Let λ be a complex number. In determining the asymptotic properties of the
exponential function exp(λx), where x ∈ R, the real part of the complex number λ
often plays the principal role because the asymptotic properties differ for Re λx > 0
and Re λx < 0 and the domains for the real part being positive or negative are in
the complex plane for λ “separated” by the set of points where Re λ = 0. In the
definition of the Lambert function by (8), the behavior of the exponential function
plays an important role as well.
Define the set of complex numbers such that Re W(z) = 0. Assuming z = x + iy
and W(z) = u + iv, from (8), we get
x + iy = W(z)eW (z)
= iveiv
= iv(cos v + i sin v) = −v sin v + iv cos v,
i.e.,
x = −v sin v, (15)
y = v cos v (16)
where v ∈ R. Analyzing the part of this curve corresponding to the principal branch
W0(x + iy), i.e.,
x = −v sin v > −
1
e
we conclude that (15), (16) is a simple closed curve for the admissible range v ∈
[−π/2, π/2]. This curve is depicted in Figure 1. From (15), (16), it is easy to deduce
that the real part of the principal branch of the Lambert function is negative for
|z| < − arctan
Re z
|Im z|
. (17)
This domain is bounded by the above curve (see Figure 1). Note that a Lambert
W function cannot be expressed in terms of elementary functions. For more details,
see [3].
1.3. Limits with principal part W0 of the Lambert function. Let k be a
nonnegative integer. Define a polynomial
Pk(x) =
k
j=0
(k + 1 − j)j
j!
xj
. (18)
Then, the formula
eBkτ
τ =
k
j=0
Bj ((k + 1 − j)τ)j
j!
= Pk(Bτ), (19)
where B0
= I, expressing the values of a delayed matrix exponential at the nodes
t = kτ, k = 0, 1, 2, . . . holds and can be simply verified using the definition of the
delayed matrix exponential.
ASYMPTOTIC PROPERTIES OF DELAYED MATRIX EXPONENTIAL FUNCTIONS 127
Re z−
π
2
Im zRe W0(z) > 0
Re W0(z) < 0
1
Figure 1. The curve Re W0(z) = 0
Let x, α and β be real numbers and let n be a positive integer. The following is
a well-known Abel’s extension of the binomial theorem (see, e.g. [1])
(x + α)n
= xn
+
n
1
α(x + β)n−1
+
n
2
α(α − 2β)(x + 2β)n−2
+ · · · +
n
α(α − β) −1
(x + β)n−
+ . . .
+
n
n − 1
α (α − (n − 1)β)
n−2
(x + (n − 1)β) + α(α − nβ)n−1
,
which, for α = 0, can be rewritten as
(x + α)n
=
n
=0
n
α(α − β) −1
(x + β)n−
(20)
and will be used in the computations below.
Lemma 1.1. Let x ∈ (−1/e, 1/e) be fixed. Then,
lim
k→∞
Pk(x)
Pk+1(x)
= E(x), (21)
lim
k→∞
Pk+1(x)
Pk(x)
= exp(W0(x)) (22)
and, for l ∈ N, we have
lim
k→∞
1
Pk+1(x)
P
(l)
k (x) −
l
=1
l
P
( )
k+1(x)E(l− )
(x) = E(l)
(x). (23)
Proof. We decompose the ratio
Pk(x)
Pk+1(x)
into the Maclaurin power series with respect to x
Pk(x)
Pk+1(x)
=
∞
=0
a x , a ∈ R,
128 JOSEF DIBL´IK AND ZDENˇEK SVOBODA
and show that the sum of the first (k + 1) terms of this expansion where k ≥ 0
equals a polynomial of k-th degree (compare (12))
Ek(x) =
k
=0
(− − 1)
( + 1)!
x , (24)
i.e.,
a =
(− − 1)
( + 1)!
, = 0, 1, . . . , k,
and
Pk(x)
Pk+1(x)
= Ek(x) +
∞
=k+1
a x (25)
or
Pk(x)
Pk+1(x)
= Ek(x) + O(xk+1
) (26)
where O is the Landau order symbol “big” O. We prove this by matching the
coefficients at identical powers n, n = 0, 1, , . . . , k of two polynomials Ek(x)Pk+1(x)
and Pk(x). The coefficient at the power xn
(0 ≤ n ≤ k) of the product
Ek(x)Pk+1(x) =
k
=0
(− − 1)
( + 1)!
x ·


k+1
j=0
(k + 2 − j)j
j!
xj


can be expressed as
n
=0
(− − 1)
( + 1)!
(k + 2 − n + )n−
(n − )!
=
n
=0
(−1)
(− − 1) −1
!
(k + 2 − n + )n−
(n − )!
=
1
n!
n
=0
(−1)
n
(− − 1) −1
(k + 2 − n + )n−
= (we use identity (20) with α = −1, β = 1, x = k + 2 − n)
=
(k + 1 − n)n
n!
and is the same as the coefficient at the power xn
of the polynomial Pk(x). Therefore,
formula (26) holds with the indicated accuracy. Formula (21) now follows from
the property
lim
k→∞
Ek(x) = E(x).
Formula (22) is a consequence of (21), (11) and (9) since
lim
k→∞
Pk+1(x)
Pk(x)
=
1
lim
k→∞
Pk(x)
Pk+1(x)
=
1
E(x)
= exp(W0(x)).
Now we will show that (23) holds. Without loss of generality, we assume k > l in
the sequel. Since power series are infinitely many times differentiable within their
ASYMPTOTIC PROPERTIES OF DELAYED MATRIX EXPONENTIAL FUNCTIONS 129
interval of convergence, from (24), we have
Ek(x) = E(x) −
∞
=k+1
(− − 1)
( + 1)!
x
and
E
(l)
k (x) = E(l)
(x) + O(xk−l+1
). (27)
Rewriting (25) as
Pk(x) = Pk+1(x)Ek(x) + Pk+1(x)
∞
=k+1
a x , (28)
differentiating (28) l-times, and using (27) we get
P
(l)
k (x) = (Pk+1(x)Ek(x))
(l)
+ Pk+1(x)
∞
=k+1
a x
(l)
=
l
=0
l
P
( )
k+1(x)E
(l− )
k (x) + O(xk−l+1
)
=
l
=0
l
P
( )
k+1(x)E(l− )
(x) + O(xk−l+1
)
or
P
(l)
k (x) −
l
=1
l
P
( )
k+1(x)E(l− )
(x) = Pk+1(x)E(l)
(x) + O(xk−l+1
).
Then,
1
Pk+1(x)
P
(l)
k (x) −
l
=1
l
P
( )
k+1(x)E(l− )
(x) = E(l)
(x) + O(xk−l+1
)
and, taking limit as k → ∞, we get formula (23).
Lemma 1.2. Let x ∈ (−1/e, 1/e) be fixed. Then,
lim
k→∞
Pk(x) exp(−kW0(x)) =
1
E(x)(1 + W0(x))
. (29)
Proof. We can decompose exp(−kW0(x))Pk(x), using (13) and (18), into the
Maclaurin power series. In the following decomposition, the first (k + 1) terms
are written exactly.
(exp(−kW0(x)))Pk(x) =
∞
=0
−k(−k − ) −1
!
x
k
=0
(k + 1 − )
!
x
=
k
l=0
xl
l
=0
−k(−k − ) −1
!
·
(k + 1 − l + )l−
(l − )!
+ O(xk+1
)
=(we use (20) with n = l, α = −k, β = 1, x = k + 1 − l)
=
k
l=0
(1 − l)l
l!
xl
+ O(xk+1
).
130 JOSEF DIBL´IK AND ZDENˇEK SVOBODA
For the limit of this product, we obtain
lim
k→∞
exp(−kW0(x))Pk(x) =
∞
l=0
(1 − l)l
l!
xl
. (30)
Now we put r = −1 in (13) and develop the Maclaurin power series of the expression
(below, the values for x = 0 are understood as limits for x → 0)
−x2 d
dx
eW0(x)
x
.
We get
− x2 d
dx
eW0(x)
x
= −x2
∞
n=0
−(n − 1)n−1
n!
(−x)n
x
= −x2
∞
n=0
(1 − n)n−1
n!
xn−1
= x2
∞
n=0
(1 − n)n
n!
xn−2
=
∞
n=0
(1 − n)n
n!
xn
. (31)
Comparing (30) with (31), we conclude that
lim
k→∞
exp(−kW0(x))Pk(x) = −x2 d
dx
eW0(x)
x
.
Using (9) and (14), we get
− x2 d
dx
eW0(x)
x
= −x2 1
W0(x)
= x2 W0(x)
W2
0 (x)
= x2 1
W2
0 (x)
·
W0(x)
x(1 + W0(x))
=
x
W0(x)(1 + W0(x))
=
1
E(x)(1 + W0(x))
. (32)
Now, formula (29) is a consequence of (30)–(32).
Remark 1. As it follows from formula (29) in Lemma 1.2, for fixed x ∈ (−1/e, 1/e),
we have
Pk(x) ∼
exp(kW0(x))
E(x)(1 + W0(x))
(33)
where k → ∞, and
lim
k→∞
Pk(x)E(x)
1 + W0(x)
exp(kW0(x))
= 1.
2. Preliminaries. Let us recall that two n×n matrices A and B are called similar
if B = P−1
∗ AP∗ for some invertible n×n matrix P∗ (for properties of matrices used
in this part, we refer, e.g. to [4, Chapter V]). Let s be a positive integer and s ≤ n.
An s × s matrix Jλ,s
Jλ,s =







λ 1 0 · · · 0
0 λ 1 · · · 0
...
...
...
...
...
0 0 0 λ 1
0 0 0 0 λ







,
ASYMPTOTIC PROPERTIES OF DELAYED MATRIX EXPONENTIAL FUNCTIONS 131
where λ is a complex number, is called a Jordan block. Any block diagonal matrix
whose blocks are Jordan blocks is called a Jordan matrix and any matrix A is similar
to an n × n Jordan matrix
J = diag(Jλ1,m1
, Jλ2,m2
, . . . , JλN ,mN
) (34)
where, for positive integers mi, i = 1, 2, . . . , mN , we have m1 + m2 + · · · + mN = n
and λi are the eigenvalues of A with multiplicities mi. The Jordan matrix J given
by (34) is unique up to a permutation of its diagonal blocks. J is called the Jordan
normal form of A and, for some suitable invertible n × n matrix P, we have
A = P−1
JP.
For an analytic function with a radius of convergence r given by the series
f(z) =
∞
h=0
ahzh
and for any matrix A with spectral radius ρ(A)
def
= max
i=1,2,...,mN
|λi| satisfying ρ(A) <
r, also the matrix
f(A) =
∞
h=0
ahAh
= P−1
diag(f(Jλ1,m1
), f(Jλ2,m2
), . . . , f(JλN ,mN
))P
is defined where the series has the same radius of convergence and the matrices
f(Jλi,mi
) =
∞
h=0
ah(Jλi,mi
)h
, i = 1, 2, . . . , N,
defined by the series with the same radius of convergence r again, satisfy:
f(Jλi,mi
) =

























f(λi)
f (λi)
1!
f (λi)
2!
· · ·
f(mi−2)
(λi)
(mi − 2)!
f(mi−1)
(λi)
(mi − 1)!
0 f(λi)
f (λi)
1!
· · ·
f(mi−3)
(λi)
(mi − 3)!
f(mi−2)
(λi)
(mi − 2)!
0 0 f(λi) · · ·
f(mi−4)
(λi)
(mi − 4)!
f(mi−3)
(λi)
(mi − 3)!
...
...
...
...
...
...
0 0 0 · · · f(λi)
f (λi)
1!
0 0 0 · · · 0 f(λi)

























.
Now we develop matrix analogies of the statements formulated in Lemma 1.1.
Let k be a nonnegative integer, λ ∈ C, and s be a positive integer. For k ≥ s, we
define an s × s matrix
Pk(Jλ,s) = (pij(k, λ, s))s
i,j=1
132 JOSEF DIBL´IK AND ZDENˇEK SVOBODA
as
Pk(Jλ,s) =
























Pk(λ)
Pk(λ)
1!
Pk (λ)
2!
· · ·
P
(s−2)
k (λ)
(s − 2)!
P
(s−1)
k (λ)
(s − 1)!
0 Pk(λ)
Pk(λ)
1!
· · ·
P
(s−3)
k (λ)
(s − 3)!
P
(s−2)
k (λ)
(s − 2)!
0 0 Pk(λ) · · ·
P
(s−4)
k (λ)
(s − 4)!
P
(s−3)
k (λ)
(s − 3)!
...
...
...
...
...
...
0 0 0 · · · Pk(λ)
Pk(λ)
1!
0 0 0 · · · 0 Pk(λ)
























