CONDITIONAL OSCILLATION
OF
HALF-LINEAR EQUATIONS
Petr Hasil
Habilitation Thesis
c Petr Hasil, Masaryk University, 2016
Preface
The nature around us can be described in many ways. One of the most accurate ways
is using equations. The type of the investigated equations depends on the analyzed
phenomenon itself. There are two fundamental types, namely, differential equations (for
continuous time) and difference equations (for discrete time). Of course, during the
last decades, the unification of both types, the so called time scale calculus, have been
developed. Chapter 2 is devoted to differential equations, then Chapter 3 deals with
difference equations, and finally the dynamic equations on time scales are treated in
Chapter 4.
Oscillation (or non-oscillation) is one of the fundamental topics from the qualitative
theory of differential and difference equations. The main idea of this notion is to count
zeros of solutions at infinity. If there exists the greatest zero of a solution, we say that this
solution is non-oscillatory. On the other hand, if the zeros of a solution tend to infinity,
the solution is said to be oscillatory.
The main goal of this thesis is to study the conditional oscillation of equations and
to find the so-called critical oscillation constants. Once we prove that an equation is
conditionally oscillatory and find the critical constant (which usually depends on the
coefficients of the studied equation), the oscillation properties of such an equation are
fully resolved with no more than one exception (the critical case). This fact means
that conditionally oscillatory equations are ideal testing equations. In particular, many
equations (which are not conditionally oscillatory) can be compared with these testing
equations using comparison theorems and, consequently, it is possible to specify their
oscillation properties.
This work is based on papers [35, 40, 41, 42, 43, 44, 45, 46] published or submitted for
publication during years 2014 and 2015. Hence, I am obliged to my coauthors, I thank
them for fruitful collaboration and look forward to solving open problems which appeared
during our conjoint work.
Also, I would like to sincerely thank Prof. Ondˇrej Doˇsl´y, who lead my first steps to
mathematical analysis, for his advice and willingness.
Last, but not least, I thank my wife, whole family and friends for their support.
Petr Hasil, 2016
2010 Mathematics Subject Classification: 39A10; 39A21; 39A70;
47B25; 47B36; 47B39.
Contents
Page
1 Introduction 1
1.1 The essentials of the used techniques . . . . . . . . . . . . . . . . . . 1
1.2 Short history overview . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Differential equations 13
2.1 Coefficients with mean values . . . . . . . . . . . . . . . . . . . . . . 13
§ 2.1.1 Results and examples . . . . . . . . . . . . . . . . . . . . . . . . . . 14
§ 2.1.2 An application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Periodic coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
§ 2.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
§ 2.2.2 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
§ 2.2.3 Main results and examples . . . . . . . . . . . . . . . . . . . . . . . 45
2.3 Sums of periodic coefficients . . . . . . . . . . . . . . . . . . . . . . . 52
§ 2.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
§ 2.3.2 Lemmata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
§ 2.3.3 Results and examples . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.4 Modified Euler equations . . . . . . . . . . . . . . . . . . . . . . . . . 64
§ 2.4.1 Equation for Pr¨ufer angle . . . . . . . . . . . . . . . . . . . . . . . 65
§ 2.4.2 Pr¨ufer angle of average function . . . . . . . . . . . . . . . . . . . . 67
§ 2.4.3 Oscillation constant . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.5 Critical case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
§ 2.5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
§ 2.5.2 Results and examples . . . . . . . . . . . . . . . . . . . . . . . . . . 85
2.6 Riemann–Weber equations . . . . . . . . . . . . . . . . . . . . . . . . 89
§ 2.6.1 Modified Pr¨ufer angle and average function . . . . . . . . . . . . . . 91
§ 2.6.2 Preliminaries and results . . . . . . . . . . . . . . . . . . . . . . . . 96
§ 2.6.3 Corollaries and examples . . . . . . . . . . . . . . . . . . . . . . . . 106
3 Difference equations 111
3.1 Coefficients with mean values . . . . . . . . . . . . . . . . . . . . . . 111
§ 3.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
iv
CONTENTS v
§ 3.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
§ 3.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4 Dynamic equations on time scales 128
4.1 Equations with periodic coefficients . . . . . . . . . . . . . . . . . . . 128
§ 4.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
§ 4.1.2 Conditional oscillation . . . . . . . . . . . . . . . . . . . . . . . . . 136
§ 4.1.3 Applications and concluding remarks . . . . . . . . . . . . . . . . . 149
A Bibliography 152
B Curriculum vitae 158
C Publications 160
Chapter
1 Introduction
1.1 The essentials of the used techniques
The topic of this thesis belongs to the oscillation theory of half-linear equations. The
main part (Chapter 2) deals with differential equations. Therefore, we recall the basics
of half-linear differential equations at this place and difference equations and dynamic
equations on time scales will be introduced in Chapters 3 and 4, respectively. Note that
we will use the standard notation R+
= (0, ∞) and Ra = [a, ∞) for arbitrary given a ∈ R.
Our main interest is to study equations of the form
[r(t)Φ(x )] + z(t)Φ(x) = 0, Φ(x) = |x|p−1
sgn x, p > 1, (1.1.1)
with continuous coefficients r > 0 and z. An equation of this form appeared for the
first time in [8] and as the basic pioneering papers in the field of half-linear differential
equations we mention [24, 60]. During the last decades, these equations have been widely
studied in the literature. A detailed description and a comprehensive literature overview
concerning the topic can be found in [21] (see also [2, Chapter 3]).
The name half-linear equations was introduced in [7]. This term is motivated by the
fact that the solution space of these equations is homogeneous (likewise in the linear
case), but it is not additive. There are several differences between linear equations and
half-linear equations. Especially, some tools widely used in the theory of linear equations
are not available for half-linear equations (e.g., see [25] for the Wronskian identity and
[23] for the Fredholm alternative). In fact, these differences are caused, more or less, by
the lack of the additivity. On the other hand, many results from the theory of linear
equations are extendable to their half-linear counterparts. Nevertheless, according to our
best knowledge, the results presented in this thesis are new even for the linear case (i.e.,
for p = 2).
Since the main tools in this thesis are based on the Riccati technique and the Pr¨ufer
angle (more precisely on their generalizations and combinations), we recall these notions
at this place. To begin with the Pr¨ufer transformation, we have to recall the half-linear
trigonometric functions as well. For more comprehensive description, we refer, e.g., to
[21, Section 1.1.2]. The half-linear sine function, denoted by sinp, is defined as the odd
2πp-periodic extension of the solution of the initial problem
[Φ (x )] + (p − 1)Φ(x) = 0, x(0) = 0, x (0) = 1, (1.1.2)
1
1.1. THE ESSENTIALS OF THE USED TECHNIQUES 2
where
πp :=
2
p
B
1
p
,
1
q
=
2Γ 1
p
Γ 1
q
pΓ 1
p
+ 1
q
=
2π
p sin π
p
. (1.1.3)
In the definition of πp, we use the Euler beta and gamma functions
B(x, y) =
1
0
τx−1
(1 − τ)y−1
dτ, x, y > 0, Γ(x) =
∞
0
τx−1
e−τ
dτ, x > 0,
and the formula
Γ(x)Γ(1 − x) =
π
sin(πx)
, x > 0,
together with the identity (the conjugacy of the numbers p and q)
1
p
+
1
q
= 1, i.e., p + q = pq. (1.1.4)
The derivative of the half-linear sine function is called the half-linear cosine function
and it is denoted by cosp. Note that the half-linear sine and cosine functions satisfy the
half-linear Pythagorean identity
| sinp t|p
+ | cosp t|p
= 1, t ∈ R. (1.1.5)
Especially, the half-linear trigonometric functions are bounded. Therefore, there exists
L > 0 such that
| cosp y|p
< L, | Φ(cosp y) sinp y | < L, | sinp y|p
< L, y ∈ R. (1.1.6)
In fact, (1.1.6) is valid for any L > 1.
Using the notion of the half-linear trigonometric functions, we can introduce the basic
half-linear Pr¨ufer transformation
x(t) = ρ(t) sinp ϕ(t), (1.1.7)
rq−1
(t)x (t) = ρ(t) cosp ϕ(t), (1.1.8)
whose modifications will be used later, and we apply it to Eq. (1.1.1) as follows. We
differentiate (1.1.7) and combine it with (1.1.8). It leads to
rq−1
(t)ρ(t) cosp ϕ(t) = ρ (t) sinp ϕ(t) + ρ(t)ϕ (t) cosp ϕ(t). (1.1.9)
Then we apply the function Φ to (1.1.8), we differentiate the result, and combine it with
Eq. (1.1.1). This results to
z(t)ρp−1
(t)Φ(sinp ϕ(t))
= (p − 1) ϕ (t)ρp−1
(t)Φ(sinp ϕ(t)) − ρ (t)ρp−2
(t)Φ(cosp ϕ(t)) .
(1.1.10)
1.1. THE ESSENTIALS OF THE USED TECHNIQUES 3
Finally, we combine the [Φ(cosp ϕ(t))ρ−1
(t)] multiple of (1.1.9) with [sinp ϕ(t)ρ1−p
(t)] multiple
of (1.1.10). It leads directly to the system of first order differential equations
ϕ (t) =
z(t)| sinp ϕ(t)|p
p − 1
+ r1−q
(t)| cosp ϕ(t)|p
,
ρ (t) = ρ(t)Φ(sinp ϕ(t)) cosp ϕ(t) r1−q
(t) −
z(t)
p − 1
.
(1.1.11)
Remark 1.1.1. The function ϕ used above is called the Pr¨ufer angle and the first equation
from (1.1.11) is referred as the equation of the Pr¨ufer angle (this equation will be very
important later). The connection of the Pr¨ufer angle to the oscillation theory is obvious
directly from (1.1.7), i.e., if the Pr¨ufer angle is bounded then there exists the greatest zero
point of the solution x. This is equivalent to the definition of oscillation of Eq. (1.1.1)
given (more correctly) below.
Now, we turn our attention to the Riccati technique. We derive the Riccati equation
associated to Eq. (1.1.1) and we show their mutual connection. To obtain the Riccati
equation, we use the transformation
w(t) = r(t)
Φ(x (t))
Φ(x(t))
, (1.1.12)
where x is a solution of Eq. (1.1.1) which is non-zero on the interval under consideration.
We simply compute the derivative of w and use Eq. (1.1.1) and (1.1.4) as follows
w (t) =
[r(t)Φ(x )] Φ(x) − (p − 1)r(t)Φ(x )|x|p−2
x
Φ2(x)
= −z(t) − (p − 1)r(t)|x |p
|x|−p
= −z(t) − (p − 1)r1−q
(t)|w(t)|q
.
We obtained the half-linear Riccati differential equation
w (t) + z(t) + (p − 1)r1−q
(t)|w(t)|q
= 0. (1.1.13)
The connection of Eq. (1.1.13) and Eq. (1.1.1) is embodied in the below given halflinear
Reid roundabout theorem 1.1.1. Nevertheless, to formulate it properly, we have to
briefly mention the notion of disconjugacy and the energy functional of Eq. (1.1.1).
Definition 1.1.1. Eq. (1.1.1) is said to be disconjugate on the closed interval [a, b] if
the solution x given by the initial condition x(a) = 0, r(a)Φ(x (a)) = 1 has no zero in
(a, b] (by a zero of a solution x we mean such a t0 that x(t0) = 0). In the opposite case
Eq. (1.1.1) is said to be conjugate on [a, b].
We recall that the Sobolev space W1,p
0 (a, b) contains absolutely continuous functions
f such that
f ∈ Lp
(a, b) =



y : (a, b) → R;