where the polynomial Pk is given by formula (18). To avoid possible ambiguities in
the following computations, we also define
Pk(Jλ,0) := (1)
where k and λ are as above. In what follows, we do not consider zero points of the
polynomial Pk, so we will assume Pk(λ) = 0. Thus, Pk(Jλ,s) is an invertible matrix.
To describe the result of the matrix product
Pk(Jλ,s) = (pk
ij(Jλ,s))s
i,j=1 := Pk(Jλ,s)(Pk+1(Jλ,s))−1
, (35)
we need to define some auxiliary determinants Mk(λ, s). The meaning of k and λ
remains the same. The integer s in the following definition satisfies s ∈ Z.
Definition 2.1. Determinants Mk(λ, s) are defined as follows.
1. If s < 0, then Mk(λ, s) := 0.
2. If s = 0, then Mk(λ, 0) := 1.
3. If s > 0, then
Mk(λ, s) :=
Pk(λ)
1!
Pk (λ)
2!
· · ·
P
(s−1)
k (λ)
(s − 1)!
P
(s)
k (λ)
s!
Pk(λ)
Pk(λ)
1!
· · ·
P
(s−2)
k (λ)
(s − 2)!
P
(s−1)
k (λ)
(s − 1)!
0 Pk(λ) · · ·
P
(s−3)
k (λ)
(s − 3)!
P
(s−2)
k (λ)
(s − 2)!
...
...
...
...
...
0 · · · 0 Pk(λ)
Pk(λ)
1!
.
Lemma 2.2. Let Mij, i, j = 1, . . . , s be minors of the matrix Pk+1(Jλ,s). Then,
a) Mij = 0 if i < j,
b) Mij = (Pk+1(λ))s−1
if i = j,
c) Mij = (Pk+1(λ))s−1+j−i
Mk+1(λ, i − j) if i > j.
ASYMPTOTIC PROPERTIES OF DELAYED MATRIX EXPONENTIAL FUNCTIONS 133
Proof. a) Let i < j. Then, Mij is the determinant of an upper triangular matrix
with the main diagonal
(Pk+1(λ), . . . , Pk+1(λ)
i−1
, 0, . . . , 0
j−i
, Pk+1(λ), . . . , Pk+1(λ)
s−j
)
and, consequently, Mij = 0.
b) Let i = j. Then, the minor Mii is the determinant of an upper triangular
matrix with the main diagonal
(Pk+1(λ), . . . , Pk+1(λ)
s−1
)
and Mij = (Pk+1(λ))s−1
.
c) Let i > j. Then, the minor Mij = (mpq)s−1
p,q=1 is the determinant of a matrix
with the following structure - its main diagonal equals
(Pk+1(λ), . . . , Pk+1(λ)
j−1
, Pk+1(λ), . . . , Pk+1(λ)
i−j
, Pk+1(λ), . . . , Pk+1(λ)
s−i
),
the elements mpq = 0 if
α) q = 1, . . . , j − 1 and p > q,
β) p = i + 1, . . . , s − 1 and p > q,
and the elements mpq where p, q = j, . . . , i − 1 generate a matrix with the determinant
Mk+1(λ, i − j). We get
Mij =
· · ·
P
(j−2)
k+1
(λ)
(j − 2)!
P
(j)
k+1
(λ)
j!
· · ·
P
(i−2)
k+1
(λ)
(i − 2)!
P
(i−1)
k+1
(λ)
(i − 1)!
P
(i)
k+1
(λ)
i!
· · ·
P
(s−1)
k+1
(λ)
(s − 1)!
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
· · · Pk+1(λ)
Pk+1(λ)
2!
· · ·
P
(i−j)
k+1
(λ)
(i − j)!
P
(i−j+1)
k+1
(λ)
(i − j + 1)!
· · · · · ·
P
(s−j+1)
k+1
(λ)
(s − j + 1)!
· · · 0
Pk+1(λ)
1!
· · ·
P
(i−j−1)
k+1
(λ)
(i − j − 1)!
P
(i−j)
k+1
(λ)
(i − j)!
· · · · · ·
P
(s−j)
k+1
(λ)
(s − j)!
· · · 0 Pk+1(λ)
.
.
.
P
(i−j−2)
k+1
(λ)
(i − j − 2)!
P
(i−j−1)
k+1
(λ)
(i − j + 1)!
· · · · · ·
P
(s−j−1)
k+1
(λ)
(s − j − 1)!
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
· · · · · · · · · 0 Pk+1(λ)
Pk+1(λ)
1!
Pk+1(λ)
2!
· · ·
P
(s−i+1)
k+1
(λ)
(s − i + 1)!
· · · · · · · · · · · · 0 0 Pk+1(λ) · · ·
P
(s−i−1)
k+1
(λ)
(s − i − 1)!
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
,
from which it follows that
Mij = (Pk+1(λ))j−1
Mk+1(λ, i−j)(Pk+1(λ))s−i
= (Pk+1(λ))s−1+j−i
Mk+1(λ, i−j).
134 JOSEF DIBL´IK AND ZDENˇEK SVOBODA
Remark 2. In the sequel, we will need to express any minor Mij of the matrix
Pk+1(Jλ,s) in terms of the determinants Mk+1(λ, i − j). For every minor Mij,
i, j = 1, . . . , s, the same formula
Mij = (Pk+1(λ))s−1+j−i
Mk+1(λ, i − j)
holds since, using Definition 2.1, we can write the statements of Lemma 2.2 as
a) Mij = 0 = (Pk+1(λ))s−1+j−i
Mk+1(λ, i − j) if i < j,
b) Mij = (Pk+1(λ))s−1
= (Pk+1(λ))s−1+j−i
Mk+1(λ, i − j) if i = j,
c) Mij = (Pk+1(λ))s−1+j−i
Mk+1(λ, i − j) if i > j.
Using Remark 2, we can express the cofactors Cij, i, j = 1, . . . , s of the matrix
Pk+1(Jλ,s) as:
Cij(Jλ,s) = (−1)i+j
Mij = (−1)i+j
(Pk+1(λ))s−1+j−i
Mk+1(λ, i − j).
Now we will continue the computation of the matrix product (35). We can
find the inverse matrix (Pk+1(Jλ,s))−1
by a well-known procedure using the adjoint
matrix whose elements can be defined through the cofactors Cij(Jλ,s), i, j = 1, . . . , s
and using the obvious formula det Pk+1(Jλ,s) = (Pk+1(λ))s
.
We get
pk
ij(Jλ,s)) =
s
=1
pi (k, λ, s)
Cj (Jλ,s)
(Pk+1(λ))s
=
s
=i
P
( −i)
k (λ)
( − i)!
Cj (Jλ,s)
(Pk+1(λ))s
=
s−i
l=0
P
(l)
k (λ)
l!
Cj,l+i(Jλ,s)
(Pk+1(λ))s
=
s−i
l=0
P
(l)
k (λ)
l!
(−1)j+l+i
Mk+1(λ, j − l − i)
(Pk+1(λ))1+j−l−i
.
Because of the properties of determinants Mk (see Definition 2.1), we have
pk
ij(Jλ,s)) = 0 if i > j,
and, for the rest of the elements pk
i,i+j(Jλ,s)) with j = 0, 1, . . . , s − i, we get
pk
i,i+j(Jλ,s)) =
s−i
l=0
P
(l)
k (λ)
l!
(−1)j+l
Mk+1(λ, j − l)
(Pk+1(λ))1+j−l
=
j
l=0
P
(l)
k (λ)
l!
(−1)j+l
Mk+1(λ, j − l)
(Pk+1(λ))1+j−l
. (36)
Due to (36), where the index i is not included in the final formula, we can define
ˆpk
j (Jλ,s) := pk
i,i+j(Jλ,s)) (37)
for any i = 1, . . . , s, j = 0, 1, . . . , s − i.
ASYMPTOTIC PROPERTIES OF DELAYED MATRIX EXPONENTIAL FUNCTIONS 135
Compute now the (1, l)-cofactor of Mk+1(λ, s). It has the form
(−1)1+l
×
Pk+1(λ) · · ·
P
(l−1)
k+1 (λ)
(l − 1)!
P
(l+1)
k+1 (λ)
(l + 1)!
· · ·
P
(s−2)
k+1 (λ)
(s − 2)!
P
(s−1)
k+1 (λ)
(s − 1)!
...
...
...
...
...
...
...
0 · · · Pk+1(λ)
Pk+1(λ)
2!
· · ·
P
(s−l)
k+1 (λ)
(s − l)!
P
(s−l+1)
k+1 (λ)
(s − l + 1)!
0 · · · 0
Pk+1(λ)
1!
· · ·
P
(s−l−1)
k+1 (λ)
(s − l − 1)!
P
(s−l)
k+1 (λ)
(s − l)!
0 · · · 0 Pk+1(λ)
...
P
(s−l−2)
k+1 (λ)
(s − l − 2)!
P
(s−l−1)
k+1 (λ)
(s − l − 1)!
...
...
...
...
...
...
...
0 · · · · · · · · · 0 Pk+1(λ)
Pk+1(λ)
1!
and equals
(−1)1+l
(Pk+1(λ))l−1
Mk+1(λ, s − l).
Applying the Laplace expansion of the the determinant Mk+1(λ, s) along the first
row, we get
Mk+1(λ, s) =
s
l=1
P
(l)
k+1(λ)
l!
(−1)1+l
(Pk+1(λ))l−1
Mk+1(λ, s − l).
This equation can be rewritten in the form
0 =
s
l=0
P
(l)
k+1(λ)
l!
(−1)1+l
(Pk+1(λ))l−1
Mk+1(λ, s − l). (38)
Using (38) for s ≥ 1, we can prove a recurring equation between the elements of
the matrix product Pk(Jλ,s)(Pk+1(Jλ,s))−1
:
Lemma 2.3. For the elements ˆpk
j (Jλ,s)) of the product Pk(Jλ,s)(Pk+1(Jλ,s))−1
,
defined by (37), and integer 1 ≤ l ≤ s − 1, we have:
P
(l)
k (λ)
l!
=
l
=0
P
( )
k+1(λ)
!
ˆpk
l− (Jλ,s) (39)
Proof. Substitute (36) for ˆpk
l− (Jλ,s) in the right-hand side of (39) to obtain:
l
=0
P
( )
k+1(λ)
!
ˆpk
l− (Jλ,n)
=
l
=0
P
( )
k+1(λ)
!
l−
i=0
P
(i)
k (λ)
i!
(−1)l− +i
Mk+1(λ, l − + i)
(Pk+1(λ))1+l− −i
=
l
=0
l−
i=0
P
( )
k+1(λ)
!
P
(i)
k (λ)
i!
(−1)l− +i
Mk+1(λ, l − − i)
(Pk+1(λ))1+l− −i
= (∗)
136 JOSEF DIBL´IK AND ZDENˇEK SVOBODA
Now we rearrange by the formula
l
=0
l−
i=0
ai =
l
i=0
l−i
=0
ai the last sum (∗) and
apply the identity (38) to get:
(∗) =
l
i=0
l−i
=0
P
( )
k+1(λ)
!
P
(i)
k (λ)
i!
(−1)l− +i
Mk+1(λ, l − − i)
(Pk+1(λ))1+l− −i
=
l
i=0
P
(i)
k (λ)
i!
l−i
=0
P
( )
k+1(λ)
!
(−1)l−2 +i+1
(Pk+1(λ))l−i
(−1)1+
(Pk+1(λ)) −1
Mk+1(λ, l − − i)
=
P
(l)
k (λ)
l!
+
l−1
i=0
P
(i)
k (λ)
i!
(−1)l+i+1
(Pk+1(λ))l−i
×
l−i
=0
P
( )
k+1(λ)
!
(−1)1+
(Pk+1(λ)) −1
Mk+1(λ, l − i − )
=0 due to (38) with s:=l−i≥1
=
P
(l)
k (λ)
l!
.
3. Main results. Based on the auxiliary results proved we can now prove the main
results of the paper.
Theorem 3.1. Let τ > 0 and let an n × n constant matrix B ≡ Θ be given. If the
eigenvalues λi, i = 1, . . . , n of the matrix B satisfy the inequality |λi|τ < 1/e, then
lim
k→∞
eBkτ
τ (eB(k+1)
τ )−1
= E(Bτ) (40)
and
lim
k→∞
eB(k+1)
τ (eBkτ
τ )−1
= exp(W0(Bτ)). (41)
Proof. First we show that (40) holds if B is replaced by a Jordan block Jλ,n.
The limits of the elements pk
ii(Jλ,n)) = ˆpk
0(Jλ,n)), i = 1, . . . , n of the product
Pk(Jλ,n)(Pk+1(Jλ,n))−1
,
as it follows from formula (36) (where j = 0) and from formula (21), are
lim
k→∞
ˆpk
0(Jλ,n)) = lim
k→∞
Pk(λ)
Pk+1(λ)
= E(λ).
Now, by induction, we prove that, for the limits of other elements ˆpk
l (Jλ,n)) =
pk
i,i+l(Jλ,n)), l = 1, . . . , n − i, we have
lim
k→∞
ˆpk
l (Jλ,n)) =
E(l)
(λ)
l!
, (42)
i.e., for k → ∞ we have
ˆpk
l (Jλ,n)) =
E(l)
(λ)
l!
+ o(1)
where o is the Landau order symbol “small” o. The assertion is proved for l = 0.
Now we assume that this assertion holds for i = 0, . . . , l where l < n − i. We use
formula (39) to express the element ˆpk
l+1(Jλ,n):
ASYMPTOTIC PROPERTIES OF DELAYED MATRIX EXPONENTIAL FUNCTIONS 137
ˆpk
l+1(Jλ,n) =
1
Pk+1(λ)
P
(l+1)
k (λ)
(l + 1)!
−
l+1
=1
P
( )
k+1(λ)
!
ˆpk
l+1− (Jλ,n)
=
1
Pk+1(λ)
P
(l+1)
k (λ)
(l + 1)!
−
l+1
=1
P
( )
k+1(λ)
!
E(l+1− )
(λ)
(l + 1 − )!
+ o(1)
=
1
(l + 1)!
1
Pk+1(λ)
P
(l+1)
k (λ) −
l+1
=1
l + 1
P
( )
k+1(λ) E(l+1− )
(λ) + o(1) .
Applying (23), we obtain:
lim
k→∞
ˆpk
l+1(Jλ,n) =
E(l+1)
(λ)
(l + 1)!
.
Consequently, formula (42) holds.
The remaining elements pk
ij(Jλ,n) of the product Pk(Jλ,n)(Pk+1(Jλ,n))−1
with
i > j (under the main diagonal) are equal to zero.
The Jordan block Jλ,n has the spectral radius ρ(Jλ,n) = |λ| and, by the assumption,
|λ|τ < 1/e. Substituting Jλ,nτ for x into (12), we conclude that there is a
matrix E(Jλ,n τ) as the value of the analytic function defined by the series (12)
with the radius of convergence r = 1/e such that
lim
k→∞
e
Jλ,nkτ
τ e
Jλ,n(k+1)τ
τ
−1
= [by (19)] = lim
k→∞
Pk(Jλ,nτ)(Pk+1(Jλ,nτ))−1
= E(Jλ,n τ).
From the representation
B = P−1
diag(Jλ1,m1
, Jλ2,m2
, . . . , JλN ,mN
)P, (43)
we directly get
eBkτ
τ = P−1
diag e
Jλ1,m1
kτ
τ , . . . , e
JλN ,mN
kτ
τ P
and
eBkτ
τ eB(k+1)τ
τ
−1
= P−1
diag e
Jλ1,m1
kτ
τ e
Jλ1,m1
(k+1)τ
τ
−1
, . . . , e
JλN ,mN
kτ
τ e
JλN ,mN
(k+1)τ
τ
−1
P
as well. Now we can obtain easily
lim
k→∞
eBkτ
τ eB(k+1)τ
τ
−1
= P−1
diag lim
k→∞
e
Jλ1,m1
kτ
τ e
Jλ1,m1
(k+1)τ
τ
−1
,
. . . , lim
k→∞
e
JλN ,mN
kτ
τ e
JλN ,mN
(k+1)τ
τ
−1
P
= P−1
diag (E(Jλ1,m1
τ), . . . , E(JλN ,mN
τ)) P = E(Bτ)
and (40) is proved.
Note that, due to formulas (10), (12), (43) and
W0(Bτ) = P−1
diag(W0(Jλ1,m1
), W0(Jλ2,m2
), . . . , W0(JλN ,mN
))P,
matrices B, E(Bτ) and W0(Bτ) mutually commute (the Jordan canonical forms
for B and W0(Bτ) have, for the same regular matrix P, diagonal blocks of the same
138 JOSEF DIBL´IK AND ZDENˇEK SVOBODA
type). Then, formula (41) is a consequence of (40) since, by using (11) and (8), we
get
lim
k→∞
eB(k+1)τ
τ eBkτ
τ
−1
= lim
k→∞
eBkτ
τ eB(k+1)τ
τ
−1 −1
= (E(Bτ))
−1
= Bτ (W0(Bτ))
−1
= exp(W0(Bτ)). (44)
The following corollary specifies the matrix C mentioned in formula (7).
Corollary 1. From Theorem 3.1 and formula (44), we have
lim
k→∞
eB(k+1)τ
τ eBkτ
τ
−1
= eCτ
where
C :=
1
τ
W0(Bτ).
Theorem 3.2. Let τ > 0 and let an n × n constant matrix B ≡ Θ be given. If the
eigenvalues λi, i = 1, . . . , n of the matrix B satisfy the inequality |λi|τ < 1/e, then
lim
k→∞
eBkτ
τ exp(−kW0(Bτ)) = Bτ (W0(Bτ)(I + W0(Bτ)))
−1
. (45)
Proof. Let n = 1. In the scalar case, (45) is a simple consequence of (29) since,
by (19) and (11),
lim
k→∞
eBkτ
τ exp(−kW0(Bτ)) = lim
k→∞
Pk(Bτ) exp(−kW0(Bτ))
= (E(Bτ)(1 + W0(Bτ)))
−1
= Bτ(W0(Bτ)(1 + W0(Bτ)))
−1
.
Let n > 1. The radius of convergence of the Maclaurin series of the function
x (W0(x)(1 + W0(x)))
−1
is r = 1/e (see formulas (10)–(12)). Since inequalities |λi|τ < 1/e, i = 1, . . . , n
imply ρ(Bτ) < 1/e, we can substitute x → Bτ into this Maclaurin decomposition
to get convergent matrix series. Its sum equals
Bτ (W0(Bτ)(I + W0(Bτ)))
−1
.
Then,
lim
k→∞
eBkτ
τ exp(−kW0(Bτ)) = lim
k→∞
Pk(Bτ) exp(−kW0(Bτ))
= Bτ (W0(Bτ)(I + W0(Bτ)))
−1
.
Let F(k) = {fij(k)}
n
i,j=1 and G = {fij(k)}
n
i,j=1
be matrices defined for all sufficiently
large k. We say that
F(k) G(k), k → ∞ (46)
if
fij(k) = gij(k)(1 + o(1)), k → ∞ (47)
where o(1) is the Landau order symbol “small” o.
ASYMPTOTIC PROPERTIES OF DELAYED MATRIX EXPONENTIAL FUNCTIONS 139
Remark 3. Let all assumptions of Theorem 3.2 be valid. From formula (45), we
get the asymptotic relation
eBkτ
τ Bτ exp(kW0(Bτ))(W0(Bτ)(I + W0(Bτ)))−1
, k → ∞ (48)
This formula can be useful, e.g., in the investigation of the asymptotic behavior
of solutions of problem (1), (2) or (4), (5) at nodes t = kτ, as can be seen from
formulas (3), (6).
The following theorem gives results on the behavior of the spectral radius ρ(·)
and spectral norm · ρ (defined for a matrix A as A ρ = ρ(AAT
)
1/2
) of the
sequence of values of delayed exponential eBkτ
τ for (discrete) k → ∞ and of delayed
exponential eBt
τ for (continuous) t → ∞.
Theorem 3.3. Let τ > 0 and let an n×n constant matrix B ≡ Θ be given. Assume
that the eigenvalues λi, i = 1, . . . , n of the matrix B satisfy inequality τ|λi| < 1/e,
i = 1, . . . , n. The following three statements are true:
(i) If all the eigenvalues λi, i = 1, . . . , n satisfy
τ|λi| < − arctan
Re λi
|Im λi|
, (49)
then
lim
k→∞
ρ eBkτ
τ = 0. (50)
(ii) If there exist an index i0 ∈ {1, . . . , n} such that
τ|λi0
| > − arctan
Re λi0
|Im λi0
|
, (51)
then
lim sup
k→∞
eBkτ
τ ρ
= ∞. (52)
(iii) If all the eigenvalues λi, i = 1, . . . , n are real and satisfy
τ|λi| > − arctan
Re λi
|Im λi|
, (53)
then
lim
t→∞
eBt
τ ρ
= ∞. (54)
Proof. To prove this theorem we use Remark 3. Figure 2 details the eigenvalue
domain for each case considered.
(i) From (49), we conclude that, for all the eigenvalues λi, i = 1, . . . , n, by (17),
Re W0(λiτ) < 0 is true, therefore,
lim
k→∞
ρ (exp (k W0(λiτ))) = 0.
It is well-known that the n roots of a polynomial of degree n depend continuously
on the coefficients and that the eigenvalues of a matrix depend continuously on the
matrix (we refer, e.g. to [9]). Then, (48) implies
lim
k→∞
ρ eBkτ
τ = 0,
so that (50) holds.
140 JOSEF DIBL´IK AND ZDENˇEK SVOBODA
Re z−
π
2τ
1
e
Im z
Re W0(z) > 0
case i case ii case iii
Re W0(z) < 0
1
Figure 2. Detailed eigenvalue domains
(ii) From assumption (51), by (17), the existence follows of at least one eigenvalue
λi0
such that Re W0(λi0
τ) > 0. Therefore,
lim sup
k→∞
ρ (exp (k W0(λi0
τ))) = ∞.
In much the same way as in part (i), by (48), we also deduce
lim sup
k→∞
ρ eBkτ
τ = ∞.
Then the conclusion of part (ii) follows from the relation between the spectral radius
and the spectral norm:
ρ(A) ≤ A ρ
for any matrix A.
(iii) Let n = 1. In the scalar case, the condition (53) implies
0 < λ1 < 1/(eτ).
The delayed exponential function eλ1t
τ is a solution of the equation
y (t) = λ y(t − τ) (55)
satisfying the initial condition
y(t) = 1, t ∈ [−τ, 0]. (56)
Since the solution y = y(t) of problem (55), (56) satisfies y(t) > 0, t ≥ −τ and
y (t) ≥ λ1 > 0 for t > 0, we have
lim
t→∞
eλ1t
τ = ∞.
Let n > 1. Then, as above, we have
0 < λi < 1/(eτ) , i = 1, . . . , n.
Let J be the Jordan canonical form of square matrix B. I.e., there is an invertible
matrix P∗ such that B = P−1
∗ JP∗. Note that the Jordan canonical form of the
delayed exponential of matrix eBt
τ has the form P−1
∗ eJt
τ P∗ and, due to this fact,
all the eigenvalues of eBt
τ are eλit
τ , i = 1, . . . , n where λi, i = 1, . . . , n are all the
ASYMPTOTIC PROPERTIES OF DELAYED MATRIX EXPONENTIAL FUNCTIONS 141
eigenvalues of B. Proceeding similarly to the scalar case, we conclude that (54)
holds.
4. Applications. In this part we make some suggestions for possible applications
of the above results.
4.1. Equation of a showering person. Systems (1) often describe mathematical
models of real-world phenomena. The solution of the initial problem (1), (2) is
given by formula (3). Investigate the long-time behavior of the solutions generated
by constant initial functions, i.e., assume ϕ(t) ≡ Cϕ for every fixed t ∈ [−τ, 0] and
Cϕ ∈ Rn
. Then, ϕ (t) ≡ θ, t ∈ [−τ, 0] where θ is the null vector. Formula (3)
becomes
y(t) = eBt
τ ϕ(−τ) = eBt
τ Cϕ. (57)
If all assumptions of Theorem 3.2 hold, by formula (48), we get the asymptotic
relation for (57) at nodes t = kτ as k → ∞
y(kτ) = eBkτ
τ Cϕ Bτ exp(kW0(Bτ))(W0(Bτ)(I + W0(Bτ)))−1
Cϕ. (58)
Consider the equation modeling the behavior of a showering person (for details
we refer, e.g., to [7, part 3.6.3])
T (t) = −γ[T(t − τ) − Td], t ∈ [0, ∞) (59)
where T is the regulated temperature of water leaving the mixer, γ > 0 and Td is
the desired temperature of water agreeable for a showering person. Setting y(t) =
T(t) − Td in (59), we get
y (t) = −γy(t − τ), t ∈ [0, ∞). (60)
Assuming the water temperature before regulation is constant, i.e. the initial condition
is given by the equation
y(t) = y0, t ∈ [−τ, 0], (61)
the solution of (60), (61) is
y(t) = e−γt
τ y0, t ∈ [−τ, ∞)
and if γτe < 1 then, by (46)–(48) and (58),
y(kτ) = e−γkτ
τ y0 = −γτ exp(kW0(−γτ))
y0(1 + o(1))
W0(−γτ)(1 + W0(−γτ))
as k → ∞. By (9), the last formula can be simplified to
y(kτ) =
y0(1 + o(1))
1 + W0(−γτ)
e(1 + k)W0(−γτ) , k → ∞.
Since, by (10),
W0(−γτ) = −γτ − (γτ)2
−
3
2
(γτ)3
+ · · · ,
we have y(kτ) > 0 and limk→∞ y(kτ) = 0. It means that the regulated temperature
T(kτ) will tend to the desired value Td as k → ∞.
The above example can be generalized, e.g., for two showering persons. Suppose
that hot and cold water is supplied in two separate pipes to a bathroom with
two showers. Inside the bathroom, each pipe branches into two pipes leading to
the shower mixers. A person taking a shower regulates the water temperature
flowing from the mixer to the sprinkler. Due to the changes in the water pressure
caused by water being regulated by two persons simultaneously, there is a mutual
142 JOSEF DIBL´IK AND ZDENˇEK SVOBODA
dependence between the temperatures T1 and T2 of the water flowing from mixer
one to sprinkler one and from mixer two to sprinkler two, respectively. Then, a
simple model modeling the behavior of two showering persons is
T1(t) = − γ11[T1(t − τ) − Td1] + γ12[T2(t − τ) − Td2], (62)
T2(t) = γ21[T1(t − τ) − Td1] − γ22[T2(t − τ) − Td2] (63)
where γij > 0, i, j = 1, 2 and Tdi, i = 1, 2 are the desired temperatures of water
agreeable for each of the two showering persons. Substituting yi(t) = Ti(t) − Tdi
in (62), (63) we get
y1(t) = − γ11y1(t − τ) + γ12y2(t − τ), (64)
y2(t) = γ21y1(t − τ) − γ22y2(t − τ). (65)
Assuming the water temperature before regulation is constant, i.e. the initial condition
is given by the relation
y1(t) = y2(t) = y0, t ∈ [−τ, 0], (66)
the solution of (64)–(66) is
y(t) = (y1(t), y2(t))T
= e−Γt
τ y0
, t ∈ [−τ, ∞) (67)
where y0
= (y0, y0)T
and
Γ =
−γ11 γ12
γ21 −γ22
.
Let the eigenvalues
λi =
1
2
−(γ11 + γ22) + (−1)i
(γ11 − γ22)2 + 4γ12γ21 , i = 1, 2
of the matrix Γ satisfy |λi|τe < 1, i = 1, 2. Then, by formula (58), at nodes t = kτ,
the solution (67) has the asymptotic behavior
y(kτ) Γτ exp(kW0(Γτ))(W0(Γτ)(I + W0(Γτ)))−1
y0
as k → ∞.
4.2. Instability of solutions. In this part we give sufficient conditions for the
instability of the system (1). In general, the instability of systems (1) will be proved
if, in every δ-neighborhood of zero initial function, there exist an initial function
generating a solution not remaining in a given ε-neighborhood of the zero solution.
In the proof of the following theorem, it is sufficient to restrict the set of initial
functions to constant initial functions only.
Theorem 4.1. Let τ > 0 and let an n × n constant matrix B ≡ Θ be given.
Assume that the eigenvalues λi, i = 1, . . . , n of the matrix B satisfy the inequality
τ|λi| < 1/e, i = 1, . . . , n. If, moreover, there exist an index i0 ∈ {1, . . . , n} such
that
τ|λi0 | > − arctan
Re λi0
|Im λi0 |
,
then the system (1) is instable.
ASYMPTOTIC PROPERTIES OF DELAYED MATRIX EXPONENTIAL FUNCTIONS 143
Proof. We will employ constant initial functions
ϕi
(t) = Ci
:= (0, . . . , 0
i−1
, 1, 0, . . . , 0
n−i
)T
, t ∈ [−τ, 0], i = 1, . . . , n.
Generated by ϕi
(t), solution yi
= yi
(t) equals
yi
(t) = eBt
τ Ci
, t ∈ [−τ, ∞), i = 1, . . . , n,
Consider a matrix equation
Y (t) = BY (t − τ), t ∈ [0, ∞) (68)
where Y (t) is an n × n matrix. Clearly, the matrix
Y (t) := (y1
(t), . . . , yn
(t)) = eBt
τ (C1
, . . . , Cn
) = eBt
τ , t ∈ [−τ, ∞)
is a solution of the system (68) satisfying Y (t) = I, t ∈ [−τ, 0]. Obviously,
Y (t) ρ = eBt
τ ρ
and by applying the well-known result on the equivalence of norms, there exists a
constant M > 0 such that, for the element-wise max norm · max of a matrix, we
have
M max
i,j=1,...,n
|yi
j(t)| = M Y (t) max ≥ Y (t) ρ = eBt
τ ρ
, t ∈ [0, ∞) (69)
where yi
j(t), j = 1, . . . , n are co-ordinates of the solution yi
(t). All assumptions
of Theorem 3.3, part (ii) are satisfied and, therefore, for t = kτ and k → ∞, by
formula (52), we have
lim sup
k→∞
eBkτ
τ ρ
= ∞.
Then, from (69), we derive
lim sup
k→∞
max
i,j=1,...,n
|yi
j(kτ)| = ∞.
This property proves the instability of the system (1).
Remark 4. A similar result on instability can be derived for the system (4) if the
following modifications are taken into account. Instead of constant initial functions
used in the proof of Theorem 4.1, initial functions as solutions of the system
ϕ (t) = Aϕ(t), t ∈ [−τ, 0]
can be used. Then, the formula (6) becomes
y(t) = eA(t+τ)
eB1t
τ ϕ(−τ), t ∈ [−τ, ∞)
where B1 = e−Aτ
B. In addition to this, additional assumptions on the matrix A
for the statement on instability must be included.
Acknowledgments. The first author was supported by the Czech Science Foundation
under the project 16-08549S. The research of the second author was carried
out under the project CEITEC 2020 (LQ1601) with financial support from the
Ministry of Education, Youth and Sports of the Czech Republic under the National
Sustainability Programme II. The work was realized in CEITEC - Central European
Institute of Technology with research infrastructure supported by the project
CZ.1.05/1.1.00/02.0068 financed from European Regional Development Fund.
144 JOSEF DIBL´IK AND ZDENˇEK SVOBODA
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for differential systems with a single delay, Nonlinear Anal., 72 (2010), 2251–2258.
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[6] D. Ya. Khusainov and G. V. Shuklin, On relative controllability in systems with pure delay,
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[7] V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional
Differential Equations, Kluwer Academic Publishers, Dordrecht, 1999.
[8] J. H. Lambert, Observationes variae in mathesin puram, Acta Helvetica, physicomathematico-anatomico-botanico-medica,
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Boston, MA, 1997.
Received for publication September 2016.
E-mail address: josef.diblik@ceitec.vutbr.cz
E-mail address: zdenek.svoboda@ceitec.vutbr.cz
Existence of Strictly Decreasing Positive Solutions
of Linear Differential Equations of Neutral Type
Josef Dibl´ık ∗
Zdenˇek Svoboda †
Abstract
The paper is concerned with a linear neutral differential equation
˙y(t) = −c(t)y(t − τ(t)) + d(t) ˙y(t − δ(t))
where c: [t0, ∞) → (0, ∞), d: [t0, ∞) → [0, ∞), t0 ∈ R and τ, δ : [t0, ∞) →
(0, r], r ∈ R, r > 0 are continuous functions. A new criterion is given for
the existence of positive strictly decreasing solutions. The proof is based
on the Rybakowski variant of a topological Wa˙zewski principle suitable
for differential equations of the delayed type. Unlike in the previous investigations
known, this time the progress is achieved by using a special
system of initial functions satisfying a so-called sewing condition. The
result obtained is extended to more general equations. Comparisons with
known results are given as well.
Keywords: Neutral equation, delay, positive solution, sewing condition
AMS 2010 classification: Primary 34K40; 34K25; 34K12.
1 Introduction
The aim of the paper is to give a criterion for the existence of positive strictly
decreasing solutions to the linear neutral differential equation
˙y(t) = −c(t)y(t − τ(t)) + d(t) ˙y(t − δ(t)) (1)
where c: [t0, ∞) → (0, ∞), d: [t0, ∞) → [0, ∞), t0 ∈ R, and τ, δ: [t0, ∞) →
(0, r], r ∈ R, r > 0 are continuous functions.
The existence of positive solutions of functional differential equations of delayed
type is a classical problem which is satisfactorily solved for various classes
of equations in numerous papers and books. We should note, however, that the
positivity of solutions to neutral differential equations is investigated to a degree
less than that of the positivity of solutions of non-neutral equations with delay.
∗Brno University of Technology, Brno, Czech Republic (E-mail josef.diblik@ceitec.
vutbr.cz)
†Brno University of Technology, Brno, Czech Republic (zdenek.svoboda@ceitec.vutbr.cz)
1
Some results on the existence of positive solutions for delayed differential
equations and their systems are summarized, e.g., in [1, 2, 3, 23, 24, 25].
Let us cite one of the nice classical implicit results on the existence of a
positive solution of a linear equation with delay ([39], see also [23, Theorem
2.1.4] and [2, Theorem 2.2.13]), which serves as a source for various explicit
sufficient positivity criteria. Consider the equation
˙y(t) + p(t)y(t − τ(t)) = 0 (2)
where p, τ : [t0, ∞) → R+, R+ := [0, ∞) are continuous functions, τ(t) ≤ t and
lim
t→∞
(t − τ(t)) = ∞. Set T0 = inf
t≥t0
{t − τ(t)}. A function y is called a solution
of (2) with respect to initial point t0 if y is defined and continuous on [T0, ∞),
differentiable on [t0, ∞), and satisfies (2) for t ≥ t0.
Theorem 1. Equation (2) has a positive solution with respect to t0 if and
only if there exists a continuous function λ(t) on [T0, ∞) such that λ(t) > 0 for
t ≥ t0 and
λ(t) ≥ p(t) exp
t
t−τ(t)
λ(s)ds , t ≥ t0. (3)
The above criterion was generalized for systems of linear and nonlinear differential
equations with bounded delay in [9] and for nonlinear systems of differential
equations with unbounded delay and with finite memory in [16]. Positive
solutions of (2) in the so-called critical case were studied, e.g., in [5, 11, 12,
17, 19, 22, 35] and an overview of some sufficient conditions to equation (2) in
the critical case is given in a recent paper [4]. Asymptotic formulas describing
two classes of asymptotically different positive solutions are analyzed, e.g.,
in [13, 14] and [15]. The problem of positive solutions is also investigated in
further numerous papers such as [6, 7, 8, 10, 20, 21, 29, 37] and the references
therein.
To describe the main result of the paper we should note that, to the best
of our knowledge, there is no extension of the implicit-type (with respect to λ)
result given by Theorem 1, where the key role is played by inequality (3), to
neutral equations of the type (1) if the solutions are understood as continuously
differentiable functions (see Definition 1) below. In this direction, we will show
that in the case of equation (1), inequality (3) can be replaced by
λ(t) ≥ c(t) exp
t
t−τ(t)
λ(s)ds + d(t)λ(t − δ(t)) exp
t
t−δ(t)
λ(s)ds , (4)
t ≥ t0, where λ: [t0 − r, ∞) → (0, ∞). Strictly speaking, Theorem 1 for p > 0
deals with strictly decreasing positive solutions. Our method gives the same
statement in this sense. Namely, inequality (4) is necessary and sufficient for
the existence of a positive and strictly decreasing solution of equation (1).
The topological (retract) method of T. Wa˙zewski [38], which was successfully
modified to retarded differential equations by K.P. Rybakowski (see, e.g., [33,
2
34]) serves as a theoretical tool to prove the main result. For a nice overview
of topological principle, we also refer to [36]. In [18] the retract principle was
modified for neutral functional differential equations. This modification should
make it possible to use this in the present paper. Even if [18] contains an
illustrative example, showing how this modification works, there is one serious
problem restricting the classes of equations suitable for considering by it. Below,
we explain the heart of the matter.
We consider a neutral functional differential system of the form
˙y(t) = f(t, yt, ˙yt) (5)
where the symbol ˙y stands for the derivative (considered, if necessary, as onesided).
Sometimes we use the symbol y as well (if there is no doubt whether
the derivative is one-sided or not).
Let C be the set of all continuous functions φ: [−r, 0] → Rn
and C1
be the
set of all continuously differentiable functions φ: [−r, 0] → Rn
. Assume t ≥ t0,
yt(θ) = y(t + θ), θ ∈ [−r, 0] and f : Er → Rn
with Er := [t0, ∞) × C × C.
We pose an initial problem for (5):
yt0
= φ, ˙yt0
= ˙φ (6)
where φ ∈ C1
. The norm of φ ∈ C is defined as φ r := max
θ∈[−r,0]
φ(θ) and, if
φ ∈ C1
, then
φ r := max
θ∈[−r,0]
φ(θ) + max
θ∈[−r,0]
φ (θ)
where · is the Euclidean norm.
In the literature there are various definitions of a solution to neutral differential
equations. In the paper, as a solution of (5), (6), we assume a continuously
differentiable function within the meaning of the following definition.
Definition 1. A continuously differentiable function y: [t0 − r, tφ) → Rn
,
tφ ∈ (t0, ∞], is a solution of (5), (6) if yt0
= φ, ˙yt0
= ˙φ and (5) is satisfied for
any t ∈ [t0, tφ).
V. Kolmanovskii and A. Myshkis [28] considered the initial-value problem
for neutral differential equations (5), (6). Although this problem should be
expected have a continuously differentiable solution on an interval [t0, tφ), in
general, this is not true. Even if the functional f and the initial function φ are
arbitrarily smooth, and the initial problem can be solved by the method of steps,
the continuous solution may, generally speaking, have jumps of the derivative
for arbitrarily large t. Such jumps will be absent if the initial function φ satisfies
the sewing condition
˙φ(0) = f(t0, φ, ˙φ). (7)
Theorem 2. [28, p.107] Let f : Er → Rn
be a continuous functional satisfying,
in some neighborhood of any point of Er, the Lipschitz condition
f(t, ψ1, χ1) − f(t, ψ2, χ2) ≤ L1 ψ1 − ψ2 r + L2 χ1 − χ2 r
3
with constants Li ∈ [0, ∞), i = 1, 2. Assume also φ ∈ C1
and the sewing condition
(7) being fulfilled. Then, there exists a tφ ∈ (t0, ∞] such that:
a) There exists a solution y of (5), (6) on [t0 − r, tφ).
b) On any interval [t0 − r, t1] ⊂ [t0 − r, tφ), t1 > t0, this solution is unique.
c) If tφ < ∞, then ˙x(t) has not a finite limit as t → t−
φ .
d) The solution y and ˙y depend continuously on φ.
For a particular case of system (5) given by
˙y(t) = f(t, yt, ˙yt)
:= f(t, y(t − h1(t)), . . . , y(t − ho(t)), ˙y(t − g1(t)), . . . , ˙y(t − g (t))),
where indices o ≥ 0 and ≥ 1, a more general result can be proved easily by
the method of steps (compare [28, pages 111, 96, and 15]).
Theorem 3. Let
f : [t0, ∞) × Ro+
→ Rn
,
hi : [t0, ∞) → (0, r], i = 1, . . . , o and gj : [t0, ∞) → (0, r], j = 1, . . . ,
be continuous functions. Assume also φ ∈ C1
and the sewing condition (7), in
the case considered having the form
˙φ(0) = f(t0, φ(−h1(t0)), . . . , φ(−ho(t0)), ˙φ(−g1(t0)), . . . , ˙φ(−g (t0))) (8)
being fulfilled. Then:
a) There exists a solution y of (5), (6) on [t0 − r, ∞).
b) On any interval [t0 − r, t1] ⊂ [t0 − r, ∞), t1 > t0, this solution is unique.
c) The solution y and ˙y depend continuously on φ.
To succeed in applying Theorem 2 (or Theorem 3) to prove the existence and
uniqueness of a continuously differentiable (by Definition 1) solution, the sewing
condition (7) (or (8)) must be fulfilled. If not, then, generally speaking, a solution
has no continuous derivative and certainly, it has no two-sided derivative for
t = t0. To define an initial function that satisfies the sewing condition is usually
not an easy task. The above weighty circumstance when applying the retract
principle to neutral functional differential equations, follows from the necessity
to satisfy the sewing condition. When the retract principle is used, it is necessary
to construct not only one initial function but a set of functions, called the
set of initial functions, satisfying several assumptions. One of the assumption is
that every function of this set must satisfy a sewing condition. So, from above
it follows that, technically, is not easy to construct such a set. In the present
paper, we perform, for the case of linear neutral differential equation (1), the
relevant construction of a set of initial functions when dealing with a criterion
for a solution to be positive. This is an important progress as, eventually, we
are able to prove that such a positive solution is continuously differentiable (in
the meaning of Definition 1).
4
The rest of the paper is structured as follows. In Part 2 we give a generalization
of the retract principle to neutral functional differential equations,
previously developed in [18]. The main result (a criterion for the existence of
a positive strictly decreasing and continuously differentiable solution of neutral
differential equation (1)) is given in Part 3 where a special construction of
a system of initial functions satisfying the sewing condition is also developed.
For a more general equation than (1), a criterion for the existence of a positive
strictly decreasing and continuously differentiable solution is formulated in
Part 4. Some open questions, corollaries and remarks as well as comparisons
with some of the previous results are listed in Part 5.
2 Retract Method
This part provides necessary background. It is mainly taken from papers [18]
and [34]. Note that the underlying ideas are based, in addition to the paper
of the founder T. Wa˙zewski [38], on the so-called Razumikhin condition in the
theory of stability, e.g., [30, 31, 32], and on Razumikhin’s type of extension of
Wa˙zewski’s principle by K.P. Rybakowski [33, 34]). Mentioned are the necessary
changes of the original versions, making it possible to prove a criterion for the
existence of positive solutions to equation (1).
If a set A ⊂ R × Rn
is given, then int A, A and ∂A denote, as usual, the
interior, the closure, and the boundary of A, respectively.
Definition 2. (compare [18, 34]) Let Λ be a topological space, let a subset
˜Ω ⊂ R × Λ be open in R × Λ, and let x be a mapping associating with every
(δ, λ) ∈ ˜Ω a function x(δ, λ): Dδ,λ → Rn
where Dδ,λ is an interval in R. Assume
(1)–(3):
(1) δ ∈ Dδ,λ.
(2) If t ∈ int Dδ,λ, then there is an open neighbourhood O(δ, λ) of (δ, λ) in ˜Ω
such that t ∈ Dδ ,λ holds for all (δ , λ ) ∈ O(δ, λ).
(3) If (δ , λ ), (δ, λ) ∈ ˜Ω, and t ∈ Dδ ,λ , t ∈ Dδ,λ, then
lim
(δ ,λ ,t )→(δ,λ,t)
x(δ , λ )(t ) = x(δ, λ)(t).
Then, (Λ, ˜Ω, x) is called a system of curves in Rn
.
Definition 3. If A ⊂ A∗
are any two sets of a topological space and π: A∗
→
A is a continuous mapping from A∗
onto A such that π(p) = p for every p ∈ A,
then π is said to be a retraction of A∗
onto A. If there exists a retraction of A∗
onto A, A is called a retract of A∗
.
Lemma 1. (compare [18, 34]) Let (Λ, ˜Ω, x) be a system of curves in Rn
.
Let ˜ω, W, Z be sets. Assume the below conditions (1)–(4):
5
(1) a) ˜ω ⊂ [t0 −r, t∗)×Rn
, t∗ > t0, the cross-section {(˜t, y) ∈ ˜ω} is an open
simply connected set for every ˜t ∈ [t0 − r, t∗), and W ⊂ ∂˜ω,