b
a
|y(t)|p
+ |y (t)|p
dt


1
p
< ∞



, f(a) = 0 = f(b).
1.1. THE ESSENTIALS OF THE USED TECHNIQUES 4
The energy functional of Eq. (1.1.1) is
F(y; a, b) =
b
a
r(t)|y |p
− z(t)|y|p
dt, y ∈ W1,p
0 (a, b).
Now, we can formulate and prove the half-linear Reid roundabout theorem.
Theorem 1.1.1 (Half-linear Reid roundabout theorem). The following statements are
equivalent.
(i) Eq. (1.1.1) is disconjugate on the interval [a, b].
(ii) There exists a solution of Eq. (1.1.1) having no zero in [a, b].
(iii) There exists a solution w of the generalized Riccati equation (1.1.13) which is defined
on the whole interval [a, b].
(iv) The energy functional F(y; a, b) is positive for every 0 ≡ y ∈ W1,p
0 (a, b).
Proof. To prove that all the statements in the theorem are equivalent, we prove the
validity of the implications
(i) ⇒ (ii) ⇒ (iii) ⇒ (iv) ⇒ (i).
The first implications (i) ⇒ (ii) is a consequence of the continuous dependence of
solutions of Eq. (1.1.1) on initial conditions. More precisely, we suppose that Eq. (1.1.1)
is disconjugate and we consider a solution ˆx of the initial value problem given by Eq. (1.1.1)
and the conditions ˆx(a) = ε, r(a)Φ(ˆx (a)) = 1 with sufficiently small ε > 0. Then the
solution ˆx is positive on [a, b]
The second implication (ii) ⇒ (iii) comes directly from the Riccati substitution
w(t) = r(t)
Φ(x (t))
Φ(x(t))
,
i.e., whenever there exists a solution of Eq. (1.1.1) with no zero, we obtain a solution w
of Eq. (1.1.13).
The third implication (iii) ⇒ (iv) can be proved by a direct computation. We suppose
that there exists a solution w of Eq. (1.1.13) which is defined for all t ∈ [a, b]. Then we
have for x ∈ W1,p
0 (a, b)
F(x;a, b) =
b
a
r(t)|x |p
− z(t)|x|p
dt
=
b
a
(w(t)|x|p
) + pr1−q
(t)
1
p
|rq−1
(t)x |p
− rq−1
(t)x Φ(x)w(t) +
1
q
|Φ(x)w(t)|q
dt
= p
b
a
r1−q
(t)
1
p
|rq−1
(t)x |p
− rq−1
(t)x Φ(x)w(t) +
1
q
|Φ(x)w(t)|q
dt ≥ 0.
1.1. THE ESSENTIALS OF THE USED TECHNIQUES 5
In the above computation, we used the definition of w and the Young inequality
Ap
p
− AB +
Bq
q
≥ 0
with A = rq−1
(t)x and B = Φ(x)w(t) where the equality holds for Φ(A) = B. Of course,
the identity F(x; a, b) = 0 holds if and only if Φ(rq−1
(t)x ) = Φ(x)w(t) which can be
rewritten to x = xΦ−1 w(t)
r(t)
. Immediately, together with the fact that x(a) = 0, we
obtain
x(t) = x(a) exp


t
a
Φ−1 w(s)
r(s)
ds

 ≡ 0.
Hence, the energy functional F(y; a, b) is positive for every 0 ≡ y ∈ W1,p
0 (a, b) and it is
equal to zero only if y ≡ 0.
To prove the final implication (iv) ⇒ (i), we use a contradiction. We suppose that
Eq. (1.1.1) is conjugate and, at the same time, there exists 0 ≡ y ∈ W1,p
0 (a, b) such that
the energy functional F(y; a, b) > 0. The fact that Eq. (1.1.1) is conjugate means that
the solution x given by the initial conditions x(a) = 0, r(a)Φ(x (a)) = 1 has at least one
zero in (a, b]. We denote this zero (or one from these zeros) by t0 and we introduce the
function
y(t) =
x(t) for t ∈ [a, t0],
0 for t ∈ [t0, b]
.
Evidently, y ∈ W1,p
0 (a, b) and using integration by parts we obtain
F(y; a, b) = F(y; a, t0) = F(x; a, t0)
= [r(t)x(t)Φ(x (t))]
t0
a −
t0
a
x(t) [r(t)Φ(x (t))] + z(t)Φ(x(t)) dt = 0.
(1.1.14)
Which is a contradiction and the proof is complete.
To introduce properly the definition of (non-)oscillation of Eq. (1.1.1), we formulate
and prove the half-linear Sturm comparison theorem. Since its usefulness in the upcoming
chapter, we state the half-linear Sturm comparison theorem as well.
Theorem 1.1.2 (Sturm separation theorem). Let t1 < t2 be two consecutive zeros of
a nontrivial solution x of Eq. (1.1.1). Then any other solution of this equation which is
not proportional to x has exactly one zero in the interval (t1, t2).
Proof. Since t1 and t2 are consecutive zeros of a solution x, we can, without loss of
generality, suppose that x(t) > 0 for t ∈ (t1, t2). Hence, we can introduce a solution
w(t) = r(t)Φ(x )
Φ(x)
of Eq. (1.1.13) defined on the interval (t1, t2). Then we have
lim
t→t+
1
w(t) = ∞, lim
t→t−
2
w(t) = −∞.
1.1. THE ESSENTIALS OF THE USED TECHNIQUES 6
Now, we suppose by contradiction that there exists a solution ˆx of Eq. (1.1.1) with no
zero between t1 and t2 which is linearly independent of the solution x. Hence, ˆx(t1) =
0 = ˆx(t2). Since the solution ˆx is non-zero for all t ∈ [t1, t2], there is a solution ˆw(t) =
r(t)Φ(ˆx )
Φ(ˆx)
of Eq. (1.1.13) which is finite at t1 and t2. Especially, ˆw(t1) < ∞ and ˆw(t2) >
−∞. Therefore, w(t0) = ˆw(t0) for some t0 ∈ (t1, t2) which contradicts the uniqueness of
solutions of Eq. (1.1.13). We note that the unique solvability of Eq. (1.1.13) comes from
the fact that it can be rewritten as
w = −z(t) − (p − 1)r1−q
(t)|w|q
which is a first order equation whose right-hand side has the Lipschitz property with
respect to w.
Definition 1.1.2. We say that Eq. (1.1.1) is non-oscillatory (more precisely, non-oscillatory
at infinity), if there exists t0 ∈ R such that Eq. (1.1.1) is disconjugate on any
interval of the form [t0, T], T > t0. Otherwise, Eq. (1.1.1) is said to be oscillatory.
Theorem 1.1.3 (Sturm comparison theorem). We consider Eq. (1.1.1) together with the
equation
[R(t)Φ(x )] + Z(t)Φ(x) = 0, (1.1.15)
where R > 0, Z are continuous functions. Let us denote t1 < t2 two consecutive zeros of
a nontrivial solution x of Eq. (1.1.1) and let the coefficients satisfy
R(t) ≤ r(t), Z(t) ≥ z(t), t ∈ [t1, t2]. (1.1.16)
Then any solution of Eq. (1.1.15) has a zero in (t1, t2) or it is a multiple of the solution
x. The last possibility is excluded if at least one of the inequalities in (1.1.16) is strict on
a set of positive measure.
Proof. Let us consider a nontrivial solution x of Eq. (1.1.1) with two consecutive zeros
t1 < t2. Then, using integration by parts (analogous to (1.1.14)), we obtain F(x; t1, t2) =
0. Combined with (1.1.16) it implies
FRZ(x; t1, t2) =
t2
t1
R(t)|x |p
− Z(t)|x|p
dt ≤ 0.
Using Theorem 1.1.1 (equivalency (i) ⇔ (iv)), we know that the solution ˆx of Eq. (1.1.15)
given by the initial conditions ˆx(t1) = 0, r(t1)Φ(ˆx (t1)) = 1 has a zero in (a, b]. Hence, by
Sturm separation theorem 1.1.2, any solution of Eq. (1.1.15) which is linearly independent
on ˆx has a zero in (t1, t2).
Next, we suppose that the only zero of ˆx in (t1, t2] is t2. Hence, there exists a solution
ˆw(t) = R(t)Φ(ˆx (t))
Φ(ˆx(t))
, t ∈ (t1, t2), of the Riccati equation associated to Eq. (1.1.15) and we
1.1. THE ESSENTIALS OF THE USED TECHNIQUES 7
can calculate the energy functional FRZ(x; t1, t2) as follows
FRZ(x; t1, t2) =
t2
t1
R(t)|x |p
− Z(t)|x|p
dt
=
t2
t1
( ˆw(t)|x|p
) + pR1−q
(t)
|Rq−1
(t)x |p
p
− Rq−1
(t)x Φ(x) ˆw(t) +
|Φ(x) ˆw(t)|q
q
dt ≥ 0.
Indeed, we used Young inequality and the facts
lim
t→t+
1
ˆw(t)|x|p
= lim
t→t+
1
x(t)R(t)Φ(ˆx )
Φ(x(t))
Φ(ˆx(t))
= 0, lim
t→t−
2
ˆw(t)|x|p
= 0,
which are consequences of the existence of the limits
lim
t→t+
1
x(t)
ˆx(t)
= lim
t→t+
1
x (t)
ˆx (t)
=
x (t1)
ˆx (t1)
, lim
t→t−
2
x(t)
ˆx(t)
=
x (t2)
ˆx (t2)
.
Altogether, we obtained that 0 ≤ FRZ(x; t1, t2) ≤ 0, i.e., FRZ(x; t1, t2) = 0. Hence,
ˆw(t) = R(t)Φ(x (t))
Φ(x(t))
(see also the proof of the implication (iii) ⇒ (iv) of Theorem 1.1.1),
i.e., the solutions x and ˆx are proportional which cannot happen if at least one of the
inequalities in (1.1.16) is strict on a set of positive measure.
Remark 1.1.2. If inequalities in (1.1.16) hold, Eq. (1.1.1) is said to be the minorant
equation of Eq. (1.1.15) and Eq. (1.1.15) is said to be the majorant equation of Eq. (1.1.1).
Then we can summarize Theorem 1.1.3 as follows. If minorant equation is oscillatory then
the original equation is oscillatory as well. If majorant equation is non-oscillatory then
the original equation is non-oscillatory as well.
The main objective of this thesis is to study the so-called conditional oscillation of
Eq. (1.1.1). It means that there exists the so-called critical oscillation constant, which is
a value dependent on coefficients r and c with the following property. Any equation of
the form (1.1.1) whose coefficients:
1) indicate a value greater than the critical one is oscillatory;
2) indicate a value less than the critical one is non-oscillatory.
We can reformulate the notion of conditional oscillation as follows. We say that the
equation
[r(t)Φ(x )] + γz(t)Φ(x) = 0, γ ∈ R, (1.1.17)
is conditionally oscillatory if there exists a constant Γ ∈ R such that Eq. (1.1.17) is nonoscillatory
for γ < Γ and oscillatory for γ > Γ. This constant Γ is called the critical
oscillation constant and forms a sharp borderline between oscillation and non-oscillation
1.2. SHORT HISTORY OVERVIEW 8
of Eq. (1.1.17). This fact is also one of the reasons why identifying of conditionally oscillatory
equations and their critical constants is an important part in the field of oscillation
theory. Using many comparison theorems (e.g., the Sturm comparison theorem 1.1.3),
we can test equations which are not conditionally oscillatory towards appropriate conditionally
oscillatory equation. This makes conditionally oscillatory equation ideal testing
equations. Of course, a natural question is a behavior of conditionally oscillatory equations
in the critical case γ = Γ. It turns out that some equations (e.g., with constant
or periodic coefficients, see [19]) are non-oscillatory in the border case. Nevertheless, the
oscillation properties of more complicated equations may be generally unsolvable (see Remark
2.1.3). In this thesis, we will solve critical cases in theorems 2.2.1, 2.3.1, and 2.5.2.
Another importance of half-linear equations lies in their connection with the partial differential
equations with p-Laplacian. Eq. (1.1.1) can be considered the equation with one
dimensional p-Laplacian and results dealing with Eq. (1.1.1) are helpful tools in the study
of more general partial differential equations (for more details and an example of such
use, see § 2.1.2).
1.2 Short history overview
In this section, we collect the milestones in the theory of the conditional oscillation with
respect to the topic of this thesis. It appears that appropriate half-linear equations for
the study of the conditional oscillation are the Euler type equations, i.e., the equations
written in the form
[r(t)Φ (x )] +
γs(t)
tp
Φ(x) = 0 (1.2.1)
with continuous coefficient s.
The conditional oscillation (as well as many other areas in the oscillation theory) of
half-linear equations originates from the oscillation theory of linear differential equations.
The first result about the conditional oscillation of the considered differential equations
was obtained by A. Kneser in [52], where the oscillation constant Γ = 1/4 was found for
the equation
x +
γ
t2
x = 0. (1.2.2)
More than one hundred years later, in [33, 70], the above result concerning Eq. (1.2.2)
was extended for the linear equations
[r(t)x ] +
γs(t)
t2
x = 0 (1.2.3)
with positive α-periodic coefficients r, s, where the critical constant is
Γ =
α2
4


α
0
dτ
r(τ)


−1 

α
0
s(τ) dτ


−1
. (1.2.4)
1.2. SHORT HISTORY OVERVIEW 9
We should also mention, at least as references, papers [53, 54, 55, 56, 71] containing
more general results (see also [31, 32]). Note that the critical case γ = Γ of Eq. (1.2.3)
was solved as non-oscillatory (see [71]).
In the field of half-linear equations, the basic critical constant
Γ =
p − 1
p
p
(1.2.5)
for the equation
[Φ (x )] +
γ
tp
Φ(x) = 0
comes from [26] (see also [27]). Then, in [34, 38], the conditional oscillation was proved
for more general equations of the form
[r(t)Φ (x )] +
γs(t)
tp
Φ(x) = 0. (1.2.6)
Especially, the critical constant of Eq. (1.2.6) with positive α-periodic functions r, s was
identified as (cf. (1.2.4), (1.2.5))
Γ1 =
α(p − 1)
p
p


α
0
r
1
1−p (τ) dτ


1−p 

α
0
s(τ) dτ


−1
(1.2.7)
in [34].
Let us turn our attention to the perturbed Euler type equations. The linear case of
such equations with periodic coefficients is studied in [54, 71]. The half-linear case is
treated in [19], where the equations
[r(t)Φ (x )] + γs(t) +
µd(t)
log2
t
Φ(x)
tp
= 0 (1.2.8)
are analyzed for positive α-periodic coefficients r, s, and d. There is proved that, in the
critical case γ = Γ1 (see (1.2.7)), Eq. (1.2.8) is oscillatory for
µ > Γ2 :=
αp
2
p − 1
p
p−1


α
0
r
1
1−p (τ) dτ


1−p 

α
0
d(τ) dτ


−1
and non-oscillatory for µ < Γ2. For further generalizations, we refer to [16, 17, 22, 39]
(see also [28]).
Another direction of researches, which is related to the one presented here, is based
on the oscillation of Euler type equations generalizing Eq. (1.2.2) in a different way. We
point out at least papers [4, 5, 73, 74, 79], where the equations of the following form (and
generalizations of this form)
x + f(t)g(x) = 0
are considered and oscillation theorems are proved.
1.3. ORGANIZATION OF THE THESIS 10
1.3 Organization of the thesis
In this section, we briefly mention only the main result of each upcoming section. As
we already mentioned above, the main part of this thesis is the analysis of half-linear
differential equations which is the topic of Chapter 2. In Section 2.1, we study Euler-type
equation (1.2.1) with coefficients r, s having mean values. From technical reasons, we
rewrite Eq. (1.2.1) into the form
r−p
q (t)Φ(x ) +
s(t)
tp
Φ(x) = 0 (1.3.1)
Since the coefficient r is considered positive, it does not mean any loss of generality.
We prove that this equation is conditionally oscillatory and we identify the borderline
of oscillation and non-oscillation. Of course, this result cover all previously known results
concerning Euler-type equations (1.2.3) and (1.2.6) whose coefficients are constant,
periodic, (asymptotically) almost periodic etc. This section is based on paper [35].
The critical case remains unsolved in Section 2.1 and for such general equations it is
not possible to solve it in general (for details see Remark 2.1.3). Therefore, Sections 2.2
and 2.3 are devoted to this problem. Section 2.2 (based on paper [40]) deals with the
study of the critical case of Eq. (1.3.1) with periodic coefficients r, s whose periods do not
have to coincide (e.g., one may be rational and the other irrational). In Section 2.3, we
solve the critical case for the equation
r(t) +
m
i=1 Ri(t)
log2
t
−p
q
Φ (x ) + s(t) +
n
i=1 Si(t)
log2
t
Φ(x)
tp
= 0 (1.3.2)
with α-periodic coefficients r, s and general periodic coefficients R1, . . . , Rm, S1, . . . , Sn
defined on some interval [a, ∞), a ∈ R+
(see [41]). Moreover, we are able to combine
results of Section 2.3 to prove that Eq. (1.3.2) is non-oscillatory if and only if
lim
t→∞