b) Z ⊂ ˜ω ∪ W, Z ∩ W is a retract of W, but not a retract of Z.
(2) There is a continuous map q: B → Λ where B = Z ∩ (Z ∪ W) such
that, for any z = (δ, y) ∈ B, (δ, q(z)) ∈ ˜Ω, and, if also z ∈ W, then
x(δ, q(z))(δ) = y.
(3) Let A be the set of all z = (δ, y) ∈ Z ∩ ˜ω such that, for fixed (δ, y) ∈ A,
there is a t > δ, t ∈ Dδ,q(z) and (t, x(δ, q(z))(t)) ∈ ˜ω.
Assume that, for every z = (δ, y) ∈ A, there is a t(z), t(z) > δ, such that:
a) t(z) ∈ Dδ,q(z) and, for all t, δ ≤ t < t(z), (t, x(δ, q(z))(t)) ∈ ˜ω,
b) (t(z), x(δ, q(z))(t(z))) ∈ W,
c) For any σ > 0, there is a t, t(z) < t ≤ t(z) + σ such that t ∈ Dδ,q(z)
and (t, x(δ, q(z))(t)) ∈ ˜ω.
(4) For any z = (δ, y) ∈ W ∩ B and all σ > 0, there is a t, δ < t ≤ δ + σ such
that t ∈ Dδ,q(z) and (t, x(δ, q(z))(t)) ∈ ˜ω.
Then, there is a z0 = (δ0, y0) ∈ Z ∩ ˜ω such that, for every t ∈ Dδ0,q(z0),
(t, x(δ0, q(z0))(t)) ∈ ˜ω. (9)
Remark 1. Let
Λ = C1
, ˜Ω ⊂ {(t, λ) ∈ [t0, ∞) × C1
such that ˙λ(0) = f(t0, λ, ˙λ)}
and function f satisfies all the assumptions of Theorem 2. In this case, through
each (t0, λ) ∈ ˜Ω, there exists a unique solution y(t0, λ) of (5) defined on its
maximal interval [t0 − r, aλ). Let Dt0,λ = [t0 − r, aλ) where aλ > t0. Then,
(Λ, ˜Ω, y) is a system of curves in Rn
within the meaning of Definition 2. A
similar remark holds when all the assumptions of Theorem 3 are satisfied.
Usually, when applying Lemma 1 to prove the existence of a solution of a
given system with the graph staying in a prescribed domain ˜ω, the form of ˜ω
should be specified. As a standard shape of such a domain, used in numerous
investigations, serves the so-called polyfacial set defined below.
Definition 4. Let p and s be nonnegative integers, p + s > 0, t∗ > t0, and
let
li : [t0 − r, t∗) → R × Rn
, i = 1, . . . , p,
mj : [t0 − r, t∗) → R × Rn
, j = 1, . . . , s
be continuously differentiable functions. The set
ω := {(t, y) ∈ [t0 − r, t∗) × Rn
, li(t, y) < 0, mj(t, y) < 0, for all i, j }
6
is called a polyfacial set provided that the cross-section
ω ∩ {(t, y): t = t∗
, y ∈ Rn
}
is an open and simply connected set for every fixed t∗
∈ [t0 − r, t∗).
When p = 0 in Definition 4, the functions li, i = 1, . . . , p are not defined.
Similarly, if s = 0, the functions mj, j = 1, . . . , s are omitted. In order to
prove the existence of a solution of (5) satisfying the property (9), a polyfacial
set ω should meet some additional requirements. We can characterize such requirements
as properties guaranteeing the properties of solutions of system (5),
formulated for the system of curves (Λ, ˜Ω, x) in Lemma 1. Because of the neutrality
of the equations, we need to be able to foresee the properties of the
derivatives of solutions as described by auxiliary inequalities.
Definition 5. (compare [18]) Let q be a nonnegative integer, t∗ > t0, and
let
ck : [t0 − r, t∗) × Rn
× Rn
→ R, k = 1, . . . , q,
be continuous functions. A polyfacial set ω is called regular with respect to
Eq. (5) and auxiliary inequalities
ck(t, y, x) ≤ 0, k = 1, . . . , q (10)
if α) – δ) below hold:
α) If (t, φ) ∈ R × C1
and (t + θ, φ(θ)) ∈ ω for θ ∈ [−r, 0), then (t, φ, ˙φ) ∈ Er.
β) If (t, φ) ∈ R × C1
, (t + θ, φ(θ)) ∈ ω for θ ∈ [−r, 0) and, moreover,
ck(t + θ, φ(θ), ˙φ(θ)) ≤ 0, θ ∈ [−r, 0), k = 1, . . . , q, (11)
then also
ck(t + θ, φ(θ), f(t, φ, ˙φ)) ≤ 0, k = 1, . . . , q. (12)
γ) For all i = 1, . . . , p, all (t, y) ∈ ∂ω for which li(t, y) = 0 and for all φ ∈ C1
for which φ(0) = y, (t + θ, φ(θ)) ∈ ω, θ ∈ [−r, 0) and
ck(t + θ, φ(θ), ˙φ(θ)) ≤ 0, θ ∈ [−r, 0), k = 1, . . . , q, (13)
it follows that:
Dli(t, y) ≡
∂li
∂t
(t, y) +
n
r=1
∂li
∂yr
(t, y) · fr(t, φ, ˙φ) > 0.
δ) For all j = 1, . . . , s, all (t, y) ∈ ∂ω for which mj(t, y) = 0 and for all
φ ∈ C1
for which φ(0) = y, (t + θ, φ(θ)) ∈ ω, θ ∈ [−r, 0) and
ck(t + θ, φ(θ), ˙φ(θ)) ≤ 0, θ ∈ [−r, 0), k = 1, . . . , q
for all θ ∈ [−1, 0) , it follows that:
Dmj(t, y) ≡
∂mj
∂t
(t, y) +
n
r=1
∂mj
∂yr
(t, y) · fr(t, φ, ˙φ) < 0.
7
If ω is a polyfacial set, then define the set W used in Lemma 1 as
W := {(t, y) ∈ ∂ω : mj(t, y) < 0, j = 1, . . . , s}. (14)
Moreover, we need to specify the properties of the mapping q in Lemma 1.
The following definition describes the admissible behavior of functions with
respect to ω. A fixed set of functions generated by this mapping and satisfying
properties gathered in the following definition is called a set of initial functions.
Definition 6 (Set of initial functions). Let Z be a subset of ω ∪ W and
let the mapping
q: B → C1
, B := Z ∩ (Z ∪ W)
be continuous. We assume that, if z = (δ, y) ∈ B, then (δ, q(z)) ∈ ˜Ω. If
moreover
1) For z ∈ Z ∩ ω, we have (δ + θ, q(z)(θ)) ∈ ω for θ ∈ [−r, 0].
2) For z ∈ W ∩ B, we have (δ, q(z)(δ)) = z and
either
2a) (δ + θ, q(z)(θ)) ∈ ω for θ ∈ [−r, 0)
or
2b) (δ + θ, q(z)(θ)) ∈ ω for θ ∈ [−r, 0) and, for all σ > 0, there is
a t = t(σ, z), δ < t ≤ δ + σ such that t is within the domain of
definition of solution x(δ, q(z)) of (5) and (t, x(δ, q(z))(t)) ∈ ω,
then such a set of functions is called a set of initial functions for (5) with respect
to ω and Z.
Finally, we will formulate the below theorem as an application of Lemma 1
for a system of neutral equations (5). Therefore, its proof is omitted.
Theorem 4. Let ω be a nonempty polyfacial set, regular with respect to (5)
and inequalities (10). Assume φ ∈ C1
and the sewing condition (7) being fulfilled.
Let a fixed t∗ ∈ (t0, ∞] exist such that:
a) There exists a solution y of (5), (6) on [t0 − r, t∗).
b) On any interval [t0 − r, t1] ⊂ [t0 − h, t∗), t1 > t0, this solution is unique.
c) If t∗ < ∞, then ˙y(t) has not a finite limit as t → t−
∗ .
d) The solution y and ˙y depend continuously on φ.
Assume that q defines a set of initial functions for (5) with respect to ω and
Z and that the derivative of every solution x(δ, q(z))(t) of (5) defined by any
z = (δ, x) ∈ B has a finite left limit at every point t provided that
(t, x(δ, q(z))(t)) ∈ ω.
8
Let, moreover, Z ∩ W be a retract of W, but not a retract of Z. Then, there
exists at least one point z0 = (δ0, x0) ∈ Z ∩ω such that a solution x(δ0, q(z0))(t)
exists on [t0 − r, t∗) and
(t, x(δ0, q(z0))(t)) ∈ ω
holds for all t ∈ [t0 − r, t∗).
3 Main Result
In this section we give a criterion (sufficient and necessary conditions) for
the existence of a positive and strictly decreasing solution of the equation (1).
Equation (1) is a particular case of equation (5) if the functional f in the
right-hand side of (5) is specified as
f(t, φ, ˙φ) := −c(t)φ(−τ(t)) + d(t) ˙φ(−δ(t)).
Such a functional f is used in the remaining part of the paper.
Theorem 5. For the existence of a positive strictly decreasing solution of (1)
on [t0 − r, ∞), a necessary and sufficient condition is that there exists a continuous
function λ: [t0 − r, ∞) → (0, ∞) such that inequality (4) holds for t ≥ t0.
Proof. Necessity. Let a continuously differentiable positive strictly decreasing
solution y = y(t) of (1) be given on [t0 − r, ∞). From (1) we conclude
˙y(t) < 0 for every t ∈ [t0, ∞). We show that y(t) can be expressed in the form
y(t) = exp −
t
t0
λ(s)ds , t ≥ t0 − r (15)
where λ satisfies all conditions formulated in the theorem. Taking the derivative
of y, we get
˙y(t) = −λ(t) exp −
t
t0
λ(s)ds , t ≥ t0 − r (16)
and, therefore,
λ(t) := −
˙y(t)
y(t)
, t ≥ t0 − r. (17)
It can be seen from (15)-(17) that λ(t) > 0 if t ≥ t0 −r. Substitute (15) into (1),
assuming t ≥ t0, and divide the equation obtained by exp −
t
t0
λ(s)ds . We
get
λ(t) = c(t) exp
t
t−τ(t)
λ(s)ds + d(t)λ(t − δ(t)) exp
t
t−δ(t)
λ(s)ds
where t ≥ t0. This means that inequality (4) holds.
9
Sufficiency. In this part we make use of Theorem 4. The proof is divided
into five steps.
Step 1. Definition of the polyfacial set ω. We set n = p = 1, s = 0,
t∗ = ∞ and
l(t, y) = l1(t, y) = y y − ν exp −
t
t0
λ(s)ds
where y ∈ R, ν > 1 is a constant and λ satisfies inequality (4). Then, the set
ω := {(t, y) ∈ [t0 − r, ∞) × R, l(t, y) < 0} (18)
is a polyfacial set within the meaning of Definition 4 since, for every fixed
t∗
∈ [t0 − r, ∞), the set
ω ∩ {(t, y): t = t∗
, y ∈ R} = (t, y): t = t∗
, 0 < y < ν exp −
t∗
t0
λ(s)ds
is open and simply connected.
Step 2. Regularity of ω. Set q = 1. Define a function
c: [t0 − r, ∞) × R × R → R
as
c(t, y, x) = x x + νλ(t) exp −
t
t0
λ(s)ds , (19)
and identify c = c1.
We show that the set ω defined by (18) is regular with respect to equation (1)
and auxiliary inequality c(t, y, x) ≤ 0 by Definition 5. Therefore, we will verify
all its assumptions α) − δ) (denoted below as α∗
) − δ∗
)).
α∗
) If (t, φ) ∈ R × C1
and (t + θ, φ(θ)) ∈ ω for θ ∈ [−r, 0), then the functional
f is defined at (t, φ, ˙φ). Thus, point α) of Definition 5 holds.
β∗
) Let (t, φ) ∈ R × C1
, (t + θ, φ(θ)) ∈ ω for θ ∈ [−r, 0) and
c(t + θ, φ(θ), ˙φ(θ)) ≤ 0, θ ∈ [−r, 0). (20)
From (19) and (20) we get
−νλ(t + θ) exp −
t+θ
t0
λ(s)ds ≤ ˙φ(θ) ≤ 0, θ ∈ [−r, 0). (21)
In addition, we have
f(t, φ, ˙φ) = −c(t)φ(−τ(t)) + d(t) ˙φ(−δ(t)) < 0 (22)
10
since c(t) > 0 and φ(−τ(t)) > 0. Now using the definition of ω (18) and
inequalities (21), (4), we get
f(t, φ, ˙φ) = − c(t)φ(−τ(t)) + d(t) ˙φ(−δ(t))
≥ − νc(t) exp −
t−τ(t)
t0
λ(s)ds
− νd(t)λ(t − δ(t)) exp −
t−δ(t)
t0
λ(s)ds
=ν exp −
t
t0
λ(s)ds −c(t) exp
t
t−τ(t)
λ(s)ds
− d(t)λ(t − δ(t)) exp
t
t−δ(t)
λ(s)ds
≥ − νλ(t) exp −
t
t0
λ(s)ds . (23)
Combining (22) and (23), we obtain
−νλ(t) exp −
t
t0
λ(s)ds ≤ f(t, φ, ˙φ) < 0. (24)
A consequence of (24) is the inequality
c(t + θ, φ(θ), f(t, φ, ˙φ))
= f(t, φ, ˙φ) f(t, φ, ˙φ) + νλ(t) exp −
t
t0
λ(s)ds ≤ 0.
Thus, point β) of Definition 5 holds.
γ∗
) Let φ ∈ C1
([−r, 0], R) be such that (t + θ, φ(θ)) ∈ ω for θ ∈ [−r, 0) and
(t, φ(0)) ∈ ∂ω. Then, either
φ(0) = 0 (25)
or
φ(0) = ν exp −
t
t0
λ(s)ds . (26)
Moreover, we assume that (13) holds, i.e.,
c(t + θ, φ(θ), ˙φ(θ))
= ˙φ(θ) ˙φ(θ) + νλ(t + θ) exp −
t+θ
t0
λ(s)ds ≤ 0, θ ∈ [−r, 0). (27)
11
Let (25) be true. We will use the properties φ(−τ(t)) > 0 (it follows from
definition (18) of the set ω) and ˙φ(−δ(t)) ≤ 0 (it is a consequence of (27))
to get
Dl(t, y) = Dl(t, 0) =
∂l
∂t
(t, 0) +
∂l
∂y
(t, 0) · f(t, φ, ˙φ)
= −ν exp −
t
t0
λ(s)ds (−c(t)φ(−τ(t)) + d(t) ˙φ(−δ(t))) > 0.
Let (26) be true. We will use the properties
φ(−τ(t)) < ν exp −
t−τ(t)
t0
λ(s)ds
(it follows from definition (18) of the set ω) and
˙φ(−δ(t)) ≥ −νλ(t − δ(t)) exp −
t−δ(t)
t0
λ(s)ds
(it is a consequence of (27)).
Then,
Dl(t, y) = Dl t, ν exp −
t
t0
λ(s)ds
=
∂l
∂t
t, ν exp −
t
t0
λ(s)ds +
∂l
∂y
t, ν exp −
t
t0
λ(s)ds f(t, φ, ˙φ)
=ν exp −
t
t0
λ(s)ds
· νλ(t) exp −
t
t0
λ(s)ds − c(t)φ(−τ(t)) + d(t) ˙φ(−δ(t))
>ν exp −
t
t0
λ(s)ds νλ(t) exp −
t
t0
λ(s)ds
−νc(t) exp −
t−τ(t)
t0
λ(s)ds − νd(t)λ(t − δ(t)) exp −
t−δ(t)
t0
λ(s)ds
≥ν2
exp −2
t
t0
λ(s)ds λ(t) − c(t) exp
t
t−τ(t)
λ(s)ds
− d(t)λ(t − δ) exp
t
t−δ(t)
λ(s)ds ≥ [ by (4) ] ≥ 0.
12
Thus, point γ) of Definition 5 holds.
δ∗
) There is no function of the type m(t, y) in the definition (18) of polyfacial
set ω.
We conclude that the set ω defined by (18) is regular by Definition 5 with
respect to equation (1) and auxiliary inequality c(t, y, x) ≤ 0.
Step 3. Using Theorem 4 - sets W and Z. To apply Theorem 4, we define
the set W in accordance with (14) as
W := {(t, y) ∈ ∂ω : mj(t, y) < 0, j = 1, . . . , s} = {(t, y) ∈ ∂ω}
since no function of the type mj, j = 1, . . . , s is used. Moreover, define
Z := {(t, y) ∈ ω ∪ W : t = t0} = {(t0, y): y ∈ [0, 1]}.
Obviously, Z ∩ W is a retract of W, but not a retract of Z.
Step 4. Using Theorem 4 - initial functions for (1). Now we will
construct a set of initial functions for (1) with respect to ω and Z such that
every initial function φ satisfies the sewing condition (7), i.e.
S(t0, φ) = 0 (28)
where
S(t0, φ) := f(t0, φ, ˙φ) − ˙φ(0) = −c(t0)φ(−τ(t0)) + d(t0) ˙φ(−δ(t0)) − ˙φ(0).
Define for any z = (t0, y) ∈ Z (recall that y ∈ [0, 1]) two initial functions
ϕmax
y , ϕmin
y ∈ C1
[−r, 0]:
ϕmax
y (s) := ν exp −
t0+s
t0
λ(u)du − ν + y,
ϕmin
y (s) :=
1
2
ks2
+ y
where, for a constant ε ∈ (0, 1),
k :=
ε
r
· min
−r≤θ≤0
λ(t0 + θ) exp −
t0+θ
t0
λ(u)du > 0.
Obviously, ϕmax
y (0) = y, ϕmin
y (0) = y. For s ∈ [−r, 0), we prove
0 < ϕmin
y (s) < ϕmax
y (s) < ν exp −
t0+s
t0
λ(u)du . (29)
13
The left-hand inequality in (29) holds since y ∈ [0, 1] and k > 0. The right-hand
inequality in (29) holds since −ν+y < 0. To prove the middle inequality in (29),
we define a function
Ψ(s) := ϕmin
y (s) − ϕmax
y (s), s ∈ [−r, 0].
Then, for s ∈ [−r, 0).
Ψ (s) =
ε
r
s min
−r≤θ≤0
λ(t0 + θ) exp −
t0+θ
t0
λ(u)du
+ νλ(t0 + s) exp −
t0+s
t0
λ(u)du
≥ −ε min
−r≤θ≤0
λ(t0 + θ) exp −
t0+θ
t0
λ(u)du
+ νλ(t0 + s) exp −
t0+s
t0
λ(u)du > 0.
Therefore,
ϕmin
y (s) − ϕmax
y (s) = Ψ(s) < Ψ(0) = 0, s ∈ [−r, 0)
and the middle inequality in (29) is proved.
Moreover, the following chain of inequalities obviously hold
0 ≥ ˙ϕmin
y (s) = ks ≥ −kr = −ε min
−r≤θ≤0
λ(t0 + θ) exp −
t0+θ
t0
λ(u)du
> −νλ(t0 + s) exp −
t0+s
t0
λ(u)du = ˙ϕmax
y (s), s ∈ [−r, 0]. (30)
We show that the values S(t0, ϕmax
y ), S(t0, ϕmin
y ) take opposite signs. Using (4),
we get
S(t0, ϕmax
y ) = − c(t0) ν exp −
t0−τ(t0)
t0
λ(s)ds − ν + y
− νd(t0)λ(t0 − δ(t0)) exp −
t0−δ(t0)
t0
λ(s)ds + νλ(t0)
= − νc(t0) exp
t0
t0−τ(t0)
λ(s)ds
− νd(t0)λ(t0 − δ(t0)) exp
t0
t0−δ(t0)
λ(s)ds
14
+ νλ(t0) + c(t0)(ν − y) > 0, (31)
and
S(t0, ϕmin
y ) = −c(t0)
k
2
(−τ(t0))2
+ y − d(t0)kδ(t0) < 0. (32)
Define a one-parameter family of functions ϕα
y depending on a parameter α ∈
[0, 1] as
ϕα
y (s) := αϕmax
y (s) + (1 − α)ϕmin
y (s), s ∈ [−r, 0]. (33)
Then, by (31) and (32),
S(t0, ϕ0
y)S(t0, ϕ1
y) = S(t0, ϕmin
y )S(t0, ϕmax
y ) < 0.
The operator
S(t0, ϕα
y ) = −c(t0) αϕmax
y (−τ(t0)) + (1 − α)ϕmin
y (−τ(t0))
+ d(t0) α ˙ϕmax
y (−δ(t0)) + (1 − α) ˙ϕmin
y (−δ(t0)) − ˙ϕα
y (0), (34)
where
˙ϕα
y (0) = α ˙ϕmax
y (0) + (1 − α) ˙ϕmin
y (0) = −ανλ(t0),
is strongly monotone with respect to α, since, due to (4),
∂
∂α
S(t0, ϕα
y ) = −c(t0) ϕmax
y (−τ(t0)) − ϕmin
y (−τ(t0))
+ d(t0) ˙ϕmax
y (−δ(t0)) − ˙ϕmin
y (−δ(t0)) + νλ(t0)
= −c(t0) ν exp −
t0−τ(t0)
t0
λ(u)du − ν + y −
k
2
(−τ(t0))2
− y
+ d(t0) −νλ(t0 − δ(t0)) exp −
t0−δ(t0)
t0
λ(u)du − k(−δ(t0)) + νλ(t0)
= ν λ(t0) − c(t0) exp −
t0−τ(t0)
t0
λ(u)du
−d(t0)λ(t0 − δ(t0)) exp −
t0−δ(t0)
t0
λ(u)du
+ νc(t0) +
k
2
c(t0)τ2
(t0)) + kd(t0)δ(t0) > 0.
Then, there exists a unique value α = αy ∈ [0, 1] such that S(t0, ϕ
αy
y ) = 0, i.e.,
the sewing condition (28) is true. This value, as can be seen in (34), is defined
by the formula
αy =
c(t0)ϕmin
y (−τ(t0)) − d(t0) ˙ϕmin
y (−δ(t0))
c(t0)Ψ(−τ(t0)) − d(t0) ˙Ψ(−δ(t0)) + νλ(t0)
15
and depends continuously on y since ϕmin
y , ϕmax
y and Ψ depend continuously on
y. Therefore, the function
ϕαy
y (s) = αyϕmax
y (s) + (1 − αy)ϕmin
y (s)
= αy ν exp −
t0+s
t0
λ(u)du − ν + y + (1 − αy)
1
2
ks2
+ y , s ∈ [−r, 0]
is continuous with respect to y as well.
Applying (30), we see that, for any function ϕ
αy
y (s), s ∈ [−r, 0], defined by (33),
we have:
˙ϕαy
y (s) =αy ˙ϕmax
y (s) + (1 − αy) ˙ϕmin
y (s)
≤ αy ˙ϕmin
y (s) + (1 − αy) ˙ϕmin
y (s) = ˙ϕmin
y (s), s ∈ [−r, 0],
˙ϕαy
y (s) =αy ˙ϕmax
y (s) + (1 − αy) ˙ϕmin
y (s)
≥ αy ˙ϕmax
y (s) + (1 − αy) ˙ϕmax
y (s) = ˙ϕmax
y (s), s ∈ [−r, 0].
Step 5. Using Theorem 4 - initial functions for (1) and mapping q.
By Definition 6, we will construct a continuous mapping q: B → C1
where the
set B is defined in Lemma 1, point (2) and, in our case, becomes
B = Z ∩ (Z ∪ W) = Z.
Then, q maps the set Z into the space of initial functions satisfying the sewing
condition. Define such a mapping q: B → C1
[−r, 0] for every z = (t0, y) ∈ B
by the formula
q(z) = q((t0, y)) = ϕαy
y . (35)
This mapping is continuous and
(t0 + θ, q(z)(θ))
= (t0 + θ, αyϕmax
y (θ) + (1 − αy)ϕmin
y (θ)) ∈ ω for θ ∈ [−r, 0),
(t0, q(z)(0)) = (t0, αyϕmax
y (0) + (1 − αy)ϕmin
y (0)) = z.
The mapping q satisfies conditions 1) and 2a) of Definition 6. All assumptions
of Theorem 4 are now fulfilled. Therefore, there exists at least one point z0 =
(t0, y0) ∈ Z ∩ ω such that a solution x(t0, q(z0))(t) of (1) exists on [t0 − r, ∞)
and
(t, x(t0, q(z0))(t)) ∈ ω (36)
holds for all t ∈ [t0 −r, ∞). Because of the shape of ω, such a solution is positive
and, by (4), it is strictly decreasing. 2
16
Remark 2. Let all assumptions of Theorem 5 be true. From its proof
(see (36) and the definition (18) of the set ω) we deduce that, if (4) holds
for t ≥ t0, then there exist a positive strictly decreasing solution y = y(t) of (1)
on [t0 − r, ∞) satisfying the inequalities
0 < y(t) < exp −
t
t0
λ(s)ds , t ∈ [t0 − r, ∞). (37)
Moreover, from formulas (11) and (12) of Definition 5, such a solution satisfies
the inequalities
c(t, y(t), ˙y(t)) ≤ 0, t ∈ [t0 − r, ∞),
i.e.,
−λ(t) exp −
t
t0
λ(s)ds ≤ ˙y(t) ≤ 0, t ∈ [t0 − r, ∞). (38)
Due to the linearity of (1), the coefficient ν is omitted in (37) and (38).
4 Generalization
Consider an equation
˙y(t) = −
m
i=1
ci(t)y(t − τi(t)) +
r
j=1
dj(t) ˙y(t − δj(t)) (39)
where ci, dj : [t0, ∞) → [0, ∞), and τi, δj : [t0, ∞) → (0, r] are continuous functions.
Moreover, assume
m
i=1 ci(t) > 0, t ∈ [t0, ∞). Obviously, equation (39)
is more general than equation (1). Now we will formulate a generalization of
Theorem 5. We omit its proof since it is similar to that of Theorem 5. Note that
the system of initial functions can be used in the proof without any changes.