t
αp


a+α
a
r(τ) dτ


p
q


a+α
a
s(τ) dτ

 − q−p
t
+
1
tp


a+t
a
n
i=1
Si(τ) dτ




a+t
a
r(τ) dτ


p
q
+
p
a+t
a
m
i=1
Ri(τ) dτ
qp+1
a+t
a
r(τ) dτ





≤
q1−p
2
,
see Theorem 2.3.4.
Then we turn our attention to another type of conditionally oscillatory equations
r(t)tp−1
Φ(x ) +
s(t)
t logp
t
Φ(x) = 0, (1.3.3)
1.3. ORGANIZATION OF THE THESIS 11
where r > 0 and s are continuous. Eq. (1.3.3) is discussed in Section 2.4 (see paper [42]).
We prove that this equation remains conditionally oscillatory and we identify the sharp
borderline. Remarkable is the fact that the obtained results cover any equation of the
form (1.3.3) with coefficients r > 0, s satisfying
lim
t→∞
t+α
t
r1−q
(τ) dτ
√
t log t
= 0, lim
t→∞
t+α
t
|s(τ)| dτ
√
t log t
= 0
for some positive α. Unfortunately, as well as above, the critical case is complicated
and remains unsolved in general. Hence, we dedicate Section 2.5 to the critical case
of Eq. (1.3.3) with periodic coefficients r, s whose period may differ, i.e., without any
common period (see [44]).
The last section in Chapter 2 deals with the behavior of the perturbed Eq. (1.3.3).
The objective of Section 2.6 is to identify the form of perturbations which preserve the
conditional oscillation of the original equation and to solve its behavior (see [45]). More
precisely, we perturb Eq. (1.3.3) in the critical case and we analyze (non-)oscillation of
the resulting equation
r1(t) +
r2(t)
[log(log t)]2
−p
q
tp−1
Φ(x )
+
1
t logp
t
s1(t) +
s2(t)
[log(log t)]2 Φ(x) = 0,
where r1 > 0 and s1 are α-periodic continuous functions and r2, s2 are continuous functions
satisfying
r1(t) +
r2(t)
[log(log t)]2 > 0, t > e,
lim
t→∞
1
√
t log t
t+α
t
|r2(u)| du = 0, lim
t→∞
1
√
t log t
t+α
t
|s2(u)| du = 0.
Chapters 3 and 4 are devoted to a presentation of the situation in the discrete case
and the case of dynamic equations on time scales. Since the discrete and time scale
methods are more complicated than in the continuous case, the results are in general not
as advanced as in the continuous case. Nevertheless, we succeeded to obtain a discrete
version of the result from Section 2.1. This result is contained in Chapter 3 (which is
based on paper [43]). We find the oscillation constant of the half-linear difference equation
∆ [rkΦ(∆xk)] +
sk
(k + 1)(p)
Φ(xk+1) = 0,
where k(p)
stands for the generalized power function, the sequence {rk} is a positive
bounded sequence such that there exists a (positive) mean value of {r1−q
k } and the sequence
{sk} has a mean value.
1.3. ORGANIZATION OF THE THESIS 12
Finally, in Chapter 4 (see [46]) are treated dynamic half-linear equations on time scales
r(t)Φ(y∆
)
∆
+
s(t)
t(p−1)σ(t)
Φ(yσ
) = 0, (1.3.4)
where σ, f∆
, t(p)
stand for the forward jump operator, ∆-derivative, and generalized power
function, respectively. The considered coefficients r, s are rd-continuous, positive, and
periodic. The obtained result shows that Eq. (1.3.4) is conditionally oscillatory and
reveals its critical constant. In spite of the fact that this result is much weaker than the
results in continuous and discrete case, as far as we know, it is the most general result
about conditional oscillation on time scales and it could be a step to this rich field.
Chapter
2 Differential equations
2.1 Equations with coefficients having mean values
In this section, we will study Eq. (1.1.1) in the form
r−p
q (t)Φ(x ) +
s(t)
tp
Φ(x) = 0, (2.1.1)
where r : Ra → R is a continuous function having mean value M(r) = 1 and satisfying
0 < r−
:= inf
t∈Ra
r(t) ≤ r+
:= sup
t∈Ra
r(t) < ∞ (2.1.2)
and s : Ra → R is a continuous function having mean value M(s) > 0.
The Riccati equation associated to Eq. (2.1.1) has the form (see (1.1.13))
w (t) +
s(t)
tp
+ (p − 1)r(t)|w(t)|q
= 0. (2.1.3)
Finally, using the substitution w(t) = −ζ(t)t1−p
, we obtain the adapted Riccati equation
ζ (t) =
1
t
[(p − 1)ζ(t) + s(t) + (p − 1)r(t)|ζ(t)|q
] , (2.1.4)
which will play a crucial role in the proof of the announced result (see the below given
Theorem 2.1.3).
To prove the main results, we will apply the Riccati technique for Eq. (1.1.1). The
fundamental connection between the non-oscillation of Eq. (1.1.1) and the solvability of
Eq. (1.1.13) is described by the following theorem.
Theorem 2.1.1. Eq. (1.1.1) is non-oscillatory if and only if there exists a function w
which solves Eq. (1.1.13) on some interval [T, ∞).
Proof. The theorem is a consequence of the Reid roundabout theorem 1.1.1.
We will also use the Sturm comparison theorem in the form given below.
Theorem 2.1.2. Let z, Z : Ra → R be continuous functions satisfying Z(t) ≥ z(t) for all
sufficiently large t. Let us consider Eq. (1.1.1) and the equation
[r(t)Φ(x )] + Z(t)Φ(x) = 0. (2.1.5)
13
2.1. COEFFICIENTS WITH MEAN VALUES 14
(i) If Eq. (1.1.1) is oscillatory, then Eq. (2.1.5) is oscillatory as well.
(ii) If Eq. (2.1.5) is non-oscillatory, then Eq. (1.1.1) is non-oscillatory as well.
Proof. The theorem is a (weaker) reformulation of Theorem 1.1.3 (see Remark 1.1.2).
Now, we recall the concept of mean values which is necessary to find an explicit
oscillation constant for general half-linear equations.
Definition 2.1.1. Let a continuous function f : Ra → R be such that the limit
M(f) := lim
t→∞
1
t
b+t
b
f(s) ds (2.1.6)
is finite and exists uniformly with respect to b ∈ Ra. The number M(f) is called the
mean value of f.
§ 2.1.1 Results and examples
To prove the announced result, we need the following lemmata.
Lemma 2.1.1. If there exists a solution of Eq. (2.1.4) on some interval [T, ∞), then
Eq. (2.1.1) is non-oscillatory.
Proof. A solution ζ of Eq. (2.1.4) on an interval [T, ∞) gives the solution w(t) = −ζ(t)t1−p
of Eq. (2.1.3) on the same interval. Thus, the lemma follows from Theorem 2.1.1.
Lemma 2.1.2. Let Eq. (2.1.1) be non-oscillatory and let there exists M > 0 such that
c
b
s(τ)
τp
dτ < M, a ≤ b < c ≤ ∞. (2.1.7)
For any solution w of Eq. (2.1.3) on [T, ∞), it holds
∞
T
r(τ) |w(τ)|q
dτ < ∞. (2.1.8)
Proof. The lemma follows, e.g., from [21, Theorem 2.2.3], where it suffices to use (2.1.2).
Lemma 2.1.3. If Eq. (2.1.1) is non-oscillatory, then there exists a solution ζ of Eq. (2.1.4)
on some interval [T, ∞) with the property that ζ(t) < A for all t ≥ T and for some A > 0.
2.1. COEFFICIENTS WITH MEAN VALUES 15
Proof. Considering Theorem 2.1.1, the non-oscillation of Eq. (2.1.1) implies that there
exists a solution w of Eq. (2.1.3) on some interval [T, ∞) which gives the solution ζ(t) =
−w(t)tp−1
of Eq. (2.1.4) on the interval. We show that this solution ζ is bounded above.
At first, we prove the convergence of the integral
∞
T
s(τ)
τp
dτ ∈ R (2.1.9)
and the inequality
sup
t≥T
tp−1
∞
t
s(τ)
τp
dτ < L for some L > 0. (2.1.10)
Evidently, it suffices to prove (2.1.9) and
lim sup
t→∞
tp−1
∞
t
s(τ)
τp
dτ < ∞. (2.1.11)
Let b > 0 be such that
t+b
t
s(τ) dτ > 0,
t+b
t
s(τ) dτ − bM(s) < b, t ≥ T, (2.1.12)
where we use directly Definition 2.1.1 (the existence of M(s) > 0). The symbol [f(·)]+
and [f(·)]−
will denote the positive and negative part of function f, respectively. We
choose t0 ≥ T. We can express
t0+kb
t0+(k−1)b
s(τ)
τp
dτ =
t0+kb
t0+(k−1)b
[s(τ)]+
τp
dτ −
t0+kb
t0+(k−1)b
[s(τ)]−
τp
dτ, k ∈ N.
For an arbitrarily given positive integer k, we have
Ik :=
t0+kb
t0+(k−1)b
s(τ)
τp
dτ ≤
1
[t0 + (k − 1)b]p
t0+kb
t0+(k−1)b
[s(τ)]+
dτ
−
1
[t0 + kb]p
t0+kb
t0+(k−1)b
[s(τ)]−
dτ
(2.1.13)
2.1. COEFFICIENTS WITH MEAN VALUES 16
if Ik > 0, and
−
t0+kb
t0+(k−1)b
s(τ)
τp
dτ ≤ −
1
[t0 + kb]p
t0+kb
t0+(k−1)b
[s(τ)]+
dτ
+
1
[t0 + (k − 1)b]p
t0+kb
t0+(k−1)b
[s(τ)]−
dτ
(2.1.14)
if Ik < 0. Using
lim
k→∞
[t0 + (k − 1)b]p
[t0 + kb]p
= 1
and using (2.1.12), (2.1.13), (2.1.14), we obtain the existence of n0 ∈ N such that it holds
t0+kb
t0+(k−1)b
s(τ)
τp
dτ <
2b[M(s) + 1]
[t0 + (k − 1)b]p
, k ≥ n0, k ∈ N.
Since t0 ≥ T is arbitrary, it also holds
t+b
t
s(τ)
τp
dτ <
2b[M(s) + 1]
tp
(2.1.15)
for all sufficiently large t.
Hence, the integral
∞
T
s(τ)τ−p
dτ is convergent. Especially,
sup
t≥T
t
T
s(τ)
τp
dτ < K for some K > 0. (2.1.16)
Moreover, we have (see (2.1.15))
lim sup
t→∞
tp−1
∞
t
s(τ)
τp
dτ = lim sup
t→∞
tp−1
∞
k=1
t+kb
t+(k−1)b
s(τ)
τp
dτ
≤ lim sup
t→∞
tp−1
∞
k=1
2b[M(s) + 1]
[t + (k − 1)b]p
< ∞.
Thus, (2.1.11) is valid, i.e., there exists L > 0 for which (2.1.10) is valid.
Integrating Eq. (2.1.3), we obtain
w(t) = w(T) −
t
T
s(τ)
τp
dτ − (p − 1)
t
T
r(τ) |w(τ)|q
dτ, t ≥ T. (2.1.17)
2.1. COEFFICIENTS WITH MEAN VALUES 17
We know that
∞
T
r(τ) |w(τ)|q
dτ < ∞, i.e. (see (2.1.2)),
∞
T
|w(τ)|q
dτ < ∞. (2.1.18)
Indeed, considering (2.1.7) together with (2.1.16), one can get (2.1.8) from Lemma 2.1.2.
From (2.1.9) and (2.1.17) it follows that there exists the limit limt→∞ w(t) ∈ R. In
addition, the convergence of the integral in (2.1.18) gives
lim
t→∞
|w(t)|q
= 0, i.e., lim
t→∞
w(t) = 0. (2.1.19)
Again, we consider arbitrarily given t0 ≥ T. We can rewrite Eq. (2.1.17) into (or see
directly Eq. (2.1.3))
w(t) = w(t0) −
t
t0
s(τ)
τp
dτ − (p − 1)
t
t0
r(τ) |w(τ)|q
dτ, t ≥ T. (2.1.20)
Putting t0 → ∞, from (2.1.9), (2.1.18), (2.1.19), and (2.1.20), we obtain
w(t) =
∞
t
s(τ)
τp
dτ + (p − 1)
∞
t
r(τ) |w(τ)|q
dτ, t ≥ T. (2.1.21)
Finally, let us denote w(t) = f1(t) + f2(t), where
f1(t) :=
∞
t
s(τ)
τp
dτ, f2(t) := (p − 1)
∞
t
r(τ) |w(τ)|q
dτ, t ≥ T.
We know that (see (2.1.10) and (2.1.21))
sup
t≥T
tp−1
| w(t) − f2(t) | < L. (2.1.22)
We denote S := {t ≥ T : w(t) < 0}. If w is positive, then the statement of the lemma is
true for all A > 0. Therefore, we can assume that S = ∅. Since f2 is non-increasing and
limt→∞ f2(t) = 0, function f2 is nonnegative. From (2.1.22) it follows
sup
t∈S
tp−1
[w(t)]−
+ f2(t) = sup
t∈S
tp−1
| w(t) − f2(t) | < L.
Hence, we have
sup
t∈S
tp−1
|w(t)| = sup
t∈S
tp−1
[w(t)]−
+ 0 ≤ sup
t∈S
tp−1
[w(t)]−
+ f2(t) < L
and, consequently, we obtain that ζ(t) = −tp−1
w(t) < L, t ≥ T. It means that the
statement of the lemma is valid for A = L.
2.1. COEFFICIENTS WITH MEAN VALUES 18
Remark 2.1.1. Let Eq. (2.1.1) be non-oscillatory. If the considered function s is positive
for all t ≥ a, then the statement of Lemma 2.1.3 is true for a negative solution ζ of
Eq. (2.1.4). See, e.g., [21, Lemma 2.2.5].
Theorem 2.1.3. Eq. (2.1.1) is oscillatory for M(s) > q−p
, and non-oscillatory for
M(s) < q−p
.
Proof. The proof is organized as follows. In the first part, we derive upper bounds for
two integrals involving function s. Then we prove the oscillatory part and, finally, the
non-oscillatory part.
At first, we use the existence of M(s) and the continuity of function s. There exists
β ≥ 1 with the property that
b+β
b
s(τ) dτ < β [M(s) + 1] , b ∈ [a, ∞), (2.1.23)
1
β
b+β
b
s(τ) dτ −
1
β + ξ
b+β+ξ
b
s(τ) dτ < 1, b ∈ [a, ∞), ξ ∈ (0, 1], (2.1.24)
and, consequently, there exists R > 0 with the property that
c+ξ
c
s(τ) dτ < R, c ∈ [a, a + β], ξ ∈ (0, 1]. (2.1.25)
We can rewrite (2.1.24) into the form
ξ
β + ξ
b+β
b
s(τ) dτ −
β
β + ξ
b+β+ξ
b+β
s(τ) dτ < β, b ∈ [a, ∞), ξ ∈ (0, 1].
Using (2.1.23), we obtain
β
β + ξ
b+β+ξ
b+β
s(τ) dτ < β + β[M(s) + 1], b ∈ [a, ∞), ξ ∈ (0, 1],
i.e.,
b+β+ξ
b+β
s(τ) dτ < 2β [M(s) + 2] , b ∈ [a, ∞), ξ ∈ (0, 1]. (2.1.26)
2.1. COEFFICIENTS WITH MEAN VALUES 19
Combining (2.1.25) and (2.1.26), we have
b+ξ
b
s(τ) dτ < S, b ∈ [a, ∞), ξ ∈ (0, 1], (2.1.27)
where S := max {R, 2β[M(s) + 2]}. Since the function y(t) = 1/t is decreasing and
positive on Ra, it holds
t2
t1
s(τ)
τ
dτ =
1
t1
t3
t1
s(τ) dτ +
1
t2
t2
t3
s(τ) dτ
for all t2 ≥ t1 ≥ a and for some t3 ∈ [t1, t2]. Analogously, for any t2 ≥ t1 ≥ a, there exists
t4 ∈ [t1, t2] such that
t2
t1
s(τ)
τ
dτ =
1
t1
t4
t1
s(τ) dτ.
Hence, from (2.1.27) it follows
b+ξ
b
s(τ)
τ
dτ <
S
b
, b ∈ [a, ∞), ξ ∈ (0, 1]. (2.1.28)
Now, we prove the oscillatory part. Let M(s) > q−p
. By contradiction, in this part
of the proof, we will suppose that Eq. (2.1.1) is non-oscillatory. Lemma 2.1.3 says that
there exists a solution ζ of Eq. (2.1.4) on some interval [T, ∞) and that ζ(t) < A for all
t ≥ T and for a certain number A > 0. Evidently, we can assume that T > 1.
We show that there exists K < −1 satisfying
ζ(t) > K, t ≥ T. (2.1.29)
On contrary, let us assume that lim inft→∞ ζ(t) = −∞. Let ζ(t) ≤ −P for all t from some
interval [t1, t2], where t2 ∈ (t1 + j − 1, t1 + j], j ∈ N, and let P > 0 be such that (see
(2.1.2))
h(x) := (p − 1)r−
|x|q
− S − (p − 1)|x| > 0, |x| ≥ P. (2.1.30)
2.1. COEFFICIENTS WITH MEAN VALUES 20
Indeed, limx→±∞ h(x) = ∞. We can assume that h is increasing for x ≥ P. Using (2.1.4),
(2.1.28), (2.1.30), it holds
ζ(t2) − ζ(t1) =
t2
t1
ζ (τ) dτ =
t2
t1
(p − 1)ζ(τ) + s(τ) + (p − 1)r(τ)|ζ(τ)|q
τ
dτ
≥
t2
t1
−(p − 1)P + (p − 1)r−
Pq
τ
dτ −
t2
t1
s(τ)
τ
dτ
≥
t1+1
t1
−(p − 1)P + (p − 1)r−
Pq
τ
dτ −
t1+1
t1
s(τ)
τ
dτ + · · ·
· · · +
t1+j−1
t1+j−2
−(p − 1)P + (p − 1)r−
Pq
τ
dτ −
t1+j−1
t1+j−2
s(τ)
τ
dτ
+
t2
t1+j−1
−(p − 1)P + (p − 1)r−
Pq
τ
dτ −
t2
t1+j−1
s(τ)
τ
dτ
>
S
t1 + 1
−
S
t1
+ · · · +
S
t1 + j − 1
−
S
t1 + j − 2
+ 0 −
S
t1 + j − 1
= −
S
t1
.
Thus, ζ(t) ≥ −P −ST−1
for all t ≥ T which proves (2.1.29). Indeed, it suffices to consider
ζ(t1) = −P. In addition, we can assume that −K > A, i.e.,
K < ζ(t) < −K, t ≥ T. (2.1.31)
Thus (see directly (2.1.4) and (2.1.28)), we have
| ζ(t2) − ζ(t1) | =
t2
t1
ζ (τ) dτ ≤
t2
t1
p − 1
τ
[|ζ(τ)| + r(τ)|ζ(τ)|q
] dτ +
t2
t1
s(τ)
τ
dτ
≤
1
t1
(p − 1)|K| (t2 − t1) + S + (p − 1)r+
|K|q
(t2 − t1)
for all t2 > t1 ≥ T, where t2 ≤ t1 + 1. The previous inequality implies
| ζ(t2) − ζ(t1) | ≤
(p − 1)|K| + S + (p − 1)r+
|K|q
t1
, t1 ≥ T, t2 ∈ (t1, t1 + 1]. (2.1.32)
Considering Definition 2.1.1 and M(s) > q−p
, there exist n ∈ N and ε > 0 such that
1
n
b+n
b
s(τ) dτ − q−p
> ε, b ∈ [a, ∞), (2.1.33)
2.1. COEFFICIENTS WITH MEAN VALUES 21
and, at the same time, such that
1 −
ε
4(p − 1)|K|q
<
1
n
b+n
b
r(τ) dτ < 2, b ∈ [a, ∞). (2.1.34)
For such an integer n, we define
ϑ(t) :=
1
n
t+n
t
ζ(τ) dτ, t ≥ T. (2.1.35)
Since
1
n
t+n
t
ζ(τ) dτ <
1
n
t+n
t
A dτ = A, t ≥ T,
we have
ϑ(t) < A, t ≥ T. (2.1.36)
Hence, to prove the first implication in the statement of the theorem, it suffices to show
that (2.1.36) is not true.
From (2.1.32) it follows
| ζ(t + τ) − ζ(t) | ≤
C
t
, t ≥ T, τ ∈ [0, n], (2.1.37)
where
C = n (p − 1)|K| + S + (p − 1)r+
|K|q
.
Especially (see (2.1.35)), (2.1.37) gives
| ϑ(t) − ζ(t) | ≤
C
t
, t ≥ T. (2.1.38)
Next, we consider the function
g1(t) := (p − 1)ζ(t) + q−p
+
p − 1
n
t+n
t
r(τ) |ζ(τ)|q
dτ, t ≥ T.
If ζ(t) ≥ 0 for some t ≥ T, then g1(t) > 0. Henceforth (in this paragraph), we consider
the case when ζ(t) < 0, t ≥ T. Let us define
f(x) := x +
q−p
p − 1
+ (−x)q
, x ≤ 0.
2.1. COEFFICIENTS WITH MEAN VALUES 22
It can be directly verified that function f has the global minimum
f −q
1
1−q = −q
1
1−q +
q−p
p − 1
+ q
1+q−1
1−q = q
1
1−q −1 + q
q−1
1−q +
q− q
q−1
q
q−1
− 1
= q
1
1−q
1
q
− 1 +
q
1+q−1
1−q
q
q−1
− 1
= q
1
1−q
1 − q
q
+
q−1
q−q+1
q−1
= q
1
1−q
1 − q
q
+
q − 1
q
= 0.
It means that f(x) ≥ 0, x ≤ 0. Especially, it gives the inequality
(p − 1)ζ(t) + q−p
+ (p − 1) |ζ(t)|q
≥ 0, t ≥ T. (2.1.39)
Considering (2.1.34) and (2.1.39), we have
g2(t) := (p − 1)ζ(t) + q−p
+
p − 1
n
t+n
t
r(τ) |ζ(t)|q
dτ ≥ −
ε
4
, t ≥ T. (2.1.40)
Applying (2.1.37) and the inequalities K < ζ(t) < A, t ≥ T, there exists ˜t ≥ T such that
| |ζ(t)|q
− |ζ(t + τ)|q
| <
ε
8(p − 1)
, t ≥ ˜t, τ ∈ [t, t + n].
Hence (see also (2.1.34)), we get
| g1(t) − g2(t) | ≤
p − 1
n
t+n
t
r(τ) | |ζ(t)|q
− |ζ(τ)|q
| dτ
≤
p − 1
n
t+n
t
r(τ)
ε
8(p − 1)
dτ < 2 (p − 1)
ε
8(p − 1)
=
ε
4
, t ≥ ˜t.
(2.1.41)
From (2.1.40) and (2.1.41), we know that
g1(t) ≥ −
ε
2
, t ≥ ˜t. (2.1.42)
Of course, (2.1.42) remains true for ζ(t) ≥ 0 as well.
Let us consider t ≥ ˜t for which
| ζ(t) − ϑ(t) | <
ε
4(p − 1)
, t ≥ t. (2.1.43)
Note that the existence of such a number t follows from (2.1.38). It is seen that (2.1.42)
and (2.1.43) imply
(p − 1)ϑ(t) + q−p
+
p − 1
n
t+n
t
r(τ) |ζ(τ)|q
dτ ≥ −
3ε
4
, t ≥ t. (2.1.44)
2.1. COEFFICIENTS WITH MEAN VALUES 23
Evidently, we can consider the solution ζ in an arbitrarily given neighborhood of +∞.
Hence, we can assume that
Sn
t
<
ε
24
, (2.1.45)
−Kn(p − 1)
t
<
ε
24
, (2.1.46)
r+
(−K)q
n(p − 1)
t
<
ε
24
. (2.1.47)
From (2.1.27) and (2.1.45), we see
1
n
t+n
t
s(τ)
τ
dτ −
1
n
t+n
t
s(τ)
t
dτ =
1
n
t+n
t
s(τ)
1
τ
−
1
t
dτ
≤
1
n
t+n
t
s(τ) dτ
n
t2
<
Sn
t2
<
ε
24t
, t ≥ t,
(2.1.48)
from (2.1.31) and (2.1.46), we have
p − 1
n
t+n
t
ζ(τ)
τ
dτ −
p − 1
n
t+n
t
ζ(τ)
t
dτ =
p − 1
n
t+n
t
ζ(τ)
1
τ
−
1
t
dτ
≤
p − 1
t2
t+n
t
|ζ(τ)| dτ <
−Kn(p − 1)
t2
<
ε
24t
, t ≥ t,
(2.1.49)
and, analogously, from (2.1.2), (2.1.31), and (2.1.47) it follows
p − 1
n
t+n
t
r(τ)
|ζ(τ)|q
τ
dτ −
p − 1
n
t+n
t
r(τ)
|ζ(τ)|q
t
dτ
=
p − 1
n
t+n
t
r(τ) |ζ(τ)|q 1
t
−
1
τ
dτ
≤
p − 1
t2
t+n
t
r(τ) |ζ(τ)|q
dτ <
r+
(−K)q
n(p − 1)
t2
<
ε
24t
, t ≥ t.
(2.1.50)
2.1. COEFFICIENTS WITH MEAN VALUES 24
For all t ≥ t, using (2.1.33), (2.1.44), (2.1.48), (2.1.49), and (2.1.50), we obtain
ϑ (t) =
1
n
t+n
t
ζ (τ) dτ =
1
n
t+n
t
(p − 1)ζ(τ) + s(τ) + (p − 1)r(τ) |ζ(τ)|q
τ
dτ
=
1
n
t+n
t
s(τ)
τ
dτ −
q−p
t
+
p − 1
n
t+n
t
ζ(τ)
τ
dτ +
q−p
t
+
p − 1
n
t+n
t
r(τ)
|ζ(τ)|q
τ
dτ
>
1
n
t+n
t
s(τ)
t
dτ −
q−p
t
+
p − 1
n
t+n
t
ζ(τ)
t
dτ +
q−p
t
+
p − 1
n
t+n
t
r(τ)
|ζ(τ)|q
t
dτ −
ε
8t
=
1
t

1
n
t+n
t
s(τ) dτ − q−p
+ (p − 1)ϑ(t) + q−p
+
p − 1
n
t+n
t
r(τ) |ζ(τ)|q
dτ −
ε
8

 >
ε
8t
.
Thus, it holds
ϑ(t) − ϑ(t) =
t
t
ϑ (τ) dτ ≥
t
t
ε
8τ
dτ, t ≥ t.
Since
lim
t→∞
t
t
ε
8τ
dτ = ∞,
we obtain that limt→∞ ϑ(t) = ∞. The contradiction with (2.1.36) proves the first impli-
cation.
In the non-oscillatory part of the proof, we consider M(s) < q−p
. Let n ∈ N and ε > 0
satisfy
1
n
b+n
b
s(τ) dτ − q−p
< −ε, b ∈ [a, ∞). (2.1.51)
Let us consider solution ζ of Eq. (2.1.4) given by ζ(T0) = −1 for some sufficiently large
T0 ≥ a. Since the right-hand side of Eq. (2.1.4) is continuous, the considered solution ζ
can be defined on an interval [T0, T1), where T0 < T1 ≤ ∞. In addition, if T1 < ∞, we
can assume that
lim sup
t→T−
1
|ζ(t)| = ∞. (2.1.52)
If T1 = ∞, then the considered solution of Eq. (2.1.4) satisfies the condition of Lemma
2.1.1. It means that it suffices to find B, C ∈ R for which
B ≤ ζ(t) ≤ C, t ∈ [T0, T1). (2.1.53)
As in the oscillatory part of the proof (see (2.1.29)), we can prove that ζ(t) > K for
some K < −1 and for all t ∈ [T0, T1). Indeed, we can analogously show that the inequality
2.1. COEFFICIENTS WITH MEAN VALUES 25
ζ(t) < −P −ST−1
0 cannot be valid for any t ∈ [T0, T1), where S is taken from (2.1.27) and
P from (2.1.30). We want to prove that T1 = ∞. On contrary, let (2.1.52) be valid for
some T1 ∈ R. Especially, solution ζ has to be positive on some interval [T2, T3] ⊂ [T0, T1)
in this case.
We denote
˜t := −
p
p − 1
1−p
= −q1−p
(2.1.54)
and we compute
(p − 1)˜t + q−p
+ (p − 1) ˜t
q
= (1 − p)q1−p
+ pq−p
= q−p
[(1 − p)q + p] = 0. (2.1.55)
We know that ζ is negative on an interval [T0, ˜T1) ⊆ [T0, T1). Let ˜T1 have the property
that ζ( ˜T1) = 0. For all t1, t2 ∈ [T0, ˜T1], t1 ≤ t2 ≤ t1 + 1, we have (see (2.1.28))
| ζ(t2) − ζ(t1) | =
t2
t1
ζ (τ) dτ ≤ (p − 1)
t2
t1
|ζ(τ)| + r+
|ζ(τ)|q
τ
dτ +
t2
t1
s(τ)
τ
dτ
≤ (p − 1)
|K| + r+
|K|q
t1
+
S
t1
=
(p − 1) [|K| + r+
|K|q
] + S
t1
.
Thus, for general t1, t2 ∈ [T0, ˜T1] satisfying t1 ≤ t2 ≤ t1 + 2n, we have
| ζ(t2) − ζ(t1) | ≤
2n ([p − 1] [|K| + r+
|K|q
] + S)
t1
. (2.1.56)
We can assume that T0 is so large that
ζ(τ) < 0, τ ∈ [t, t + 2n] ⊂ [T0, T1),
if ζ(t) ≤ ˜t (see (2.1.54)). Namely (ζ(T0) = −1 < ˜t), we can define the function
ϑ(t) :=
1
n
t+n
t
ζ(τ) dτ
for all t ∈ [T0, T0 + n) and for all t ≥ T0 + n when ζ(t − n) ≤ ˜t. Especially, let T0 be so
large that ϑ(T0) < ˜t.
We repeat that we assume the positivity of ζ which implies the inequality ϑ(t) > ˜t for
t from some interval. The continuity of ϑ gives the existence of t > T0 such that
ϑ(t) = ˜t, ϑ (t) ≥ 0. (2.1.57)
From (2.1.56) it follows that, for any δ > 0, one can choose T0 such that
| ζ(t2) − ζ(t1) | < δ, T0 ≤ t1 ≤ t2 ≤ t1 + n ≤ ˜T1.
2.1. COEFFICIENTS WITH MEAN VALUES 26
Thus, we can assume that
ϑ(t) − ζ(τ) < δ, τ ∈ t, t + n . (2.1.58)
Consequently, let
|ϑ(t)|q
− |ζ(τ)|q
<
ε
8(p − 1)
, τ ∈ t, t + n . (2.1.59)
At the same time, we can assume that n ∈ N was chosen in such a way that it is valid
1
n
b+n
b
r(τ) dτ − 1 < min
ε
4(p − 1)
, 1 , b ∈ [a, ∞). (2.1.60)
Using (2.1.59) and (2.1.60), we have

(p − 1)ϑ(t) + q−p
+
p − 1
n
t+n
t
r(τ) |ζ(τ)|q
dτ

 − (p − 1)ϑ(t) + q−p
+ (p − 1) ϑ(t)
q
= (p − 1)
1
n
t+n
t
r(τ) |ζ(τ)|q
dτ − ϑ(t)
q
+
1
n
t+n
t
r(τ) ϑ(t)
q
dτ −
1
n
t+n
t
r(τ) ϑ(t)
q
dτ
≤ (p − 1)
1
n
t+n
t
r(τ) |ζ(τ)|q
dτ −
1
n
t+n
t
r(τ) ϑ(t)
q
dτ
+ (p − 1) ϑ(t)
q
−
1
n
t+n
t
r(τ) ϑ(t)
q
dτ
≤ (p − 1)
1
n
t+n
t
r(τ) |ζ(τ)|q
− ϑ(t)
q
dτ + (p − 1) 1 −
1
n
t+n
t
r(τ) dτ <
ε
4
+
ε
4
=
ε
2
.
Since (see (2.1.55), (2.1.57))
(p − 1)ϑ(t) + q−p
+ (p − 1) ϑ(t)
q
= (p − 1)˜t + q−p
+ (p − 1) ˜t
q
= 0,
we have
(p − 1)ϑ(t) + q−p
+
p − 1
n
t+n
t
r(τ) |ζ(τ)|q
dτ <
ε
2
. (2.1.61)
Let T0 be so large that (see (2.1.27) and also (2.1.48))
1
n
t+n
t
s(τ)
τ
dτ −
1
n
t+n
t
s(τ)
t + n
dτ <
ε
8(t + n)
, t ≥ T0, (2.1.62)
2.1. COEFFICIENTS WITH MEAN VALUES 27
and (see (2.1.2) together with (2.1.58) and also (2.1.50))
p − 1
n
t+n
t
r(τ) |ζ(τ)|q
τ
dτ −
p − 1
n
t+n
t
r(τ) |ζ(τ)|q
t + n
dτ <
ε
8(t + n)
. (2.1.63)
Considering (2.1.51), (2.1.61), (2.1.62), and (2.1.63), we obtain
ϑ (t) =
1
n
t+n
t
ζ (τ) dτ =
1
n
t+n
t
(p − 1)ζ(τ) + s(τ) + (p − 1)r(τ) |ζ(τ)|q
τ
dτ
<
p − 1
n
t+n
t
ζ(τ)
t + n
dτ +
1
n
t+n
t
s(τ)
t + n
dτ +
p − 1
n
t+n
t
r(τ)
|ζ(τ)|q
t + n
dτ +
ε
4(t + n)
=
1
t + n