Theorem 6. For the existence of a positive strictly decreasing solution
of (39) on [t0 − r, ∞), a necessary and sufficient condition is that there exists
a continuous function λ: [t0 − r, ∞) → (0, ∞) such that the inequality
λ(t) ≥
m
i=1
ci(t)exp
t
t−τi(t)
λ(s)ds +
r
j=1
dj(t)λ(t − δj(t)) exp
t
t−δj (t)
λ(s)ds
holds for t ≥ t0. Moreover, if this inequality holds, then there exists a positive
strictly decreasing solution y = y(t) of (39) on [t0 − r, ∞) satisfying inequalities
(37) and (38).
5 Concluding discussions
From the proof of Theorem 5, we conclude that a positive solution (if inequality
(4) holds) is generated by a function from a one-parameter family of
17
functions ϕ
αy
y , defined by formula (35) where the parameter y ∈ [0, 1]. More
specifically, as it follows from points (2) and (3) of Lemma 1, we can restrict
the values of the parameter y only to values y ∈ (0, 1). In this connection, the
following open problem arises.
Open Problem 1. How to compute a value (values) of parameter y =
y∗
∈ (0, 1) such that the initial function ϕ
αy∗
y∗ determines a positive solution of
equation (1) (or (39)) indicated in Theorem 5?
A solution to this open problem can have certain importance, e.g., in numerical
computations.
Because of the linearity of considered equations and the existence of a positive
solution, we conclude that there exists a one-parameter family of linearly
dependent positive solutions of equation (1) on interval [t0 − r, ∞).
It is easy to explain, that there exists a one-parameter family of linearly
independent positive solutions of equation (1) on [t0 − r, ∞). Looking again at
the proof of Theorem 5, we emphasize that the definition of the function ϕmin
y
depends (through the constant k) on a parameter ε ∈ (0, 1). Therefore, each
function in the system of initial functions ϕ
αy
y where y ∈ (0, 1), relevant to a
choice of ε, is linearly independent on an interval [t0 − r, t0] of every function in
the system of initial functions ϕ
αy
y constructed for a different choice of ε. Consequently,
positive solutions defined by different initial functions, being linearly
independent on interval [t0 − r, t0] are linearly independent positive solutions of
equation (1) on [t0 − r, ∞). One cannot, however, conclude that such a type of
linear independence on the interval [t0 − r, ∞) implies the existence of a oneparameter
family of linearly independent positive solutions of equation (1) on
every interval [t1 −r, ∞) where t1 ≥ t0. This assertion can be wrong due to, e.g.,
the effect of solution pasting (we refer to [26, Part 3.5]). A similar discussion
applies to the function ϕmax
y and the parameter ν. Nevertheless, we formulate
the following open problem connected with this topic.
Open Problem 2. Indicate sufficient conditions for the existence of at least
a one-parameter family of linearly independent positive solutions of equation (1)
(or (39)) on every interval [t1 − r, ∞) where t1 ≥ t0.
Obviously, Theorem 5 is a generalization of Theorem 1 to neutral differential
equations. Now, we will restrict our discussion only to equation (1) and its
special cases although it is easy to formulate corresponding remarks to more
general equation (39) and its special cases.
Let the functions c(t), d(t) and delays τ(t), δ(t) in equation (1) be constant,
i.e., c(t) ≡ c = const, d(t) ≡ d = const, τ(t) ≡ τ = const, δ(t) ≡ δ = const and
equation (1) becomes
˙y(t) = −cy(t − τ) + d ˙y(t − δ). (40)
Then, Theorem 5 is formulated as
18
Theorem 7. For the existence of a positive strictly decreasing solution
of (40) on [t0 − r, ∞), a necessary and sufficient condition is that there exists
a continuous function λ: [t0 − r, ∞) → (0, ∞) such that inequality
λ(t) ≥ c exp
t
t−τ
λ(s)ds + dλ(t − δ) exp
t
t−δ
λ(s)ds (41)
holds for t ≥ t0.
From Theorem 7 and formula (41) where λ(t) ≡ λ = const, we immediately
get the following corollaries. These criteria are well-known, we refer, e.g., to [24,
Theorem 5.2.10, Corollary 5.2.11], [25, Theorem 6.7.1]. Similar criteria can be
found, e.g., in [1, Corollary 6.5], [2, Theorem 3.5.3] and [23, Theorem 3.2.3].
Corollary 1. For the existence of a positive strictly decreasing solution
of (40) on [t0 − r, ∞) it is sufficient the existence of a positive constant λ
such that inequality
λ ≥ ceλτ
+ λdeλδ
(42)
holds.
For the choice λ = 1/τ or λ = 1/δ in (42), we get
Corollary 2. For the existence of a positive strictly decreasing solution
of (40) on [t0 − r, ∞) it is sufficient that either inequality
1 > ceτ + deδ/τ
(43)
or inequality
1 > cδeτ/δ
+ de (44)
hold.
Corollaries 1, 2 can be improved in view of Remark 2 (formulas (37), (38))
in the sense that if inequalities (42), (43), (44) are valid, then on [t0, ∞) there
exist a positive solution vanishing for t → ∞ and having negative and vanishing
for t → ∞ continuous derivative.
Remark 3. In the paper we regard solutions of equation (1) as continuously
differentiable functions satisfying the given equation everywhere. As noted, e.g.,
in [28, p. 107] it leads to some complications, since the sewing condition must be
valid for continuously differentiable initial functions. In the proof of Theorem 5,
a modification of the retract principle suitable for neutral differential equations
was used. This principle, to be successfully applied, needs not only one initial
function, but a whole family of initial functions satisfying the sewing condition.
Therefore, the crucial moment of the proof was a special construction of such a
family of initial functions.
To compare our results with, e.g., those given in [1, Theorem 6.1] we emphasize
that the definition of a solution substantially differs (a solution is defined as
an absolutely continuous function satisfying the equation almost everywhere).
19
In [1, 3, 23, 24, 25] part of the results is devoted to the existence of positive
solutions of neutral equations having, e.g., the form
(y(t) + P(t)y(t − τ)) + Q(t)y(t − σ) = 0, t ≥ t0
under various conditions for P and Q. The substantial difference is that the
delays in the equation, unlike those in our investigation, are constant. Thus,
the results derived in the cited sources are, in principle, not applicable to equation
(1).
Acknowledgements
The authors greatly appreciate the work of the anonymous referee, whose
comments and suggestions have helped to improve the paper in many aspects.
The first author was supported by the Czech Science Foundation under the
project 16-08549S. The research of the second author was carried out under the
project CEITEC 2020 (LQ1601) with financial support from the Ministry of
Education, Youth and Sports of the Czech Republic under the National Sustainability
Programme II. The work of the first author was realized in CEITEC
- Central European Institute of Technology with research infrastructure supported
by the project CZ.1.05/1.1.00/02.0068 financed from European Regional
Development Fund.
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22
137 (2012) MATHEMATICA BOHEMICA No. 2, 239–248
ASYMPTOTIC PROPERTIES OF ONE DIFFERENTIAL EQUATION
WITH UNBOUNDED DELAY
Zdeněk Svoboda, Brno
(Received October 15, 2009)
Abstract. We study the asymptotic behavior of the solutions of a differential equation
with unbounded delay. The results presented are based on the first Lyapunov method,
which is often used to construct solutions of ordinary differential equations in the form of
power series. This technique cannot be applied to delayed equations and hence we express
the solution as an asymptotic expansion. The existence of a solution is proved by the retract
method.
Keywords: asymptotic expansion, retract method
MSC 2010: 34E05
1. Introduction
The first method of Lyapunov is a well known technique used to study the asymptotic
behavior of ordinary differential equations in the form of a linear system with
perturbation. This method uses the solution in the form of a convergent power
series, for details see [1]. The results for equations in the implicit form [2] or for
integro-differential equations [8] were derived by modifying the first method of Lyapunov.
The existence of solutions with a certain asymptotic form were proved in the
results cited using Wa˙zewski’s topological method. For analogous representations
of solutions for a retarded differential equation, see [6], [7]. The perturbation has a
polynomial form in both cases. In this paper, we study an equation in the form
(1.1) ˙y(t) = −a(t)y(t) +
∞
|i|=2
ci(t)
n
j=1
y(ξj(t))
ij
239
where i = (i1, . . . , in) is a multiindex, ij 0 are integers and |i| =
n
j=1
ij. The
continuous functions ξj(t) satisfy t > ξj(t) r0 for all t ∈ [t0, ∞) and the function
ξ(t), which is defined as ξ(t) = min
1 i n
ξi(t), is nondecreasing for t t0. Therefore, all
asymptotic relations such as the Landau symbols o, O and the asymptotic equivalence
∼ will be considered for t → ∞. This fact will not be pointed out in the sequel.
The function a(t) satisfies the following conditions:
(C1) a(t) is continuous and positive on the interval [t0, ∞) and 1/a(t) = O(1),
(C2) (t − ξ(t))a(t) = o (A(t)) where the functions A(t), a(t) are defined as A(t) =
t
t0
a(u) du, a(t) = max
u t
(a(u)).
Further conditions for continuous functions ci(t): [t0, ∞) → R will be given later.
In order to apply the first method of Lyapunov to the equation (1.1) we assume the
solution in the form of a formal series
(1.2) y(t, C) =
∞
n=1
fn(t)ϕn
(t, C)
where ϕ(t, C) is the solution of the homogeneous equation ˙y(t) = −a(t)y(t) given by
the formula ϕ(t, C) = C exp(−A(t)), the function f1(t) ≡ 1, and the functions fk(t)
for k = 2, . . . , n are particular solutions of a certain system of auxiliary differential
equations. Using Wa˙zewski’s topological method in the form as used in [3] and [4]
for differential equations with unbounded delay and finite memory, we prove the
existence of a solution yn(t, C) ∼ Yn(t, C) =
n
k=1
fk(t)ϕk
(t, C).
2. Preliminaries
Lemma 2.1. Let a function a(t) satisfy conditions (C1), (C2). Then
(2.1) A(t) ∼ A(ξi
(t)) as t → ∞ for any integer i ∈ N
where ξ1
(t) = ξ(t), and for i > 1, the functions ξi
(t) are defined by
ξi+1
(t) = ξ(ξi
(t)).
P r o o f. First, we see that, by virtue of condition (C2), the assertion is true for
i = 1:
t
ξ(t)
a(u) du (t − ξ(t))a(t) = o(A(t)) and lim
t→∞
A(ξ(t))
A(t)
= 1 − lim
t→∞
t
ξ(t)
a(u) du
A(t)
= 1.
240
The assumption ξ(t) → ∞ for t → ∞ implies that there exists a constant ξ(∞) and
condition (C2) is not satisfied. If ξ(t) → ∞ for t → ∞, then ξi
(t) → ∞ for t → ∞,
too. Now we use the assertion for i = 1 substituting ξi
(t) for t and the proof follows
by induction.
R e m a r k 2.1. Note that condition (C1) implies the divergence of the integral
∞
t0
a(u) du, which has two consequences.
First, the function ϕ(t, C) satisfies the relation ϕk
(t, C) = o ϕl
(t, C) for k > l,
which guarantees that the sequence {ϕn
(t, C)}∞
n=1 is asymptotic.
Second, the divergence implies the relation 1/A(t) = o(1) which is suitable for
asymptotic estimation.
In order to specify the asymptotic behavior of the solution of the auxiliary equations
we consider the equation
(2.2) ˙y(t) = na(t)y(t) + f(t)
where n > 0 is a constant and the properties of the function f(t) are described by a
function k(t), a constant K, and the relations
(F1) lim
t→∞
f(t) exp (τk(t)) = 0 for all τ < K,
(F2) lim
t→∞
|f(t)| exp (τk(t)) = ∞ for all τ > K.
The asymptotic behavior of the solution of equation (2.2) depends on the relation
between the functions k(t) and na(t).
Lemma 2.2. Let either k(s) − k(t) = o(
s
t
na(u) du) or k(s) − k(t) =
O(
s
t na(u) du) and K = 0 where K is the constant used in assumptions (F1),
(F2). Now if the function f(t) satisfies assumption (F1), then there exists at least
one solution Y (t) of equation (2.2) satisfying also assumption (F1). If the function
f(t), moreover, satisfies assumption (F2), then the solution Y (t) also satisfies
assumption (F2).
P r o o f. We may rewrite assumptions (F1), (F2) for the function f(t) satisfying
them so that, for sufficiently large t and constants τ1, τ2 > 0, the function f(t)
satisfies the inequality
exp ((K − τ2)k(t)) |f(t)| exp ((K + τ1)k(t)) ,
and also, for the desired solution Y (t) =
∞
t
−f(s) exp
s
t
−na(u) du ds, we have
estimates of the solution of equation (2.2)
exp((K + τ1)k(t))
∞
t
exp − (K + τ1)(k(t) − k(s)) −
s
t
na(u) du ds |Y (t)|
exp((K − τ2)K(t))
∞
t
exp − (K − τ2)τ(k(s) − k(t)) −
s
t
na(u) du ds.
241
Now utilizing the assumptions of this lemma, we see that the asymptotic behavior
of exponents involved in both integrands are the same as the asymptotic behavior
of the function
s
t
na(u) du. As the function (na(t))−1
is bounded, the integral
s
t
na(u) du is divergent for s → ∞ and the integrals on both sides of the inequalities
are convergent and there exist constants A1, A2 such that
A1 exp ((K − τ2)k(t)) |Y (t)| A2 exp ((K + τ1)k(t)) .
Assumption (F1) implies the second inequality, which ensures the convergence and
thus the existence of the integral defining Y (t) which is the solution of the given
equation.
To make the specification of the coefficients of the power series which is the product
of the power series raised to a power easier, we use the following notation: s =
(s1, . . . , sn) is an ordered n-tuple of sequences sj = sk
j
∞
k=1
of nonnegative integers
with a finite sum |sj| =
∞
k=1
sk
j , and we denote s! =
n
j=1
∞
k=1
sk
j !, i(s)! =
n
j=1
|si|!, V (s) =
n
j=1
∞
k=1
ksk
j , i(s) = (|s1|, . . . , |sn|). For any ordered n-tuple of sequences (of numbers
or functions) C = (c1, . . . , cn) where cj = {ck
j }∞
k=1, we denote Cs
=
n
j=1
∞
k=1
ck
j
sk
j
where ck
j
0
= 1 for every ck
j . Then it is possible to write
n
j=1
∞
k=1
ck
j xk
ij
=
∞
k=|i|
xk
i(s)=i
V (s)=k
i(s)!
s!
Cs
where the symbol
i(s)=i
V (s)=k
denotes the sum over all s such that V (s) = k, i(s) = i and,
for empty set of s, this symbol equals 0.
3. Main results
We assume that the formal solution of equation (1.1) is expressed in the form (1.2)
where ϕ(t, C) is the general solution of the equation ˙y(t) = −a(t)y(t). Consequently,
ϕ(t, C) = C exp(−A(t)) where C = 0 is a constant, f1(t) = 1 and fk(t), k 2 for the
time being are unknown functions. Substituting y(t) in equation (1.1) and matching
the coefficients at the same powers ϕk
(t, C), we obtain an auxiliary system of linear
differential equations
(3.1) ˙fk(t) = (k − 1)a(t)fk(t) +
∞
|i|=2
ci(t)
i(s)=i
V (s)=k
i(s)!
s!
Fs
242
where F(t) is the n-tuple of sequences {fk(ξi(t)) exp (k(A(t) − A(ξi(t))))}
∞
k=1 i.e.
F(t) = . . . {fk(ξi(t)) exp (k(A(t) − A(ξi(t))))}
∞
k=1 , . . .). The facts V (s) = k 2 and
|i(s)| 2 imply sl
i = 0 for l k. Moreover, the auxiliary system (3.1) is recurrent.
Theorem 3.1. For the functions ci(t), let lim
t→∞
ci(t) exp(−τA(t)) = 0 for all
positive τ. Then there exists a sequence {fk(t)}∞
k=1 of solutions of the auxiliary
system (3.1)
(3.2) fk(t) =
∞
t
−a(s) exp −
s
t
(k − 1)a(u) du
∞
|i|=2
ci(t)
i(s)=i
V (s)=k
|i(s)|!
i(s)!
Fs
ds
such that lim
t→∞
fk(t) exp(−τA(t)) = 0 for all τ.
P r o o f. Formula (3.2) can be obtained by integrating the system (3.1). When
applying Lemma 2.2, we put k(t) = A(t). Condition (C2) proves that for the function
y(t) satisfying assumption (F1) of Lemma 2.2, the function y(ξj
(t)) satisfies this
assumption, too. Therefore, the sum and the product of functions verifying assumption
(F1) of Lemma 2.1 satisfy the assumptions of Lemma 2.2. Using Lemma 2.2,
we can then easily show the convergence of (3.2) and the desired property.
R e m a r k 3.1. An assertion analogous to the one of Theorem 3.1 with the property
described by assumption (F2) of Lemma 2.2 cannot be proved as the sum of
functions verifying the assumption (F2) need not satisfy this assumption.
Let · denote the maximum norm on C0
[r∗
, t0]. Moreover, we denote
yk(t) =
k
l=1
fl(t)ϕl
(t, C),
k
(t) =
∞
|i|=2
ci(t)
i(α)=i
V (α)=k
i(α)!
α!
Fα
.
Theorem 3.2. Let the assumptions of Theorem 3.1 hold and let
lim
t→∞
f−1
k+1(t) exp(−τA(t)) = 0
where τ < 1 is a constant. We denote r∗
= min
t t0
(ξ(t)). Then for every C = 0 and
ψ ∈ C0
[r∗
, t0], ψ 1, ψ(t0) = 0, there exists a solution yC(t) of equation (1.1) such
that
(3.3) |yC(t) − yk(t)| σ|fk+1(t)ϕk+1
(t, C)|
for t ∈ [tC, ∞) where the functions fk(t) are solutions (3.2) of system (3.1), σ > 1 is
a constant. tC is a function of the parameter C and of σ, k.
243
P r o o f. The existence of the solution yC(t) is proved by Theorem 1 in [3], which
is based on the retract method and the second method of Lyapunov. A sufficient
condition for the existence of a solution of the equation with unbounded delay and
finite memory is described there. The theory of this type of equations (referred to
as p-type retarded functional differential equation) is given in [5]. In this case we
put p(t, ϑ) = t + ϑ(t − ψ(t)) and the function on the right hand side of the equation
f(t, yt): R × C0
[−1, 0] → R is defined by the formula:
f(t, ψ) = −a(t)ψ(p(t, 0)) +
∞
|i|=2
ci(t)
n
l=1
ψil
(p(t, ϑil
(t)))
where ϑil
(t) = −(t − ξil
(t))/(t − ξ(t)). The set ω used in Theorem 1 is defined as
ω = {(y, t): yk(t) − σ|fk+1|(t)ϕk+1
< y < yk(t) + σ|fk+1(t)|ϕk+1
, t > tC}.
Note that the numbers p, n used in Theorem 1 in [3] equal 1 and, consequently,
the indices of functions δ, ̺ are omitted, i.e., δ = yk(t) + σ|fk+1|(t)ϕk+1
(t, C) and
̺ = yk(t) − σ|fk+1|(t)ϕk+1
(t, C). We verify the inequalities
δ′
(t) > f(t, π) and ̺′
(t) < f(t, π)
where π ∈ C([p(t, −1), t], R) is such that (θ, π(θ)) ∈ ω for all θ ∈ [p(t, −1), t) and
π(t) = δ(t) or π(t) = ̺(t), respectively, for a sufficiently large t. As the sequence
{ϕk
(t, C)}∞
k=1 is asymptotic, we can rearrange the terms in these inequalities with
respect to the powers of the functions ϕk
(t, C). We verify the first inequality.
First, for sufficiently large t, fk+1ϕk+1
(t, C) = 0 and the derivative δ′
(t) exists:
δ′
(t) =
k
l=1
(f′
l (y) − la(t)fl(t)) ϕl
(t, C)
+ σ sign(fk+1(t)) f′
k+1(t) − (k + 1)a(t)fk+1(t) ϕk+1
(t, C).
Second, for π(t) = δ(t) there exist suitable positive constants such that
f(t, πt) = − a(t) yk(t) + σ|fk+1(t)|ϕk+1
(t, C)
+
∞
|i|=2
ci(t)
n
l=1
yk(t) + Klσ|fk+1(t)|ϕk+1
(t, C)
il
.
Since the system (3.1) is recurrent, the coefficients at ϕl
(t, C) after substituting
y(t, C) in the form (1.2) and y(t) = yk(t) ± σ|fk+1|(t)ϕk+1
(t, C) in the sum
244
∞
|i|=2
ci(t) y(ξ(t))
i
coincide for l = 1, . . . , k + 1, i.e.
f(t, πt) = − a(t)
k
l=1
fl(t)ϕl
(t, C) + σ|fk+1|(t)ϕk+1
(t, C)
+
k+1
j=1
j
(t)ϕ(t, C)j
+ ϕ(t, C)k+2
R(t)
where R(t) is a function satisfying lim
t→∞
R(t) exp(−τ
t
t0
du/g(u)) = 0 for all positive
τ.
Now we can evaluate the sign of the difference δ′
(t) − f(t, πt) (with π(t) = δ(t)):
δ′
(t) − f(t, πt) =
k
l=1
f′
l (y) −
(l − 1)fl(t)
g(t)
−
l
(t) ϕl
(t, C)
+ σ sign(fk+1(t)) f′
k+1(t) −
kfk+1(t)
g(t)
−
k+1
(t) ϕk+1
(t, C) − ϕ(t, C)k+2
R(t).
The functions fk(t) are solutions of (3.1) for l = 1, . . . , k. Therefore, the minimal
power of ϕ(t, C) in the difference δ′
(t) − f(t, πt) is k + 1. Moreover, the term
ϕ(t, C)k+2
R(t) and higher powers are very small for sufficiently large t, the sign of
this difference is given by the factor at the power ϕ(t, C)k+1
, i.e.
sign(δ′
(t) − f(t, πt)) = σ sign(fk+1(t)) f′
k+1(t) −
kfk+1(t)
g(t)
−
k+1
(t)
= σ sign(fk+1(t))
k+1
(t) −
k+1
(t) = σ sign(fk+1(t))
k+1
(t).
Due to definition (3.2) of fk+1(t), we obtain sign(δ′
(t) − f(t, πt)) = −1 and the
inequality δ′
(t) > f(t, πt) holds, too. A similar consideration for the difference ̺′
(t)−
f(t, πt) (with π(t) = ̺(t)) gives ̺′
(t) < f(t, πt). Now we may use Theorem 1 in [3]
to obtain the existence of a solution satisfying the estimate (3.6).
Theorem 3.3. Let the assumptions of Theorem 3.1 be satisfied and let there exist
a sequence {Kk}∞
k=1, K0 = 1 such that the assumptions of Theorem 3.2 are satisfied
for every Kk, i.e., lim
t→∞
f−1
Kk
(t) exp(−τA(t)) = 0. Then there exists an asymptotic
expansion of the solution yC(t) in the form
yC(t) ≈
∞
k=1
Fk(t), where Fk(t) =
Kk−1
l=Kk−1
fl(t)ϕl
(t, C)
and fl(t) are solutions of (3.2).
245
P r o o f. Since the assumptions of Theorem 3.2 are fulfilled for every Kk, there
exists a solution yC(t) satisfying the inequality in this theorem. Then the existence
of an asymptotic expansion follows from the fact that the sequence {Fk}∞
is asymptotic,
i.e., lim
t→∞
Fk+1(t)/Fk(t) = 0 and the assertion is proved.
E x a m p l e 1. We study the asymptotic properties of the solutions of the equation
˙y(t) = −y cos(ty(ξ(t)) = −y(t) +
∞
k=1
(−1)k+1 t2k
y(t)(y(ξ(t)))2k
(2k)!
on the interval [1, ∞) for two various delays r1(t) = r > 0, i.e., ξ1(t) = t − r, and
r2(t) = ln t, i.e., ξ2(t) = t − ln t. In this case we have a(t) = 1, A(t) = t − 1,
s = (s1, s2), c(1,2k) = (−1)k+1
t2k
/(2k)! (for other multiindices ci = 0). If we denote
F = ({fi(t)}∞
i=1, {fi(ξ(t))ei(t−ξ(t))
}∞
i=1), the system of auxiliary differential equations
of the form
˙fk(t) = (k − 1)fk(t) +
∞
i=1
(−1)i+1 t2i
(2i)!
hi(s)=(1,2i)
V (s)=k
i(s)!
s!
Fs
has a particular solution f2k = 0. First, f2(t) = 0 is due to ˙f2(t) = f2(t). We will
prove by induction that the equation for the function f2k has the form ˙f2k(t) = f2k(t),
therefore, the odd (|i(s)| = 1 + 2l) sum of odd exponents (due to the induction
hypothesis) is not even (2k) and every product on the right-hand side of the auxiliary
equation contains zero multiplicands (f2i). The asymptotic form of the solutions
f2k+1 depends on the delay ri(t) but the property f2k−1(t) ∼ f2k−1(ξ(t)) holds for
both ri(t).
First, for r1(t) the solutions have the asymptotic form f2k+1 = t2k
(c2k+1 +O(1/t)),
where c1 = 1 and c2k+1 are given by the recurrent formula
c2k+1 =
1
2k
∞
i=1
(−1)i
(2i)! i(s)=(1,2i)
V (s)=2k+1
Cs1
Cs2
r , where C = {ci}∞
i=1, Cr = {ci exp(ir)}∞
i=1.
Second, we have the relation exp(k(A(t) − A(ξ(t)))) = exp(k ln t) = tk
for the
delay r2(t) and the function f3 satisfies the equation ˙f3(t) = 2f3(t) + 1
2 t4
and we
obtain the solution f3(t) = t4
(−1
4 + O(1/t)). Applying induction for the solutions
f2k+1 in the form f2k−1(t) = tp(k)
(d(k) + O(1/t)), we see that the main power
of t in the sum on the right hand side of the equation for f2k−1 is at the product
t2
f1(t)f1(ξ(t))tf2k−3(ξ(t))t2k−3
= t2k+p(k−1)
(d(k − 1) + O(1/t)) and we obtain the
equation ˙f2k+1(t) = 2kf2k+1(t)+t2k+p(k−1)
(d(k−1)+O(1/t)). The solution f2k−1(t)
246
has the asymptotic form f2k+1 = −t2k+p(k−1)
(d(k − 1)/2k + O(1/t)) . The constants
d(k) and p(k) satisfy the recurrent formulas d(k) = −d(k−1)/2k, p(k) = p(k−1)+2k,
otherwise d(k) = (−1)k−1
2−k
/(k − 1)! and p(k) = (k + 2)(k − 1). By Theorem 3.3,
we obtain the existence of a pair of asymptotic expansions y1(t), y2(t) of the solutions
for two different delays r1(t), r2(t):
y1(t) ≈
∞
k=1
t2(k−1)
c2k−1e(2k−1)t
C2k−1
,
y2(t) ≈
∞
k=1
(−1)k−1
t(k+2)(k−1)
2k(k − 1)!
e(2k−1)t
C2k−1
.
R e m a r k 3.2. This example shows a fundamental dependence of the asymptotic
properties of the expansion on the magnitude of the delay. For a small delay
(r1(t) → 0), the expansion y1(t) converges to the expansion of the solution of an
ordinary equation ˙y(t) = −y cos(ty(t)). For a sufficiently large delay r2(t) = ln(t),
the expansion y2(t) is the same as for the equation ˙y(t) = −y(t)+t2
y(t)y2
(t−ln t)/2,
i.e., the expansions for the perturbation with infinite sum and for the perturbation
with only the first summand are the same.
A c k n o w l e d g e m e n t. This paper was supported by the Grant 201/08/0469
of the Czech Grant Agency (Prague) and by the Czech Ministry of Education in
the frame of project MSM002160503 Research Intention MIKROSYN New Trends
in Microelectronic Systems and Nanotechnologies.
References
[1] L. Cezari: Asymptotic Behaviour and Stability Problems in Ordinary Differenal Equations.
Springer, 1959.
[2] J. Diblík: Asymptotic behavior of solutions of a differential equation partially solved
with respect to the derivative. Sib. Math. J. 23 (1983), 654–662 (In English. Russian
original.); translation from Sib. Mat. Zh. 23 (1982), 80–91. zbl
[3] J. Diblík, Z. Svoboda: An existence criterion of positive solutions of p-type retarded functional
differential equations. J. Comput. Appl. Math. 147 (2002), 315–331. zbl
[4] J. Diblík, N. Koksch: Existence of global solutions of delayed differential equations via
retract approach. Nonlinear Anal., Theory Methods Appl. 64 (2006), 1153–1170. zbl
[5] V. Lakshmikantham, L. Wen, B. Zhang: Theory of Differential Equations with Unbounded
Delay. Kluwer Academic Publishers, Dordrecht, 1994. zbl
[6] Z. Svoboda: Asymptotic behaviour of solutions of a delayed differential equation. Demonstr.
Math. 28 (1995), 9–18. zbl
[7] Z. Svoboda: Asymptotic integration of solutions of a delayed differential equation.
Sborník VA Brno řada B 1 (1998), 7–18.
247
[8] Z. Šmarda: The existence and asymptotic behaviour of solutions of certain class of the
integro-differential equations. Arch. Math., Brno 26 (1990), 7–18. zbl
Author’s address: Zdeněk Svoboda, Faculty of Electrical Engineering and Communication,
Brno University of Technology, Brno, Czech Republic, e-mail: svobodaz@feec.
vutbr.cz.
248
UDC 517.9
REPRESENTATION OF SOLUTIONS OF LINEAR DIFFERENTIAL
SYSTEMS OF SECOND-ORDER WITH CONSTANT DELAYS*
ЗОБРАЖЕННЯ РОЗВ’ЯЗКIВ ЛIНIЙНИХ ДИФЕРЕНЦIАЛЬНИХ
СИСТЕМ ДРУГОГО ПОРЯДКУ IЗ СТАЛИМИ ЗАПIЗНЕННЯМИ
Z. Svoboda
Brno Univ. Technology
Brno, Czech Republic
e-mail: svobodaz@feec.vutbr.cz
We derive representations for solutions to initial-value problems for n-dimensional second-order differential
equations with delays,
x (t) = 2Ax (t − τ) − (A2
+ B2
)x(t − 2τ),
and
x (t) = (A + B)x (t − τ) − ABx(t − 2τ),
by means of special matrix delayed functions. Here A and B are commuting (n × n)-matrices and τ > 0.
Moreover, a formula connecting delayed matrix exponential with delayed matrix sine and delayed matrix
cosine is derived. We also discuss common features of the two considered equations.
Знайдено зображення розв’язкiв задач iз початковими умовами для диференцiальних рiвнянь
другого порядку розмiрностi n iз запiзненнями
x (t) = 2Ax (t − τ) − (A2
+ B2
)x(t − 2τ)
та
x (t) = (A + B)x (t − τ) − ABx(t − 2τ),
при цьому використано спецiальнi матричнi функцiї iз запiзненням. Тут A i B — комутативнi
матрицi розмiрностi n × n i τ > 0. Також отримано формулу, що зв’язує експоненцiальну
матрицю з запiзненням з sin- та cos-матрицями iз запiзненням. Також розглянуто загальнi властивостi
обох розглядуваних рiвнянь.
1. Introduction. Recently, much attention was paid to a new formalization of the well-known
method of steps in the theory of linear differential equations with constant coefficients and a
single delay. Such a formulation was given in [1, 2] utilizing what is called a delayed matrix
exponential, which is a matrix polynomial on every interval. After papers [1, 2] were published,
this formalization was widely applied, e.g., in boundary-value problems, control problems and
stability problems, modification to discrete equations was performed, generalizations to the
case of several delays were developed, etc. (see [3 – 27]). Some of these results are collected in
the book [28].
We recall the definition of a delayed matrix exponential. Let (n × n)-matrices Θ, I and A
be the zero matrix, the unit matrix, and a general constant matrix, respectively and θ be the
∗
The paper was supported by the grant FEKT-S-14-2200 of Faculty of Electrical Engineering and Communication,
Brno University of Technology.
c Z. Svoboda, 2016
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130 Z. SVOBODA
(n × 1)-zero vector. Let τ > 0. A delayed matrix exponential eAt
τ , t ∈ R, is defined as
eAt
τ =
t/τ +1
s=0
As (t − (s − 1)τ)s
s!
, (1)
where · is the floor function. The delayed matrix exponential equals the unit matrix on [−τ, 0]
and represents a fundamental matrix of a homogeneous linear system with a single delay,
˙x(t) = Ax(t − τ). (2)
In [1], a representation of solution of the Cauchy initial problem (2), (3), where
x(t) = ϕ(t), −τ ≤ t ≤ 0, (3)
and ϕ: [−τ, 0] → Rn is continuously differentiable, is given in the integral form
x(t) = eAt
τ ϕ(−τ) +
0
−τ
eA(t−τ−s)
τ ϕ (s) ds. (4)
The advantage of the representation formula (4), as compared with the well-known representation
formulas (e.g. [29 – 32]), consists in that it uses explicitly the given fundamental
matrix (1) and, consequently, provides us with an explicit analytical formula for a solution of
problem (2), (3).
The purpose of the present paper is to give representations of solutions to two initial-value
problems. The first one is
x (t) − 2Ax (t − τ) + (A2
+ B2
)x(t − 2τ) = θ, t ≥ τ, (5)
x(i)
(t) = ξ(i)
(t), i = 0, 1, t ∈ [−τ, τ], (6)
where the (n×n)-matrices A, B commute, i.e., AB = BA, the matrix B is regular, and function
ξ : [−τ, τ] → Rn is assumed to be twice continuously differentiable.
The second one is the problem (6), (7) where
x (t) − (A + B)x (t − τ) + ABx(t − 2τ) = θ, t ≥ τ, (7)
with the matrices A and B commuting but regularity of B is not assumed.
The paper is organized as follows. A representation of the solution to the problem (5), (6) is
developed in Section 2 while the problem (6), (7) is considered in Section 3. The last Section 4 is
devoted to some relations between special matrix functions describing some common features
of the considered problems.
2. Representation of the solution to problem (5), (6). Consider a linear system,
z (t) = Cz(t − τ), t ≥ 0, (8)
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REPRESENTATION OF SOLUTIONS OF LINEAR DIFFERENTIAL SYSTEMS . . . 131
where C is a (2n × 2n)-matrix defined by (n × n)-commuting matrices A and B as
C :=
A B
−B A
(9)
and z is a (2n × 1)-vector. Let z =
x
y
where x, y are (n × 1)-vectors.
We show that, if the vector-valued function z : [−τ, ∞) → R2n is a solution to system (8)
on the interval [0, ∞), then the vector-valued function x: [−τ, ∞) → Rn is a solution to the
second-order system (5) on the interval [τ, ∞). This follows from the following transformations.
The system (8) can be written as
x (t) = Ax(t − τ) + By(t − τ),
(10)
y (t) = −Bx(t − τ) + Ay(t − τ),
where t ≥ 0 and
Ax (t) − By (t) = (A2
+ B2
)x(t − τ). (11)
Differentiating (10) and using (11), we derive
x (t) = Ax (t − τ) + By (t − τ) = 2Ax (t − r) − Ax (t − τ) + By (t − τ) =
= 2Ax (t − τ) − (A2
+ B2
)x(t − 2τ). (12)
Obviously, (12) is equivalent to (5). Comparing the domains of z and x we see that the above
statement holds. The connection between systems (8) and (5) is used to prove the following
result.
Theorem 1. Let AB = BA and the matrix B be invertible. Then the solution of the initialvalue
problem (5), (6) can be expressed as
x(t) = Re e(A+iB)t
τ − Im e(A+iB)t
τ B−1
A ξ(−τ) + Im e(A+iB)t
τ B−1
ξ (0)+
+
0
−τ
Re e(A+iB)(t−τ−s)
τ ξ (s)+
+ Im e(A+iB)(t−τ−s)
τ B−1
(ξ (s + τ) − Aξ (s)) ds (13)
where t ≥ τ.
Proof. The strategy of the proof is the following. We will find a solution of a related initialvalue
problem for the system (8) in a suitable form. Then separating components for the vector
x, we will get the representation (13).
First, we compute the powers Ck, k ∈ N. Let us represent the matrix C as
C = A2nI2n + B2nJ2n,
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132 Z. SVOBODA
where
A2n :=
A Θ
Θ A
, I2n :=
I Θ
Θ I
, J2n :=
Θ I
−I Θ
, B2n :=
B Θ
Θ B
are (2n×2n)-matrices (note that the matrix J2
2n = J2n ×J2n can be viewed as a matrix analogue
to the complex unit since J2
2n = −I2n). Then
Ck
= (A2nI2n + B2nJ2n)k
=
k
s=0
k
s
As
2nIs
2nBk−s
2n Jk−s
2n =
= I2nRe (A2n + iB2n)k
+ J2nIm (A2n + iB2n)k
,
where i is the imaginary unit. This relation can easily be verified if we show that the general
terms on both sides are identical, i.e., if
k
s
As
2nIs
2nBk−s
2n Jk−s
2n = I2nRe
k
s
As
2nIs
2nik−s
Bk−s
2n + J2nIm
k
s
As
2nIs
2nik−s
Bk−s
2n
or
As
2nBk−s
2n Jk−s
2n = Re As
2nik−s
Bk−s
2n + J2nIm As
2nik−s
Bk−s
2n .
In each of the four possible cases, i.e., for
Jk−s
2n = J2n and ik−s
= i,
Jk−s
2n = −I2n and ik−s
= −1,
Jk−s
2n = −J2n and ik−s
= −i,
Jk−s
2n = I2n and ik−s
= 1,
we get an equality. From the definition of the delayed matrix exponential (1), we deduce
eCt
τ =
t/τ +1
s=0
Cs (t − (s − 1)τ)s
s!
=
=
t/τ +1
s=0
(I2nRe (A2n + iB2n)s
+ J2nIm(A2n + iB2n)s
)
(t − (s − 1)τ)s
s!
=
=
t/τ +1
s=0
Re (A2n + iB2n)s (t − (s − 1)τ)s
s!
+
+
t/τ +1
s=0
J2nIm (A2n + iB2n)s (t − (s − 1)τ)s
s!
=
=
t/τ +1
s=0
Re
A + iB Θ
Θ A + iB
s
(t − (s − 1)τ)s
s!
+
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REPRESENTATION OF SOLUTIONS OF LINEAR DIFFERENTIAL SYSTEMS . . . 133
+
t/τ +1
s=0
J2nIm
A + iB Θ
Θ A + iB
s
(t − (s − 1)τ)s
s!
=
=
t/τ +1
s=0
Re
(A + iB)s Θ
Θ (A + iB)s
(t − (s − 1)τ)s
s!
+
+
t/τ +1
s=0
J2n Im
(A + iB)s Θ
Θ (A + iB)s
(t − (s − 1)τ)s
s!
=
= Re
t/τ +1
s=0
(A + iB)s Θ
Θ (A + iB)s
(t − (s − 1)τ)s
s!
+
+ Im
t/τ +1
s=0
Θ (A + iB)s
−(A + iB)s Θ
(t − (s − 1)τ)s
s!
=
=
Re e
(A+iB)t
τ Im e
(A+iB)t
τ
−Im e
(A+iB)t
τ Re e
(A+iB)t
τ
. (14)
Now consider the initial-value problem
z(t) = ϕ∗
(t), −τ ≤ t ≤ 0,
for system (8), related to system (5) where the function
ϕ∗
=
ϕ∗
x
ϕ∗
y
: [−τ, 0] → R2n
is continuously differentiable as specified below. Since z =
x
y
, we set ϕ∗
x(t) ≡ ξ(t), t ∈
∈ [−τ, 0]. Next, we will specify ϕ∗
y. From (10), due to invertibility of the matrix B, we get
y(t − τ) = B−1
(x (t) − Ax(t − τ)), t ≥ 0,
or
y(t) = B−1
(x (t + τ) − Ax(t)), t ≥ −τ.
Consequently,
ϕ∗
y(t) ≡ B−1
(ξ (t + τ) − Aξ(t)), t ∈ [−τ, 0].
Now we utilize the formula (4) where the matrix A is replaced with C, and ϕ with ϕ∗. Utilizing
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134 Z. SVOBODA
(14), we get
z(t) = eCt
τ ϕ∗
(−τ) +
0
−τ
eC(t−τ−s)
τ ϕ∗
(s) ds =
=
Re e
(A+iB)t
τ Im e
(A+iB)t
τ
−Im e
(A+iB)t
τ Re e
(A+iB)t
τ
ξ(−τ)
B−1(ξ (0) − Aξ(−τ))
+
+
0
−τ
Re e
(A+iB)(t−τ−s)
τ Im e
(A+iB)(t−τ−s)
τ
−Im e
(A+iB)(t−τ−s)
τ Re e
(A+iB)(t−τ−s)
τ
×
×
ξ (s)
B−1(ξ (s + τ) − Aξ (s))
ds. (15)
The solution x(t) of the initial problem (5), (6) is obtained by separating the first n coordinates
from (15), i.e., the formula (13) holds.
Theorem 1 is proved.
3. Representation of the solution to problem (7), (6). In this section, we will derive a representation
of the solution to the problem (7), (6). Together with equation (7), we consider the
linear system (8) where C, in this case, is a (2n × 2n)-matrix defined by
C :=
A I
Θ B
. (16)
It is easy to see that, for k ∈ N,
Ck
=