(p − 1)ϑ(t) + q−p
+
p − 1
n
t+n
t
r(τ) |ζ(τ)|q
dτ
+
1
n
t+n
t
s(τ) dτ − q−p
+
ε
4

 < −
ε
4(t + n)
.
This contradiction (see (2.1.57)) means that (2.1.53) is true for B = K and C = 0.
Since (2.1.52) cannot be valid for any T1 < ∞, the considered solution ζ exists on interval
[T0, ∞). We repeat that the non-oscillation of Eq. (2.1.1) actually follows from
Lemma 2.1.1.
The following theorem is a version of Theorem 2.1.3 which is ready for applications
to the half-linear equations written in the form common in the literature.
Theorem 2.1.4. Let c : Ra → R be a continuous function, for which mean value M (c1−q
)
exists and for which it holds
0 < inf
t∈Ra
c(t) ≤ sup
t∈Ra
c(t) < ∞, (2.1.64)
and let d : Ra → R be a continuous function having mean value M(d). Let
Γ := q−p
M c1−q 1−p
= q−p

 lim
t→∞
1
t
a+t
a
c1−q
(τ) dτ


1−p
.
Consider the equation
[c(t)Φ(x )] +
d(t)
tp
Φ(x) = 0. (2.1.65)
Eq. (2.1.65) is oscillatory if M(d) > Γ, and non-oscillatory if M(d) < Γ.
2.1. COEFFICIENTS WITH MEAN VALUES 28
Proof. Let M(d) > 0. Eq. (2.1.65) can be rewritten into the form
c1−q
(t)
−p
q
Φ(x ) +
d(t)
tp
Φ(x) = 0,
i.e.,
c1−q
(t)
M(c1−q)
−p
q
Φ(x ) +
[M(c1−q
)]
p
q
d(t)
tp
Φ(x) = 0. (2.1.66)
Eq. (2.1.66) has the form of Eq. (2.1.1) for
r(t) =
c1−q
(t)
M(c1−q)
, s(t) = M(c1−q
)
p
q
d(t), t ≥ a.
Note that M(r) = 1 and M(s) > 0 and that (2.1.2) follows from (2.1.64). Thus (see
Theorem 2.1.3), Eq. (2.1.65) is oscillatory for
M(s) = M(c1−q
)
p
q
M(d) = M(c1−q
)
p−1
M(d) > q−p
, i.e., M(d) > Γ,
and non-oscillatory if the opposite inequality M(d) < Γ holds.
It remains to consider the case when M(d) ≤ 0. Of course, there exists k > 0 such
that 0 < M(d + k) = M(d) + k < Γ. We know that the equation
[c(t)Φ(x )] +
d(t) + k
tp
Φ(x) = 0
is non-oscillatory. Now, it suffices to use Theorem 2.1.2, (ii).
Remark 2.1.2. For reader’s convenience, we consider Eq. (2.1.65) (instead of Eq. (2.1.1))
in Theorem 2.1.4. The form of Eq. (2.1.65) shows how the presented result improves
the known ones. Especially, we get new results in two important cases, when function s
changes sign and when it is unbounded. For details, we refer to paper [38].
Remark 2.1.3. For M(d) = Γ, it is not possible do decide whether Eq. (2.1.65) is oscillatory
or non-oscillatory for general functions c, d satisfying the conditions from the statement of
Theorem 2.1.4. It follows, e.g., from the main results of [17, 19]. One of the most studied
classes of functions which have mean values is formed by almost periodic functions. Based
on the constructions from [76], it is conjectured in [38] that the case M(d) = Γ is not
generally solvable (in the sense whether it is oscillatory or non-oscillatory) even for almost
periodic coefficients of Eq. (2.1.65). It means that there exist almost periodic functions
c, d such that M(d) = Γ and Eq. (2.1.65) is oscillatory. At the same time, there exist
another set of almost periodic functions c, d satisfying M(d) = Γ such that Eq. (2.1.65)
is non-oscillatory. Note that the case of periodic functions c, d with the same period was
proved to be non-oscillatory (see [17, 19]).
To illustrate Theorem 2.1.4, we mention at least two examples.
2.1. COEFFICIENTS WITH MEAN VALUES 29
Example 2.1.1. For A > 1/2, B, C > 0, and q = 3, let us consider the equation
Φ(x )
A + cos(Bt) sin(Bt)
+
arcsin(cos t) + [cos(Ct) sin(Ct)]2
√
t3 + t2 + t + 1
Φ(x) = 0. (2.1.67)
Eq. (2.1.67) has the form of Eq. (2.1.65) for
c(t) =
1
A + cos(Bt) sin(Bt)
, d(t) =
arcsin(cos t) + [cos(Ct) sin(Ct)]2
1 + 1
t
+ 1
t2 + 1
t3
.
It can be directly verified that
M c1−q
= M [cos(Bt) sin(Bt) + A]2
=
1 + 8A2
8
and that
M(d) = M [cos(Ct) sin(Ct)]2
=
1
8
.
Hence, if 29
< 33
(1 + 8A2
) then Eq. (2.1.67) is oscillatory and if 29
> 33
(1 + 8A2
) then
it is non-oscillatory; i.e., Eq. (2.1.67) is oscillatory for A ∈ (κ, ∞) and non-oscillatory for
A ∈ (1/2, κ), where
κ = (83/33 − 1)/8
.
= 1.498 455 995.
Since d is oscillatory, the other related results in the literature give no conclusion for
Eq. (2.1.65).
Example 2.1.2. Let K ∈ R and k = 0 be arbitrarily given. We define the function
d : R1 → R by
d(t) := K + k (t − n) 3n
, t ∈ n, n +
1
2n
, n ∈ N;
d(t) := K + k n +
2
2n
− t 3n
, t ∈ n +
1
2n
, n +
2
2n
, n ∈ N;
d(t) := K, t ∈ n +
2
2n
, n + 1 , n ∈ N.
It is seen that lim supt→∞ |d(t)| = ∞ and that M(d) = K. Analogously, for L > 0 and
l > −L, we define c : R1 → R as
c(t) := L + l (t − n) 2n, t ∈ n, n +
1
2n
, n ∈ N;
c(t) := L + l n +
1
n
− t 2n, t ∈ n +
1
2n
, n +
1
n
, n ∈ N;
c(t) := L, t ∈ n +
1
n
, n + 1 , n ∈ N.
We have (2.1.64) and M(c1−q
) = L1−q
. Thus, Eq. (2.1.65) for the above given functions
c, d is oscillatory if K > q−p
L, and non-oscillatory if K < q−p
L. Since d is not bounded,
the other related results in the literature give no conclusion for this equation.
2.1. COEFFICIENTS WITH MEAN VALUES 30
Theorem 2.1.4 gives a new result for linear equations as well. Therefore, we mention
the following corollary.
Corollary 2.1.1. Let r, s : Ra → R be continuous functions having mean values M(r),
M(s) and let (2.1.2) hold. Then, the equation
x
r(t)
+
s(t)
t2
x = 0
is oscillatory if 4M(r)M(s) > 1, and non-oscillatory if 4M(r)M(s) < 1.
Example 2.1.3. Using Corollary 2.1.1, we can decide about the oscillation and nonoscillation
of several equations. Let a, b ∈ R and m, n ∈ N be relative prime. Let
f, g : R1 → R be arbitrary positive continuous functions with the property that
lim
t→∞
f(t) = 1, lim
t→∞
g(t)
t2
= 1.
The equations
[f(t)x ] + γ
| sin (
√
mt + a) | + sin (
√
nt + b)
g(t)
x = 0,
[f(t)x ] + γ
| cos (
√
mt + a) | + sin (
√
nt + b)
g(t)
x = 0,
[f(t)x ] + γ
| sin (
√
mt + a) | + cos (
√
nt + b)
g(t)
x = 0,
[f(t)x ] + γ
| cos (
√
mt + a) | + cos (
√
nt + b)
g(t)
x = 0
are oscillatory for 8γ > π and non-oscillatory for 8γ < π. Note that this simple criterion
does not follow from known ones.
In the following corollary, we mention the case of negative coefficient r.
Corollary 2.1.2. Let r, s : Ra → R be continuous functions such that function s has
mean value M(s) and that there exists mean value R := M(|r|1−q
). Let
−∞ < inf
t∈Ra
r(t) ≤ sup
t∈Ra
r(t) < 0
be fulfilled. The equation
[r(t)Φ(x )] +
s(t)
tp
Φ(x) = 0
is oscillatory if M(s) < −q−p
R1−p
, and non-oscillatory if M(s) > −q−p
R1−p
.
Proof. To prove the corollary, it suffices to use Theorem 2.1.4 and the fact that the
solution space of half-linear equations is homogeneous.
2.1. COEFFICIENTS WITH MEAN VALUES 31
One can prove a number of consequences using different types of comparison theorems.
For example, applying Theorem 2.1.4 and [21, Theorem 2.3.1], we immediately obtain the
next new result.
Corollary 2.1.3. Let r, s : Ra → R be continuous functions having mean values M(r),
M(s). Let (2.1.2) hold and let
Γ := q−p
[M (r)]1−p
= q−p

 lim
t→∞
1
t
a+t
a
r(τ) dτ


1−p
.
Let us consider the equation
r−p
q (t)Φ(x ) + y(t)Φ(x) = 0, (2.1.68)
where y : Ra → R is a continuous function satisfying
∞
a
y(τ) dτ < ∞.
(i) If there exists t0 ≥ a for which
∞
t
s(τ)
τp
dτ ≤
∞
t
y(τ) dτ, t ≥ t0,
and if M(s) > Γ, then Eq. (2.1.68) is oscillatory.
(ii) If there exists t0 ≥ a for which
∞
t
s(τ)
τp
dτ ≥
∞
t
y(τ) dτ , t ≥ t0,
and if M(s) < Γ, then Eq. (2.1.68) is non-oscillatory.
In the following example, we demonstrate, that the results are applicable even if M(s)
does not exists.
Example 2.1.4. Let p > 1 and X, Y, Z1, Z2 > 0. Putting
s(t) := X + Y (t − 2n
) 2n
, t ∈ 2n
, 2n
+
1
4
, n ∈ N;
s(t) := X + Y 2n
+
1
2
− t 2n
, t ∈ 2n
+
1
4
, 2n
+
1
2
, n ∈ N;
s(t) := X − Y t − 2n
−
1
2
2n
, t ∈ 2n
+
1
2
, 2n
+
3
4
, n ∈ N;
s(t) := X − Y (2n
+ 1 − t) 2n
, t ∈ 2n
+
3
4
, 2n
+ 1 , n ∈ N;
s(t) := X, t ∈ R2 \
n∈N
[2n
, 2n
+ 1] ,
2.1. COEFFICIENTS WITH MEAN VALUES 32
we define the continuous function s : R2 → R. Evidently, mean value M(s) does not
exist. Since
0 <
2n+1
2n
s(τ) − X
τp
dτ =
2n+1
2n
s(τ) − X
τp
dτ <
Y
16
2n
(2n)p
−
2n
(2n + 1)p
=
Y · 2n
16
(2n
+ 1)p
− (2n
)p
(2n + 1)p(2n)p
<
Y
16
·
1
(2n)p−1
, n ∈ N,
(2.1.69)
we have
0 <
∞
2
s(τ)
τp
dτ <
Y
16
∞
n=1
1
(2p−1)n
+
∞
2
X
τp
dτ < ∞. (2.1.70)
Considering
lim
n→∞
(2n+1)p−(2n)p
(2n+1)p(2n)p
1
2[n+1]p
= 0
and (2.1.69), there exists a positive continuous function h : R2 → R with the property
that
2n+1
t
h(τ)
τp
dτ >
2n+1
t
s(τ) − X
τp
dτ , t ∈ 2n
, 2n+1
, n ∈ N, (2.1.71)
and that limt→∞ h(t) = 0. Using (2.1.71), we obtain
∞
t
X − h(τ)
τp
dτ <
∞
t
s(τ)
τp
dτ <
∞
t
X + h(τ)
τp
dτ, t ≥ 2. (2.1.72)
In addition, M(X + h) = M(X − h) = X and
∞
2
h(τ)
τp
dτ < ∞. (2.1.73)
It is also seen that ∞
t
X − h(τ)
τp
dτ > 0 (2.1.74)
for all sufficiently large t. The equations
(|sin [Z1t]| + |cos [Z1t]| + |sin [Z2t]| + |cos [Z2t]|)−p
q Φ(x ) +
X − h(t)
tp
Φ(x) = 0,
(|sin [Z1t]| + |cos [Z1t]| + |sin [Z2t]| + |cos [Z2t]|)−p
q Φ(x ) +
X + h(t)
tp
Φ(x) = 0
2.1. COEFFICIENTS WITH MEAN VALUES 33
are oscillatory for X > q−p
(π/8)p−1
and non-oscillatory for X < q−p
(π/8)p−1
(see Theorem
2.1.4). Indeed,
M [|sin (Z1t)| + |cos (Z1t)| + |sin (Z2t)| + |cos (Z2t)|]
p(q−1)
q =
8
π
.
Therefore (see Corollary 2.1.3 together with (2.1.70), (2.1.72), (2.1.73), (2.1.74)), we know
that the equation
(|sin [Z1t]| + |cos [Z1t]| + |sin [Z2t]| + |cos [Z2t]|)−p
q Φ(x ) +
s(t)
tp
Φ(x) = 0
is oscillatory if X > q−p
(π/8)p−1
, and non-oscillatory if X < q−p
(π/8)p−1
.
§ 2.1.2 An application
In this paragraph, we use our main result to derive a theorem related to elliptic partial
differential equations with p-Laplacian and the power type non-linearity
div A(x) u p−2
u + C(x)Φ(u) = 0, (2.1.75)
where x = (xi)n
i=1 ∈ Rn
, A is an elliptic n × n matrix function with differentiable components,
and C is a H¨older continuous function. As a solution of Eq. (2.1.75) in Ω ⊆ Rn
,
we understand a differentiable function u such that A(x) u(x)
p−2
u(x) is also differentiable
and u satisfies (2.1.75) in Ω.
The following notation is used. We consider the usual Euclidean norm
b =
n
i=1
b2
i
1
2
,
the induced matrix norm
A = sup
b =0
Ab
b
,
and λmin(x), λmax(x) stands for the smallest and largest eigenvalue of matrix A(x), respectively.
From the fact that A(x) is positive definite, it follows that A(x) = λmax(x).
Denote Ω(r0) := {x ∈ Rn
: ||x|| ≥ r0}. We say that a solution u of Eq. (2.1.75) is
oscillatory if it has a zero in Ω(r) for every r ≥ r0. Eq. (2.1.75) is said to be oscillatory
if every solution of this equation is oscillatory. Otherwise, Eq. (2.1.75) is said to be
non-oscillatory.
Recall that in general we distinguish two types of oscillation in the theory of (2.1.75):
the (weak) oscillation defined in the previous paragraph and the so called (strong) nodal
oscillation, which is based on nodal domains (i.e., bounded domains such that the equation
possesses a nontrivial solution which vanishes on the boundary of this domain). Concerning
this concept of oscillation, Eq. (2.1.75) is said to be nodally oscillatory if every
2.1. COEFFICIENTS WITH MEAN VALUES 34
solution has a nodal domain outside of any ball in Rn
, and to be nodally non-oscillatory in
the opposite case. It is known that the nodal oscillation implies oscillation. The opposite
implication has been proved only in the linear case p = 2 (see [61]) and remains an open
question in the half-linear multidimensional case. We also refer to [72] which relates the
weak oscillation of linear PDE’s (and thus nodal oscillation) with the finiteness of negative
spectrum of the Laplace operator, which is of interest in physical applications.
Since the results of this section are based on the method of Riccati type equation, in
the oscillation criteria we essentially prove a nonexistence of eventually positive solution,
i.e., we deal with weak oscillation.
In [20, 57], there is proved a theorem which allows to deduce the oscillation of certain
half-linear partial differential equations from the oscillation of ordinary differential equations
if A(x) is either the identity matrix or a scalar multiple of the identity matrix. This
theorem has been later extended in [30] (see also [59]) as follows.
Theorem 2.1.5. Let us define the functions
˜r(t) :=



x =t
λmax(x)
λmin(x)
p−1
λmax(x) dS for p > 2;
x =t
λmax(x) dS for 1 < p ≤ 2,
˜c(t) :=
x =t
C(x) dS.
(2.1.76)
If the equation
˜r(t)Φ(u ) + ˜c(t)Φ(u) = 0
is oscillatory, then Eq. (2.1.75) is oscillatory as well.
We can easily apply the oscillation part of Theorem 2.1.4 and Theorem 2.1.5 to obtain
the following result.
Theorem 2.1.6. Let ˜r(t) and ˜c(t) be defined by (2.1.76). Suppose that
0 < lim inf
t→∞
˜r(t) ≤ lim sup
t→∞
˜r(t) < ∞
and that M(˜r1−q
) and M(˜c(t)tp
) exist. If
M(˜c(t)tp
) > q−p
M(˜r1−q
)
1−p
,
then Eq. (2.1.75) is oscillatory.
Remark 2.1.4. Note that in contrast to Theorem 2.1.4, we lack the non-oscillation part
in Theorem 2.1.5 and consequently also in Theorem 2.1.6, because there is a principal
problem with non-oscillation criteria for partial differential equations via the Riccati
method. A detailed discussion related to the relationship of the Riccati equation and the
non-oscillation of second order equations (in the multidimensional case) can be found in
[15].
2.2. PERIODIC COEFFICIENTS 35
2.2 Non-oscillation of equations with periodic
coefficients in critical case
In the previous section, we found the critical oscillation constant for Eq. (2.1.1) with
coefficients r and s having mean values. Despite that the critical case cannot be resolved
in full generality, there remains still a solvable open problem. It is not known whether
Eq. (2.1.1) with positive α-periodic coefficient r and β-periodic coefficient s is oscillatory
or not in the critical case (r and s do not need to have any common period, e.g., α = 1,
β =
√
2). In this section, we prove that Eq. (2.1.1) is non-oscillatory in this case. We
point out that coefficient s can change its sign (in contrast with the situation common in
the literature) and we remark that, according to our best knowledge, the result presented
in this section is new in the half-linear case as well as in the linear one (i.e., for p = 2).
To prove this result, we have to use another method than in Section 2.1.
§ 2.2.1 Preliminaries
In this paragraph, we mention the used form of studied equations together with the
corresponding Riccati equation, and the concept of the modified Pr¨ufer angle. These
tools will be applied in § 2.2.2 and § 2.2.3.
We study the equation
r−p
q (t)Φ (x ) +
s(t)
tp
Φ(x) = 0, (2.2.1)
where r, s : Ra → R, Ra := [a, ∞), a ≥ e (e denotes the base of the natural logarithm log).
Henceforth, let function r be bounded and positive and s be such that lim supt→∞ |s(t)| <
∞. For further use, we denote
r+
:= sup{r(t); t ∈ Ra}, s+
:= sup{|s(t)|; t ∈ Ra}. (2.2.2)
Let us recall that via the Riccati transformation
w(t) = r−p
q (t)Φ
x (t)
x(t)
,
where x is a nontrivial solution of Eq. (2.2.1) and function w is well defined whenever
x(t) = 0, we obtain the Riccati equation
w +
s(t)
tp
+ (p − 1)r(t)|w|q
= 0 (2.2.3)
associated to Eq. (2.2.1).
Using the notion of the half-linear trigonometric functions recalled in Section 1.1, we
can introduce the modified half-linear Pr¨ufer transformation
x(t) = ρ(t) sinp ϕ(t), x (t) =
r(t)ρ(t)
t
cosp ϕ(t). (2.2.4)
2.2. PERIODIC COEFFICIENTS 36
Denote v(t) = tp−1
w(t), where w is a solution of Eq. (2.2.3). Considering the transformation
given by (2.2.4), we get
v = Φ
cosp ϕ
sinp ϕ
. (2.2.5)
From the fact that sinp solves the equation in (1.1.2), we have
v = (1 − p) 1 +
cosp ϕ
sinp ϕ
p
ϕ . (2.2.6)
On the other hand, applying the Riccati equation (2.2.3), we obtain
v = tp−1
w = (p − 1)tp−2
w + tp−1
w =
p − 1
t
v −
s(t)
p − 1
− r(t)
cosp ϕ
sinp ϕ
p
. (2.2.7)
Putting (2.2.6) and (2.2.7) together and using (2.2.5), we have
(1 − p) 1 +
cosp ϕ
sinp ϕ
p
ϕ =
p − 1
t
Φ
cosp ϕ
sinp ϕ
−
s(t)
p − 1
− r(t)
cosp ϕ
sinp ϕ
p
. (2.2.8)
Then, by a direct calculation starting with (2.2.8) and taking into account the Pythagorean
identity (1.1.5), we obtain the equation for the Pr¨ufer angle ϕ associated to Eq. (2.2.1)
as
ϕ =
1
t
r(t)| cosp ϕ|p
− Φ(cosp ϕ) sinp ϕ + s(t)
| sinp ϕ|p
p − 1
. (2.2.9)
For details, we can also refer to [19].
§ 2.2.2 Auxiliary results
To prove the announced result, we will use the following lemmata. The first four of them
deal with Eq. (2.2.9).
Lemma 2.2.1. For a solution ϕ of Eq. (2.2.9) on [a, ∞), it holds
lim sup
t→∞
ϕ(t)
log t
< ∞,
i.e., there exists N > 0 with the property that
|ϕ(t)| < N log t, t ≥ a.
2.2. PERIODIC COEFFICIENTS 37
Proof. Considering (2.2.2) and (1.1.6), one can directly calculate
lim sup
t→∞
ϕ(t) − ϕ(t0)
log t
≤ lim sup
t→∞

 1
log t
t
t0
|ϕ (τ)| dτ


≤ lim sup
t→∞
1
log t
t
t0
1
τ
r(τ)| cosp ϕ(τ)|p
+ |Φ(cosp ϕ(τ)) sinp ϕ(τ)| + |s(τ)|
| sinp ϕ(τ)|p
p − 1
dτ
≤ lim sup
t→∞

 1
log t
t
t0
1
τ
r+
L + L +
s+
L
p − 1
dτ

 = K lim sup
t→∞
log t − log t0
log t
= K,
where t0 ∈ Ra is arbitrarily given and
K := r+
L + L +
s+
L
p − 1
. (2.2.10)
Lemma 2.2.2. If ϕ is a solution of Eq. (2.2.9) on [a, ∞), then the function ψ : Ra → R
defined by
ψ(t) :=
t+
√
t
t
ϕ(τ)
√
τ
dτ, t ≥ a, (2.2.11)
satisfies
|ϕ(t + s) − ψ(t)| ≤
C log t
√
t
, t ≥ a, s ∈ 0,
√
t , (2.2.12)
for some C > 0.
Proof. At first, we consider the function
˜ψ(t) :=
1
√
t
t+
√
t
t
ϕ(τ) dτ, t ≥ a,
2.2. PERIODIC COEFFICIENTS 38
and we estimate its difference from ψ. For t ≥ a, we have
˜ψ(t) − ψ(t) =
1
√
t
t+
√
t
t
ϕ(τ) dτ −
t+
√
t
t
ϕ(τ)
√
τ
dτ
≤
t+
√
t
t
|ϕ(τ)|
1
√
t
−
1
√
τ
dτ
≤
t +
√
t −
√
t
t
t+
√
t
t
N log τ dτ ≤
t +
√
t −
√
t
√
t
N log t +
√
t ,
where N is taken from the statement of Lemma 2.2.1. Evidently, it holds
lim
t→∞
t +
√
t −
√
t =
1
2
, lim
t→∞
log t +
√
t
log t
= 1.
Thus, there exists K > 0 for which
ψ(t) − ˜ψ(t) ≤
K log t
√
t
, t ≥ a. (2.2.13)
Since ϕ is continuous, we have that, for any t ≥ a, there exists t0 ∈ t, t +
√
t such
that ˜ψ(t) = ϕ(t0). Hence, we have
ϕ(t + s) − ˜ψ(t) = |ϕ(t + s) − ϕ(t0)| ≤
t+
√
t
t
|ϕ (τ)| dτ
≤
1
t



t+
√
t
t
r(τ)| cosp ϕ(τ)|p
+ |Φ(cosp ϕ(τ)) sinp ϕ(τ)| dτ
+
t+
√
t
t
| sinp ϕ(τ)|p
p − 1
|s(τ)| dτ