Ak
k−1
i=0
Ak−1−i
Bi
Θ Bk


 .
For a simple formalization of the delayed exponential eCt
τ , we define a matrix function e
(A,B)t
τ
as
e(A,B)t
τ =
t/τ
s=0
(t − (s − 1)τ)s
s!
s
i=0
As−i
Bi
.
The following formula can be verified directly by utilizing the definitions of special matrix
functions,
eCt
τ =
eAt
τ e
(A,B)t
τ
Θ eBt
τ
. (17)
Let us eliminate y(t) from system (8) with the matrix C given by (16). We get the system
x (t) =Ax(t − τ) + y(t − τ), (18)
y (t) =By(t − τ), (19)
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REPRESENTATION OF SOLUTIONS OF LINEAR DIFFERENTIAL SYSTEMS . . . 135
where t ≥ 0. Differentiating (18) and, subsequently, using both subsystems (18) and (19), we
derive the equation
x (t) = Ax (t − τ) + y (t − τ) = Ax (t − τ) + B(x (t − τ) − Ax(t − 2τ))
and, after some simplification, we get equation (7).
Theorem 2. Let AB = BA. Then the solution of Cauchy initial problem (7), (6) has the form
x(t) = eAt
τ ξ(−τ) + e(A,B)t
τ (ξ (0) − Aξ(−τ))+
+
0
−τ
eA(t−τ−s)
τ ξ (s) + e(A,B)(t−τ−s)
τ (ξ (s + τ) − Aξ (s)) ds (20)
where t ≥ τ.
Proof. Using (18) and (19), we derive
y(t) = y(−τ) +
t
−τ
y (s) ds = y(−τ) +
t
−τ
B(x (s) − Ax(s − τ)) ds =
= x (0) − Ax(−τ) +
t
−τ
B(x (s) − Ax(s − τ)) ds. (21)
Consider the initial-value problem
z(t) = ϕ∗
(t), −τ ≤ t ≤ 0,
for system (8) with the matrix C given by (16), i.e., for the system (18), (19) where
ϕ∗
=
ϕ∗
x
ϕ∗
y
: [−τ, 0] → R2n
is continuously differentiable as specified below. Since z =
x
y
, we set ϕ∗
x(t) ≡ ξ(t), t ∈
∈ [−τ, 0]. Next, we will specify ϕ∗
y. From (18), we obtain
y(t − τ) = x (t) − Ax(t − τ), t ≥ 0,
or
y(t) = x (t + τ) − Ax(t), t ≥ τ.
Consequently,
ϕ∗
y(t) ≡ ξ (t + τ) − Aξ(t), t ∈ [−τ, 0],
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136 Z. SVOBODA
and
ϕ∗
(t) =
ξ(t)
ξ (t + τ) − Aξ(t)
. (22)
Now we utilize formula (4) with the matrix A replaced by C in the form (16) and with ϕ replaced
with ϕ∗ given by (22). Utilizing (17), we get
z(t) = eCt
τ ϕ∗
(−τ) +
0
−τ
eC(t−τ−s)
τ ϕ∗
(s)ds =
=
eAt
τ e
(A,B)t
τ
Θ eBt
τ
ξ(−τ)
ξ (0) − Aξ(−τ)
+
+
0
−τ
e
A(t−τ−s)
τ e
(A,B)(t−τ−s)
τ
Θ e
B(t−τ−s)
τ
ξ (s)
ξ (s + τ) − Aξ (s)
ds.
By separating the first n components, we get formula (20).
Theorem 2 is proved.
4. Concluding remarks. 4.1. Relation between special delayed matrix functions. In the paper
[19], other delayed matrix functions called the delayed matrix sine Sinτ At and delayed matrix
cosine Cosτ At, where A is an (n × n)-matrix, are defined on R as
Sinτ At =
t/τ +1
s=0
(−1)s
A2s+1 (t − (s − 1)τ)2s+1
(2s + 1)!
(23)
and
Cosτ At =
t/τ +1
s=0
(−1)s
A2s (t − (s − 1)τ)2s
(2s)!
. (24)
The delayed matrix sine and cosine are fundamental matrices of a homogeneous second-order
linear system with a single delay,
x (t) = −A2
x(t − τ), (25)
making it possible to simply express the solutions to initial-value problems. In [19], the solution
of the Cauchy initial-value problem (3), (25), assuming that the matrix A is regular, is given in
the form
x(t) = (Cosτ At) ϕ(−τ) + A−1
(Sinτ At) ϕ (−τ)+
+ A−1
0
−τ
(Sinτ A(t − τ − ξ)) ϕ (ξ)dξ. (26)
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REPRESENTATION OF SOLUTIONS OF LINEAR DIFFERENTIAL SYSTEMS . . . 137
The equation (5) turns into (25) if we set A = Θ and then replace B with A and τ with τ/2.
Therefore, analysing formulas (4) and (26), and the formula
eCt
τ/2 =
2t/τ +1
s=0
Cs (t − (s − 1)τ/2)s
s!
=
Re eiAt
τ/2 Im eiAt
τ/2
−Im eiAt
τ/2 Re eiAt
τ/2
, (27)
obtained from (14) with the above-mentioned modifications and with
C =:
Θ A
−A Θ
, (28)
we conclude that there exists a relation between the delayed matrix exponential and the delayed
sine and cosine matrices. The next theorem provides us with such a relation.
Theorem 3. The formula
eCt
τ/2 =
Cosτ A(t − τ/2) Sinτ A(t − τ)
−Sinτ A(t − τ) Cosτ A(t − τ/2)
(29)
holds for every t ∈ R.
Proof. Let us compare the definitions (23) and (24) with the elements of the delayed matrix
exponential eCt
τ/2 expressed by (27) where C is given by (28), i.e., with Re eiAt
τ/2 and Im eiAt
τ/2. Next
we will use the formula
Im
m
s=0
(iA)s
=
(m−1)/2
u=0
(−1)u
A2u+1
which holds for an arbitrary integer m. Let k be an integer and t ∈ [kτ, (k + 1)τ). Then
2t/τ + 1 − 1
2
=
2t/τ
2
= k
and
Im eiAt
τ/2 = Im


2t/τ +1
s=0
(iA)s (t − (s − 1)τ/2)s
s!

 =
= Im
2k+1
s=0
(iA)s (t − (s − 1)τ/2)s
s!
=
=
k
u=0
(−1)u
A2u+1 (t − (2u + 1 − 1)τ/2)2u+1
(2u + 1)!
=
=
k
u=0
(−1)u
A2u+1 (t − uτ)2u+1
(2u + 1)!
. (30)
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138 Z. SVOBODA
Moreover, for t ∈ [kτ, (k + 1)τ), the definition (23) yields
Sinτ A(t − τ) =
k
s=0
(−1)s
A2s+1 (t − τ − (s − 1)τ)2s+1
(2s + 1)!
=
k
s=0
(−1)s
A2s+1 (t − sτ)2s+1
(2s + 1)!
. (31)
Comparing (30) and (31), we get
Sinτ A(t − τ) = Im eiAt
τ/2 (32)
for every t ∈ R.
Next, we use the formula
Re
m
s=0
(iA)s
=
m/2
u=0
(−1)u
A2u
.
Let t ∈ [(2k − 1)τ/2, (2k + 1)τ/2). Then
2t/τ + 1
2
= k
and
Re eiAt
τ/2 = Re


2t/τ +1
s=0
(iA)s (t − (s − 1)τ/2)s
s!

 =
= Re


( 2t/τ +1)/2
s=0
(iA)s (t − (s − 1)τ/2)s
s!

 =
= Re
k
s=0
(iA)s (t − (s − 1)τ/2)s
s!
=
=
k
u=0
(−1)u
A2u (t − (2u − 1)τ/2)2u
(2u)!
. (33)
Moreover, for t ∈ [(2k − 1)τ/2, (2k + 1)τ/2), the definition (24) yields
Cosτ A(t − τ/2) =
(t−τ/2)/τ +1
s=0
(−1)s
A2s (t − τ/2 − (s − 1)τ)2s
(2s)!
=
=
k
s=0
(−1)s
A2s (t − (2s − 1)τ/2)2s
(2s)!
(34)
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1
REPRESENTATION OF SOLUTIONS OF LINEAR DIFFERENTIAL SYSTEMS . . . 139
and, comparing (33) and (34), we get
Cosτ A(t − τ/2) = Re eiAt
τ/2 (35)
for every t ∈ R. Now it is easy to see that, by (27), (32), and (35), the formula (29) holds.
Theorem 3 is proved.
4.2. On classes of formally solvable equations. The problems (5), (6) and (6), (7), considered
in the paper, are special cases of a general problem,
x (t) + Px (t − τ) + Qx(t − 2τ) = θ, t ≥ τ,
(36)
x(i)
(t) = ξ(i)
(t), i = 0, 1, t ∈ [−τ, τ],
where P, Q are constant (n×n)-matrices provided that there exists an (n×n)-matrix Λ satisfying
the equation
Λ2
+ PΛ exp(−τΛ) + Q exp(−2τΛ) = Θ. (37)
We assume that a solution of (36) can be found in the form
x(t) = exp(Λt) (38)
where Λ is a suitable constant (n × n)-matrix. By substituting (38) into (36), we get
Λ2
exp(2Λt) + PΛ exp(Λ(t − τ)) + Q exp(Λ(t − 2τ)) = Θ
and further simplification yields equation (37). Let Y = exp(2Λτ) be a new unknown matrix.
Then, equation (37) can be written as
Y 2
+ PY + Q = Θ. (39)
The matrices A and B of the system (5) (i.e., P = −2A and Q = A2 + B2) generate complex
conjugate roots of (39),
Y1,2 = A ± iB,
and the matrices A and B of the system (7) (i.e., P = −A − B and Q = AB) generate real
roots of (39),
Y1 = A, Y2 = B.
The systems (5) and (7) are equivalent to system (8) with the matrix C defined by (9) or by
(16). Matrices on the right-hand sides of (9) and (16) can be viewed as “Jordan"forms of the
matrix C. From this point of view, the last case of the “Jordan"form of the block matrix C is
C :=
A Θ
Θ B
. (40)
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1
140 Z. SVOBODA
This matrix defines a pair of independent subsystems. This case is trivial since it is not possible to
eliminate n variables to obtain a second-order system. The above analysis leads to a conclusion
that the classes of the equations considered formally cover all the possible cases of the roots
of the quadratic equation (39) and “Jordan"forms (9), (16) and (40). For the first two of these
three cases, we derived a representation of solutions of initial-value problems. Representations
of solutions of initial-value problems in the third case was derived, as was mentioned above,
in [1].
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Received 22.12.15
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1
Electronic Journal of Qualitative Theory of Differential Equations
2017, No. 89, 1–15; https://doi.org/10.14232/ejqtde.2017.1.89 www.math.u-szeged.hu/ejqtde/
Asymptotic unboundedness of the norms of
delayed matrix sine and cosine
Zdenˇek Svoboda
CEITEC - Central European Institute of Technology, Brno University of Technology
Purkyˇnova 656/123, 612 00 Brno Czech Republic
Received 7 March 2017, appeared 12 December 2017
Communicated by Mihály Pituk
Abstract. In the paper, the asymptotic properties of recently defined special matrix
functions called delayed matrix sine and delayed matrix cosine are studied. The asymptotic
unboundedness of their norms is proved. To derive this result, a formula is used
connecting them with what is called delayed matrix exponential with asymptotic properties
determined by the main branch of the Lambert function.
Keywords: delay, delayed matrix functions, Lambert function, unboundedness.
2010 Mathematics Subject Classification: 34K06, 34K07.
1 Introduction
Recently, a new formalization has been developed of the well-known method of steps [12,13]
for solving the initial-value problem for linear differential equations with constant coefficients
and a single delay through special matrix functions called delayed matrix functions [6,15,20].
Using this method, representations have been found of solutions of homogeneous and nonhomogeneous
systems, and some stability and control problems were solved in [5,16]. Also,
a generalization has been developed to discrete systems and applied in [4,21].
Let A be a nonzero n × n constant matrix, τ > 0 and let · be the floor function.
The delayed matrix exponential, defined in [15], is a matrix polynomial on every interval
[(k − 1)τ, kτ), k = 0, 1, . . . , defined by
eAt
τ =
t/τ +1
∑
s=0
As (t − (s − 1)τ)s
s!
. (1.1)
The delayed matrix exponential equals to zero matrix Θ if t < −τ, the unit matrix I on [−τ, 0],
and is the fundamental matrix of a homogeneous linear system with a single delay
˙x(t) = Ax(t − τ). (1.2)
Email: zdenek.svoboda@ceitec.vutbr.cz
2 Z. Svoboda
For the proof, we refer to [15]. In [15], too, a representation is derived of the solution of the
Cauchy initial problem (1.2), (1.3), where
x(t) = ϕ(t), −τ ≤ t ≤ 0, (1.3)
and ϕ: [−τ, 0] → Rn is continuously differentiable.
Fundamental matrix (1.1) serves as a nice illustration of the general definition of a fundamental
matrix to linear functional differential systems of delayed type [12,13]. For system (1.2),
this definition reduces to (details are omitted)
X(t) =



A
t
−τ
X(u − τ)du + I, for almost all t ≥ −τ,
Θ, −2τ ≤ t < −τ
(1.4)
and its step-by-step application gives
X(t) = eAt
τ , t ≥ −2τ.
With its usefulness, the delayed matrix exponential stimulated the search for other delayed
matrix functions capable of simply expressing solutions of some linear differential systems
with constant coefficients. In [6], solutions of a homogeneous second-order linear system
with single delay
¨x(t) = −A2
x(t − τ). (1.5)
are expressed through delayed matrix functions called the delayed matrix sine Sinτ At and
delayed matrix cosine Cosτ At defined for t ∈ R as
Sinτ At =
t/τ +1
∑
s=0
(−1)s
A2s+1 (t − (s − 1)τ)2s+1
(2s + 1)!
(1.6)
and
Cosτ At =
t/τ +1
∑
s=0
(−1)s
A2s (t − (s − 1)τ)2s
(2s)!
. (1.7)
Matrices (1.6) and (1.7) are related to the 2n × 2n fundamental matrix X (t) of 2n-dimensional
system
˙y(t) = Ay(t − τ/2),
where
A :=
Θ A
−A Θ
, y :=
y1
y2
,
equivalent with (1.5) through the substitution x(t) = y1(t). In much the same way as above,
we can derive (for details we refer to [24])
X (t) = eAt
τ/2 =
Cosτ A(t − τ/2) Sinτ A(t − τ)
−Sinτ A(t − τ) Cosτ A(t − τ/2)
.
The paper aims to prove the asymptotic unboundedness of the norms of delayed matrix sine
and delayed matrix cosine. This is done by utilizing relations between these functions and the
delayed matrix exponential. The proof is based on the properties of the main branch of the
Lambert function.
Therefore, we at first describe the necessary properties of the delayed exponential of a
matrix and the Lambert function in Part 2. Then, in Part 3, the main result on the asymptotic
properties of delayed matrix sine and delayed matrix cosine is proved.
Unboundedness of the norms of delayed matrix sine and cosine 3
2 Delayed matrix exponential and Lambert function
To explain clearly the relationship between delayed linear differential equations and Lambert
function, we first consider the scalar case. Let n = 1, A = (a). Then, the fundamental matrix
to the scalar case of the system (1.2), i.e., of
˙x(t) = ax(t − τ) (2.1)
is defined by (1.1) as
eat
τ =
t/τ +1
∑
s=0
as (t − (s − 1)τ)s
s!
.
and its values at nodes t = kτ, k = 0, 1, . . . are
eakτ
τ =
k+1
∑
s=0
as (kτ − (s − 1)τ)s
s!
=
k
∑
s=0
as (k + 1 − s)sτs
s!
= 1 + a
kτ
1!
+ a2 (k − 1)2τ2
2!
+ · · · + ak−1 2k−1τk−1
k!
+ ak τk
k!
.
Assume that there exists a real solution c of a transcendental equation
c = ae−cτ
, (2.2)
i.e., that there exists a solution x(t) = ect of (2.1). Moreover, assume that, for a real root c
of (2.2), we have
eakτ
τ ∼ eckτ
= 1 + c
kτ
1!
+ c2 k2τ2
2!
+ · · · + cn knτn
n!
+ · · ·
when k → ∞. Then,
e
a(k+1)τ
τ
eakτ
τ
∼
ec(k+1)τ
eckτ
= ecτ
, k → ∞. (2.3)
Analyzing equation (2.3), provided it is valid, we can expect that, in a general case, the sequence
of values of delayed matrix exponential at nodes t = kτ, k → ∞ is approximately
represented by a “geometric progression” with the ordinary exponential of a constant matrix
serving as a “quotient” factor.
It is reasonable to expect that such a constant matrix can be expressed by the principal
branch of the Lambert function since (2.2) can be rewritten as
cτecτ
= aτ (2.4)
or as
cτ = W(aτ) (2.5)
where W is the well-known Lambert W-function [3] (its properties given below are taken from
this paper), defined as the inverse function to the function
z = f (w) = wew
, (2.6)
i.e., w = W(z). If z = x + iy and w = u + iv, then (2.6) yields
4 Z. Svoboda
x + iy = (u + iv)eu+iv
(2.7)
and
x = eu
(u cos v − v sin v), y = eu
(u sin v + v cos v). (2.8)
The Lambert W-function is multi-valued (except for the point z = 0). For real z = x > −1/e
and w = u > −1, equation (2.6) defines a single-valued function w = W0(x). The function
W0(x) can be extended to the whole complex plane as a holomorphic function W0(z) except
for the values x < −1/e and y = 0. The extension w = W0(z) is called the principal branch of
the Lambert function.
The range of values of the principal branch W = W0(z) is bounded by a parametric curve
[3, p. 343]
=
−v
tan v
+ iv, −π < v < π (2.9)
and equals to the domain
L := (u, v) ∈ C: u ≥ −1, |v| ≤ |v∗
| < π where
−v∗
tan v∗
= u .
For more details about the Lambert W-function, see [3].
The asymptotic properties of exp(W0(z)) are, in principle, determined by the real part of
W0(z). Let z = x + iy and
W0(x + iy) = Re W0(x + iy) + i Im W0(x + iy) = u + iv.
The set of complex numbers z = x + iy such that Re W0(z) = u = 0, i.e., (see (2.7), (2.8)),
x + iy = iv exp(iv)
is a closed curve ˜:
x = −v sin v, y = v cos v (2.10)
where, as it is clear from the definition of L, |v∗| = π/2 for u = 0 and |v| ≤ π/2. We have (as
a consequence of (2.8))
Re W0(z) < 0
if z lies within the interior of this curve and
Re W0(z) > 0 (2.11)
for numbers z of its exterior. From (2.10) it follows easily that the exterior domain to ˜ is
specified by the inequality
|z| > − arctan
Re z
|Im z|
. (2.12)
Lemma 2.1. For complex numbers z = x + iy, z = 0 with x ≥ 0,
| Im W0(z)| <
π
2
. (2.13)
Unboundedness of the norms of delayed matrix sine and cosine 5
Proof. First, from (2.9) and definition of L, we obtain inequality |v| = |Im W0(z)| < π, there-
fore,
v sin v > 0. (2.14)
Secondly, for w = u + iv = W0(z), the inequality u < 0 implies |v| < π/2 (see the definition of
L) and, in this case, (2.13) holds. This guarantees that sign(u cos v) = sign u. Applying (2.8)
and the assumption that x is nonnegative, we obtain
eu
(u cos v − v sin v) = x ≥ 0 ⇒ u ≥ 0 ⇒ arg W0(z)Im W0(z) ≥ 0.
This fact also implies
| arg W0(z) + Im W0(z)| = | arg W0(z)| + |Im W0(z)|. (2.15)
Equation (2.6) yields
z = wew
= W0(z)eW0(z)
.
Therefore,
arg z = arg W0(z) + Im W0(z)
and, due to relation, (2.15) we also have
| arg z| = | arg W0(z)| + |Im W0(z)|. (2.16)
For z = 0 with non-negative real parts, we have Re W0(z) > 0 by (2.11), from (2.14), we deduce
arg W0(z) = 0, Im W0(z) = 0, and, utilizing (2.16), we also have
π/2 ≥ | arg z| = | arg W0(z)| + |Im W0(z)| > |Im W0(z)|.
Reverting to equation (2.3), we can expect that, in some cases, there exists a constant n × n
matrix C such that
lim
k→∞
e
A(k+1)τ
τ (eAkτ
τ )−1
= eCτ
, (2.17)
provided that the matrices eAkτ
τ are nonsingular (this property will be assumed throughout
the paper). One of such cases is analysed in [23] where the following is proved.
Theorem 2.2. Let λj, j = 1, . . . , n be the eigenvalues of the matrix A and let its Jordan canonical form
be
diag(λ1, . . . , λn) = D−1
AD (2.18)
where D is a regular matrix. If
|λj| < 1/(eτ),
j = 1, . . . , n, then the sequence
e
A(k+1)τ
τ (eAkτ
τ )−1
, k → ∞
converges, (2.17) holds and
eCτ
= D exp (diag(W0(λ1τ), . . . , W0(λnτ)) D−1
. (2.19)
Note that from (2.19) we immediately get explicit form of C since
Cτ = D (diag(W0(λ1τ), . . . , W0(λn, τ)) D−1
and
C = D diag (W0(λ1τ)/τ, . . . , W0(λnτ)/τ) D−1
.
6 Z. Svoboda
3 Main result
The asymptotic properties of the delayed matrix sine and cosine can be deduced from the
relations with the delayed exponential of a matrix. We give relevant formulas that are similar
to the well-known Euler identity. Namely, for an arbitrary n × n matrix A and t ∈ R, we have
Sinτ A(t − τ) = Im eiAt
τ/2 =
1
2i
eiAt
τ/2 − e−iAt
τ/2 (3.1)
and
Cosτ A t −
τ
2
= Re eiAt
τ/2 =
1
2
eiAt
τ/2 + e−iAt
τ/2 . (3.2)
Formulas (3.1), (3.2) can be proved directly using the definitions of eAt
τ , Sinτ At and Cosτ At
given by formulas (1.1), (1.6) and (1.7) (for the proof we refer to [24]). Below, we use the
spectral norm of a matrix defined as
A S = λmax(A∗ A) (3.3)
where A∗ denotes the conjugate transpose of A and λmax is the largest eigenvalue of the matrix
A∗ A. The main result of the paper follows.
Theorem 3.1. Let λj, j = 1, . . . , n be the eigenvalues of the matrix A and let its Jordan canonical form
be given by (2.18). If |λj| < 1/(eτ), j = 1, . . . , n and there exists at least one j = j∗ ∈ {1, . . . , n}
such that λj∗ = 0, then
lim sup
t→∞
Cosτ At S = ∞
and
lim sup
t→∞
Sinτ At S = ∞.
Proof. We will only prove the assertion for Cosτ At as the proof for Sinτ At is analogous. Using
equation (3.2), we derive the assertion of the theorem utilizing the asymptotic properties of
the delayed exponential of matrix eiAt
τ/2. From the assumption (2.18), we readily get
(iA)k
= D diag((iλ1)k
, . . . , (iλn)k
)D−1
, k ≥ 0
and, using the associativity, we may express eiAkτ/2
τ/2 (with the aid of definition (1.1)) as
eAikτ/2
τ/2 = D diag eλ1ikτ/2
τ/2 , . . . , eλnikτ/2
τ/2 D−1
. (3.4)
For a natural number we define
Fk (A) := e
Ai(k+ )τ/2
τ/2 (eAikτ/2
τ/2 )−1
.
By Theorem 2.2 (formula (2.17)) and by (2.19), we have
lim
k→∞
F1
k (A) = D exp (diag(W0(λ1iτ/2), . . . , W0(λniτ/2)) D−1
. (3.5)
From
Fk (a) = ∏
l=1
F1
k−l−1(A),
we obtain
Unboundedness of the norms of delayed matrix sine and cosine 7
lim
k→∞
Fk (A) = lim
k→∞
∏
l=1
Fk (A) = ∏
l=1
lim
k→∞
Fk (A)
= D exp (diag(W0(λ1iτ/2), . . . , W0(λniτ/2)) D−1
.
Imagine, for a while, that the matrix A is a 1 × 1 matrix, i.e., A = (a). Then, from (3.5) (with
λ = a, D := (1)), we get
F1
k (a) = (exp(W0(aiτ/2))) (1 + va(k)) (3.6)
where k is an arbitrary natural number and v = va(k) is a real discrete function such that
lim
k→∞
va(k) = 0. (3.7)
Applying formula (3.6) times, we obtain
Fk (A) = (exp(W0(aiτ/2))) ∏
l=1
(1 + va(k − 1 + l)).
Now we can derive a similar formula in the case of an n × n matrix A. First, utilizing (3.6),
we obtain:
F1
k (A) = D diag e
λ1i(k+1)τ/2
τ/2 , . . . , e
λni(k+1)τ/2
τ/2 D−1
× D diag eλ1ikτ/2
τ/2
−1
, . . . , eλnikτ/2
τ/2
−1
D−1
= D diag e
λ1i(k+1)τ/2
τ/2 eλ1ikτ/2
τ/2
−1
, . . . , e
λni(k+1)τ/2
τ/2 eλnikτ/2
τ/2
−1
D−1
= D diag ((exp(W0(λ1iτ/2))) (1 + vλ1
(k)), . . .
. . . , (exp(W0(λniτ/2))) (1 + vλn
(k))) D−1
= D diag (exp(W0(λ1iτ/2)), . . . , exp(W0(λniτ/2))) D−1
× D diag ((1 + vλ1
(k)), . . . , (1 + vλn
(k))) D−1
= D diag (exp(W0(λ1iτ/2)), . . . , exp(W0(λniτ/2))) D−1
M(k)
(3.8)
where the matrix M(k) is defined as
M(k) := D diag((1 + vλ1
(k)), . . . , (1 + vλn
(k)))D−1
.
Denote
eW0(iA)τ/2
:= D diag (exp(W0(λ1iτ/2)), . . . , exp(W0(λniτ/2))) D−1
.
This matrix commutes with M(k) since
eW0(iA)τ/2
M(k) = D diag(exp(W0(λ1iτ/2)), . . . , exp(W0(λniτ/2)))D−1
× D diag((1 + v1(k)), . . . , (1 + vn(k)))D−1
= D diag((1 + v1(k)), . . . , (1 + vn(k)))D−1
× D diag(exp(W0(λ1iτ/2)), . . . , exp(W0(λniτ/2)))D−1
= M(k)eW0(iA)τ/2
.
8 Z. Svoboda
Utilizing (3.4), (3.6), and (3.8), we derive
Fk (A) = e
Ai(k+ )τ/2
τ/2 (e
Ai(k+ −1)τ/2
τ/2 )−1
· · · e
Ai(k+2)τ/2
τ/2 (e
Ai(k+1)τ/2
τ/2 )−1
e
Ai(k+1)τ/2
τ/2 (eAikτ/2
τ/2 )−1
= D diag e
λ1i(k+ )τ/2
τ/2 e
λ1i(k+ −1)τ/2
τ/2
−1
, . . . , e
λni(k+ )τ/2
τ/2 e
λni(k+ −1)τ/2
τ/2
−1
D−1
× D diag e
λ1i(k+ −1)τ/2
τ/2 e
λ1i(k+ −2)τ/2
τ/2
−1
, . . .
. . . , e
λni(k+ −1)τ/2
τ/2 e
λni(k+ −2)τ/2
τ/2
−1
D−1
· · ·
× D diag e
λ1i(k+1)τ/2
τ/2 eλ1ikτ/2
τ/2
−1
, . . . , e
λni(k+1)τ/2
τ/2 eλnikτ/2
τ/2
−1
D−1
= eW0(iA)τ/2
M(k + − 1)eW0(iA)τ/2
M(k + − 2) · · · eW0(iA)τ/2
M(k)
= eW0(iA)τ/2
−1
∏
l=0
M(k + l).
(3.9)
It is easy to see that the values of functions eλlikτ/2
τ/2 , exp( W0(λliτ/2)) (l = 1, . . . , n) and
the values of the same functions with complex conjugate arguments are complex conjugate
too. Applying this fact to Cosτ A ((k + − 1)τ/2) = Re e
iA(k+ )τ/2
τ/2 (see (3.2)), we get (utilizing
(3.4), (3.9)):
Re e
iA(k+ )τ/2
τ/2 =
1
2
e
iA(k+ )τ/2
τ/2 + e
−iA(k+ )τ/2
τ/2
=
1
2
D diag eλ1ikτ/2
τ/2 , . . . , eλnikτ/2
τ/2 D−1
eW0(iA)τ/2
−1
∏
l=0
M(k + l)
+ D diag e−λ1ikτ/2
τ/2 , . . . , e−λnikτ/2
τ/2 D−1
eW0(−iA)τ/2
−1
∏
l=0
M(k + l)
=
1
2
D diag eλ1ikτ/2
τ/2 exp( W0(λ1iτ/2))
+ e−λ1ikτ/2
τ/2 exp(− W0(λ1iτ/2)), . . . , eλnikτ/2
τ/2 exp( W0(λniτ/2))
+ e−λnikτ/2
τ/2 exp(− W0(λniτ/2)) D−1
−1
∏
l=0
M(k + l)
= D diag Re eλ1ikτ/2
τ/2 exp( W0(λ1iτ/2)) , . . .
. . . , Re eλnikτ/2
τ/2 exp( W0(λniτ/2)) D−1
−1
∏
l=0
M(k + l)
= D diag Re eλ1ikτ/2
τ/2 exp( W0(λ1iτ/2))
−1
∏
l=0
(1 + vλ1
(k + l)), . . .
. . . , Re eλnikτ/2
τ/2 exp( W0(λniτ/2))
−1
∏
l=0
(1 + vλn
(k + l)) D−1
. (3.10)
Now we use the well-known formula Re(z1z2) = |z1||z2| cos(arg z1 + arg z2) for complex
numbers z1, z2. Set
z1 = z1(k, λl) := eλlikτ/2
τ/2 , z2 = z2(λl) := exp( W0(λliτ/2)),
Unboundedness of the norms of delayed matrix sine and cosine 9
where l ∈ {1, . . . , n}, and denote
α1(k, λl) := arg z1(k, λl) = arg eλlikτ/2
τ/2 ,
α2(λl) := arg z2(λl) = arg (exp( W0(λliτ/2))) .
From the facts that the spectral radius is less or equal any matrix norm, the following inequality
for the spectral norm holds
Cosτ A ((k + − 1)τ/2) S ≥ ρ (Cosτ A ((k + − 1)τ/2))
= ρ Re e
iA(k+ )τ/2
τ/2 = ρk+ .
(3.11)
The similar matrices have same spectra and the spectral radii. The spectrum of diagonal
matrix consists to elements of the diagonal and using (3.10), we obtain
ρk = max
j=1,...,n
Re e
λjikτ/2
τ/2 exp( W0(λjiτ/2))
−1
∏
l=0
(1 + vλj
(k + l))
≥ (1 + v∗
(k)) max
j=1,...,n
Re e
λjikτ/2
τ/2 exp( W0(λjiτ/2))
(3.12)
where
v∗
(k) := min
j=1,...,n; l=0,..., −1
vλj
(k + l)
and, by (3.7),
lim
k→∞
v∗
(k) = 0. (3.13)
Applying (3.11) and (3.12) we obtain the inequality
Cosτ A ((k + − 1)τ/2) S ≥ (1 + v∗
(k)) max
j=1,...,n
Re e
λjikτ/2
τ/2 exp( W0(λjiτ/2))
≥ (1 + v∗
(k)) max
j=1,...,n
e
λjikτ/2
τ/2 exp( W0(λjiτ/2)) |cos (α1(k, λl) + α2(λl))| .
Assume that j = j∗ ∈ {1, . . . , n} is fixed and that the eigenvalue λj∗ = 0 of the matrix A
is real. Then, the number z∗ = iλj∗ τ/2 lies in the exterior domain of ˜ since inequality (2.12)
holds, i.e.,
|z∗
| = |iλj∗ τ/2| > − arctan
Re z∗
|Im z∗|
= − arctan 0 = 0 (3.14)
and, by (2.11),
Re W0(z∗
) = Re W0(iλj∗ τ/2) > 0. (3.15)
Now assume that j = j∗ ∈ {1, . . . , n} is fixed and that the eigenvalue λj∗ = 0 of the
matrix A is a complex number. Since λj∗ is an eigenvalue of A as well, we can assume that
λj∗ = x − iy where y > 0. Then, the number z∗ = iλj∗ τ/2 lies in the exterior domain of ˜ since
inequality (2.12) holds, i.e.,
|z∗
| = |iλj∗ τ/2| =
τ
2
|ix + y| =
τ
2
x2 + y2 > − arctan
Re z∗
|Im z∗|
= − arctan
y
|x|
where arctan (y/|x|) > 0. Then, by (2.11),
Re W0(z∗
) = Re W0(iλj∗ τ/2) > 0. (3.16)
10 Z. Svoboda
From (3.15) and (3.16), it follows that there exists an eigenvalue λj∗ of A and a constant C such
that
Re W0(iλj∗ τ/2) > C > 0. (3.17)
Utilizing (3.1), (3.2) (where A := (λj∗ ) and t = kτ/2) we derive
e
λj∗ ikτ/2
τ/2 = Cosτλj∗ (k − 1)τ/2) + i Sinτλj∗ (k/2 − 1)τ. (3.18)
Let k = k∗ be such that
Cosτλj∗ (k∗
− 1)τ/2) = 0. (3.19)
It is easy to see that such a k∗ always exists and note that it can be assumed greater than an
arbitrarily given sufficiently large positive integer. Then (3.18), implies
α1(k∗
, λj∗ ) = ±
π
2
. (3.20)
By (2.13), we have |α2(λj∗ )| < π/2. With regard to α2(λj∗ ), we consider two cases below:
a) Let α2(λj∗ ) = 0. Then, each interval [π/2 + 2sπ, π/2 + 2sπ + π], where s = 0, 1, . . . ,
contains at least two elements of an equidistant sequence
{α1(k∗
, λj∗ ) + nα2(λj∗ )}∞
n=−∞
and, in each interval, there exists an element of this sequence αs such that
|αs
− π/2| >
π
4
, |αs
− π/2 − π| >
π
4
and
| cos(αs
)| >
√
2/2. (3.21)
b) Let α2(λj∗ ) = 0. Then, (3.20) implies
| cos αs
| = | cos α1(k∗
, λj∗ )| = 0. (3.22)
Therefore, in both cases a) and b), there exists a sequence of positive integers { l}∞
l=1 such that
liml→∞ = ∞ and (due to (3.17), (3.21) and (3.22)) for all sufficiently large l
| exp( lW0(iλj∗ τ/2))|| cos(α1(k∗
, λj∗ ) + lα2(λj∗ ))| > M exp( lCτ/2) (3.23)
where
M :=



√
2
2
, if α2(λj∗ ) = 0,
| cos α1(k∗
, λj∗ )|, if α2(λj∗ ) = 0
and C is a constant satisfying 0 < C < C. Moreover, from (3.13), it follows that, for every
sufficiently large k, there exists a constant C0 satisfying 0 < C0 < C such that
1 + v∗
(k) > exp(−C0τ/2). (3.24)
Unboundedness of the norms of delayed matrix sine and cosine 11
From (3.12), (3.23), (3.24), we can derive
Cosτ A ((k∗
+ l − 1)τ/2) S ≥ (1 + v∗
(k∗
)) l e
λj∗ ik∗τ/2
τ/2
× exp( lW0(λj∗ iτ/2)) cos α1(k∗
, λj∗ ) + α2(λj∗ )
≥ exp(− lC0τ/2) e
λj∗ ik∗τ/2
τ/2 M exp( lCτ/2)
= M e
λj∗ ik∗τ/2
τ/2 exp( l(C − C0)τ/2).
Finally, we conclude
lim sup
t→∞
Cosτ At S ≥ lim
l→∞
Cosτ A ((k∗
+ l − 1)τ/2) S
≥ lim
l→∞
M e
λj∗ ik∗τ/2
τ/2 exp( l(C − C0)τ/2)
= ∞.
An analogous assertion can also be obtained for Sinτ At. The scheme of the proof in this
case remains the same with the following minor modifications. In (3.10) the imaginary parts
of the complex expressions considered is used instead of their real parts. The relation (3.10)
turns into
Sinτ A ((k + − 2)τ/2) = D diag Im eλ1ikτ/2
τ/2 exp( W0(λ1iτ/2))
−1
∏
l=0
(1 + vλ1
(k + l)), . . .
. . . , Im eλnikτ/2
τ/2 exp( W0(λniτ/2))
−1
∏
l=0
(1 + vλn
(k + l)) D−1
and the estimation (3.12) has the form
Sinτ A ((k + − 2)τ/2) S
≥ (1 + v∗
(k)) max
j=1,...,n
e
λjikτ/2
τ/2 exp( W0(λjiτ/2)) |sin (α1(k, λl) + α2(λl))| .
In (3.19), Sinτ instead of Cosτ is used and the constant M must be redefined as
M :=



√
2
2
, if α2(λj∗ ) = 0,
| sin α1(k∗
, λj∗ )|, if α2(λj∗ ) = 0.
4 Concluding remarks
In this part, we discuss some connections with previous results and facts. The author is
grateful to the referee for drawing attention to several topics which are discussed below.
i) Relationship with a linear ordinary non-delayed system. In the paper, properties of delayed
matrix exponential and the Lambert W-function are used to prove that spectral norms
of delayed matrix sine and delayed matrix cosine are unbounded for t → ∞. This property is
proved under the assumption that the spectral radius ρ(A) of the matrix A is less that 1/(eτ).
12 Z. Svoboda
Many papers bring results on so-called special solutions of delayed differential systems (we
refer, e.g., to [1,2,7–11,14,17–19,22] and to the references therein) approximating, in a certain
sense, all solutions of a given system. One of the conditions guaranteeing the existence of
special solutions is often (restricted to system (1.2)) the inequality
A < 1/(eτ)
where · is an arbitrary norm. The totality of all special solutions is only an n-parameter
family where n equals the number of equations of the system. Moreover, it is often stated
that, in such a case, some properties (such as stability properties) of solutions of the initial
system are the same as those for solutions of a corresponding system of ordinary differential
equations.
Because of the well-known inequality ρ(A) ≤ A , it is generally not possible from an
assumed inequality ρ(A) < 1/(eτ) to deduce A < 1/(eτ). Nevertheless, for the spectral
norm (3.3) used in the paper, we get (under the conditions of Theorem 3.1),
ρ(A) = A S < 1/(eτ).
It means that, in a way, the properties of solutions of (1.2) are close, in a meaning, to properties
of an ordinary differential system and (1.2) is asymptotically ordinary. I.e., every solution of
system (1.2) is asymptotically close to a solution of a system of ordinary differential equations.
The construction of such a linear non-delayed system is described, e.g., in [1, Theorem 2.4]
(see also the Summary part in [17]). However, to find such a system is, in general, not an
easy task. The formula defining the matrix of ordinary differential system ([1, formula (2.8)]
or [17, formula (2.10)]) is a series of recurrently defined matrices and to find its sum is not
always possible (we refer to [7, Theorem 1.2], [17, part 4]).
In the case of a constant matrix, the fundamental matrix Xo(t) of the corresponding ordinary
differential system equals an ordinary matrix exponential Xo(t) = exp(Λ0t) where the
matrix Λ0 is a unique solution of the matrix equation
Λ = A exp(−Λτ)
such that Λ0 τ < 1 (see the proof of statement (i) of the Theorem in [17]). So, an analysis of
the asymptotic behavior of the solutions of system (1.2) reduces, in a meaning, to an analysis
of the asymptotic behavior of solutions of a system of ordinary differential equations x = Λ0x,
i.e., analysis of the properties of the matrix Λ0. Tracing the proof of Theorem 3.1, we can assert
that the investigation of properties of the matrix Λ0 is, in our case, performed by using the
properties of Lambert W-function defined in Part 2 (see also the motivation example (2.1) and
formulas (2.2)–(2.5)).
ii) Existence of a root of characteristic equation with positive real part. Let n = 1 and
A = (a) in (1.5). Then, the characteristic equation (derived by substituting x = exp(λt))
equals
λ2
= −a2
exp(−τλ) (4.1)
and is equivalent with
λτ
2
exp
λτ
2
= ±
iaτ
2
.
Utilizing the Lambert W-function, the last equation can be written as (see (2.4), (2.5))
λτ
2
= W ±
iaτ
2
,
Unboundedness of the norms of delayed matrix sine and cosine 13
therefore, all roots of (4.1) are values of the Lambert function. For
z = z± = ±iaτ/2,
inequality (2.12), which determines the domain of the points for which the principal branch
of the Lambert function W0 has positive real parts (inequality (2.11)), holds (see also (3.14),
(3.15)). Thus, we conclude that the unboundedness of the delayed matrix sine and cosine is
related to the existence of a root of characteristic equation with positive real part.
iii) Asymptotic behavior of the fundamental matrix solution by using the characteristic
equation. As noted in the Introduction, the general definition of a fundamental matrix to
linear functional differential systems of delayed type in [12,13] yields (in the simple case of the
matrix of the system with single delay being a constant matrix) a delayed matrix exponential
by formula (1.4). Delayed matrix sine and cosine can be expressed through delayed matrix
exponential by formulas (3.1), (3.2). Therefore, both Theorem 2.2 and Theorem 3.1, formulate
the asymptotic properties of the relevant fundamental matrix solutions depending on the
properties of the eigenvalues of the matrix A and, consequently, through the properties of
the roots of the characteristic equation described by the Lambert W-function. It is an open
question if the method used in the paper can be extended to matrices A with Jordan canonical
forms different from (2.18) in order to get further results on the behavior of the fundamental
matrix solution.
Acknowledgements
The author would like to thank the referee for helpful suggestions incorporated in this paper.
This research was carried out under the project CEITEC 2020 (LQ1601) with financial support
from the Ministry of Education, Youth and Sports of the Czech Republic under the National
Sustainability Programme II.
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