≤
1
t
t+
√
t
t
Lr+
+ L +
Ls+
p − 1
dτ ≤
K
√
t
, t ≥ a, s ∈ 0,
√
t ,
(2.2.14)
where K is given in (2.2.10) (r+
, s+
are defined in (2.2.2) and L is from (1.1.6)). Combining
(2.2.13) and (2.2.14), we obtain (2.2.12) for C = K + K.
2.2. PERIODIC COEFFICIENTS 39
Remark 2.2.1. From the above lemmata, it follows that there exists U > 0 for which
|ψ(t)| < U log t, t ≥ a, (2.2.15)
where ψ is defined in (2.2.11) for a solution ϕ of Eq. (2.2.9) on [a, ∞).
Lemma 2.2.3. Let ϕ be a solution of Eq. (2.2.9) on [a, ∞). Then, there exist P, > 0
such that the function ψ : Ra → R defined in (2.2.11) satisfies the inequalities
ψ ≤
1
t



| cosp ψ|p
√
t
t+
√
t
t
r(τ) dτ − Φ(cosp ψ) sinp ψ
+
| sinp ψ|p
(p − 1)
√
t
t+
√
t
t
s(τ) dτ +
P
t



(2.2.16)
and
ψ ≥
1
t



| cosp ψ|p
√
t
t+
√
t
t
r(τ) dτ − Φ(cosp ψ) sinp ψ
+
| sinp ψ|p
(p − 1)
√
t
t+
√
t
t
s(τ) dτ −
P
t


 .
(2.2.17)
Proof. For arbitrarily given t > a, we have
ψ (t) = 1 +
1
2
√
t
ϕ t +
√
t
t +
√
t
−
ϕ(t)
√
t
=
1
2
√
t
·
ϕ t +
√
t
t +
√
t
+
t+
√
t
t
ϕ(τ)
√
τ
dτ
=
1
2
√
t
·
ϕ t +
√
t
t +
√
t
+
t+
√
t
t
ϕ (τ)
τ
1
2
−
ϕ(τ)
2τ
3
2
dτ
=
t+
√
t
t
1
τ
3
2
r(τ)| cosp ϕ(τ)|p
− Φ(cosp ϕ(τ)) sinp ϕ(τ)
+
| sinp ϕ(τ)|p
p − 1
s(τ) dτ +
1
2
√
t
·
ϕ t +
√
t
t +
√
t
−
t+
√
t
t
ϕ(τ)
2τ
3
2
dτ.
(2.2.18)
Since
lim
t→∞
t +
√
t
3
2
− t
3
2
t
=
3
2
,
2.2. PERIODIC COEFFICIENTS 40
there exists V > 0 for which
t +
√
t
3
2
− t
3
2
t
5
2
<
V
t
3
2
, t ≥ a. (2.2.19)
Thus, it holds (see again (2.2.2), (1.1.6), (2.2.10))
t+
√
t
t
1
τ
3
2
r(τ)| cosp ϕ(τ)|p
− Φ(cosp ϕ(τ)) sinp ϕ(τ) +
| sinp ϕ(τ)|p
p − 1
s(τ) dτ
−
1
t
3
2
t+
√
t
t
r(τ)| cosp ϕ(τ)|p
− Φ(cosp ϕ(τ)) sinp ϕ(τ) +
| sinp ϕ(τ)|p
p − 1
s(τ) dτ
≤
t+
√
t
t
r+
L + L +
s+
L
p − 1
1
t
3
2
−
1
τ
3
2
dτ ≤ K
t +
√
t
3
2
− t
3
2
t
5
2
≤
KV
t
3
2
, t ≥ a.
(2.2.20)
We have (see (2.2.12) in Lemma 2.2.2 and (2.2.15) in Remark 2.2.1)
1
2
√
t
·
ϕ t +
√
t
t +
√
t
−
ψ(t)
2t
=
1
2t
ϕ t +
√
t
1 + 1√
t
− ψ(t)
≤
1
2t


ϕ t +
√
t − ψ(t)
1 + 1√
t
+ ψ(t)

1 −
1
1 + 1√
t




≤
1
2t

 ϕ t +
√
t − ψ(t) + |ψ(t)|
1 + 1√
t
− 1
1 + 1√
t


≤
1
2t

C log t
√
t
+
U log t
√
t
·
1
1 + 1√
t
1 + 1√
t
+ 1

 ≤
Q1
t
4
3
(2.2.21)
2.2. PERIODIC COEFFICIENTS 41
for some Q1 > 0 and for all t ≥ a. We also have (see again (2.2.12), (2.2.15) with (2.2.19))
ψ(t)
2t
−
t+
√
t
t
ϕ(τ)
2τ
3
2
dτ =
t+
√
t
t
ψ(t)
2t
3
2
−
ϕ(τ)
2τ
3
2
dτ
≤
t+
√
t
t
ψ(t)
2t
3
2
−
ψ(t)
2τ
3
2
dτ +
t+
√
t
t
ψ(t)
2τ
3
2
−
ϕ(τ)
2τ
3
2
dτ
≤
U log t
2
t+
√
t
t
1
t
3
2
−
1
τ
3
2
dτ +
t+
√
t
t
C log t
√
t 2τ
3
2
dτ
≤
U log t
2
·
t +
√
t
3
2
− t
3
2
t
5
2
+
C log t
2t
3
2
≤
(V U + C) log t
2t
3
2
≤
Q2
t
4
3
(2.2.22)
for a number Q2 > 0 and for all t ≥ a. Considering (2.2.21) and (2.2.22), we get
1
2
√
t
·
ϕ t +
√
t
t +
√
t
−
t+
√
t
t
ϕ(τ)
2τ
3
2
dτ ≤
Q1 + Q2
t
4
3
, t ≥ a. (2.2.23)
It means (see also (2.2.18) and (2.2.20)) that it suffices to consider the expression
1
t



1
√
t
t+
√
t
t
r(τ)| cosp ϕ(τ)|p
− Φ(cosp ϕ(τ)) sinp ϕ(τ) +
| sinp ϕ(τ)|p
p − 1
s(τ) dτ



and that, to prove the statement of the lemma, it suffices to obtain the following set of
inequalities
| cosp ψ(t)|p
√
t
t+
√
t
t
r(τ) dτ −
1
√
t
t+
√
t
t
r(τ)| cosp ϕ(τ)|p
dτ ≤
A1 log t
√
t
, (2.2.24)
Φ(cosp ψ(t)) sinp ψ(t) −
1
√
t
t+
√
t
t
Φ(cosp ϕ(τ)) sinp ϕ(τ) dτ ≤
A2
t
, (2.2.25)
| sinp ψ(t)|p
√
t
t+
√
t
t
s(τ) dτ −
1
√
t
t+
√
t
t
s(τ)| sinp ϕ(τ)|p
dτ ≤
A3 log t
√
t
(2.2.26)
for some constants A1, A2, A3 > 0, for a number > 0, and for all t ≥ a.
2.2. PERIODIC COEFFICIENTS 42
Since the half-linear trigonometric functions are continuously differentiable and periodic,
there exists B > 0 with the property that
| cosp y|p
− | cosp z|p
≤ B|y − z|, y, z ∈ R, (2.2.27)
cosp y − cosp z ≤ B|y − z|, y, z ∈ R, (2.2.28)
| sinp y|p
− | sinp z|p
≤ B|y − z|, y, z ∈ R, (2.2.29)
sinp y − sinp z ≤ B|y − z|, y, z ∈ R. (2.2.30)
If p ≥ 2, then function Φ has the Lipschitz property, i.e., there exists B ≥ 2 for which
|Φ(y) − Φ(z)| ≤ B|y − z|, y, z ∈ (−L, L). (2.2.31)
If p ∈ (1, 2), then
yp−1
− zp−1
≤ |y − z|p−1
, y, z ∈ [0, L), (2.2.32)
and
|y|p−1
+ |z|p−1
≤ 2|y + z|p−1
, y, z ∈ [0, L). (2.2.33)
Considering (2.2.32) and (2.2.33), for p ∈ (1, 2), we have
|Φ(y) − Φ(z)| ≤ 2|y − z|p−1
, y, z ∈ (−L, L). (2.2.34)
Thus, for all p > 1, (2.2.31) and (2.2.34) give
|Φ(y) − Φ(z)| ≤ B2L|y − z|ρ
, y, z ∈ (−L, L), (2.2.35)
where ρ := min{1, p − 1} and where we use
|y − z| ≤ 2L|y − z|ρ
, y, z ∈ (−L, L). (2.2.36)
Altogether, it holds (see (1.1.6), (2.2.28), (2.2.30), (2.2.35), and (2.2.36))
|Φ(cosp y) sinp y − Φ(cosp z) sinp z | ≤ |Φ(cosp y) sinp y − Φ(cosp z) sinp y|
+ |Φ(cosp z) sinp y − Φ(cosp z) sinp z| ≤ L |Φ(cosp y) − Φ(cosp z)|
+ Lp−1
|sinp y − sinp z| ≤ 2L2
BBρ
|y − z|ρ
+ 2Lp
B|y − z|ρ
(2.2.37)
for all y, z ∈ R and p > 1. Of course, (2.2.37) guarantees the existence of B > 0 such that
|Φ(cosp y) sinp y − Φ(cosp z) sinp z| ≤ B |y − z|ρ
, y, z ∈ R. (2.2.38)
Inequality (2.2.24) follows directly from (see (2.2.2), (2.2.12), and (2.2.27))
1
√
t
t+
√
t
t
r(τ) (| cosp ψ(t)|p
− | cosp ϕ(τ)|p
) dτ
≤
1
√
t
t+
√
t
t
r(τ)B|ψ(t) − ϕ(τ)| dτ ≤
r+
BC log t
√
t
, t ≥ a.
(2.2.39)
2.2. PERIODIC COEFFICIENTS 43
Applying (2.2.12) and (2.2.38), we have
Φ(cosp ψ(t)) sinp ψ(t) −
1
√
t
t+
√
t
t
Φ(cosp ϕ(τ)) sinp ϕ(τ) dτ
≤
1
√
t
t+
√
t
t
|Φ(cosp ψ(t)) sinp ψ(t) − Φ(cosp ϕ(τ)) sinp ϕ(τ)| dτ
≤
1
√
t
t+
√
t
t
B |ψ(t) − ϕ(τ)|ρ
dτ ≤
1
√
t
t+
√
t
t
BCρ
logρ
t
t
ρ
2
dτ =
BCρ
logρ
t
t
ρ
2
, t ≥ a,
i.e., (2.2.25) is true for some A2 > 0 and ∈ (0, ρ/2). Analogously as in (2.2.39) (consider
(2.2.2), (2.2.12), and (2.2.29)), one can obtain (2.2.26) using
| sinp ψ(t)|p
√
t
t+
√
t
t
s(τ) dτ −
1
√
t
t+
√
t
t
s(τ)| sinp ϕ(τ)|p
dτ
≤
1
√
t
t+
√
t
t
|s(τ)|B|ψ(t) − ϕ(τ)| dτ ≤
s+
BC log t
√
t
, t ≥ a.
From the above calculations, we get (2.2.16) and (2.2.17) for a number P > 0 and any
such that ∈ (0, ρ/2) = (0, min{p − 1, 1}/2) and < 1/3 (see (2.2.23)).
Lemma 2.2.4. Let function r be α-periodic and s be β-periodic for arbitrary α, β > 0.
Let ϕ be a solution of Eq. (2.2.9) on [a, ∞). Then, there exist P > 0 and ˜ > 0 such that
the function ψ : Ra → R defined by (2.2.11) satisfies the inequality
ψ ≤
1
t
| cosp ψ|p
M(r) − Φ(cosp ψ) sinp ψ + M(s)
| sinp ψ|p
p − 1
+
P
t˜
. (2.2.40)
Proof. From Lemma 2.2.3 (see (2.2.16)), we know that ψ satisfies the inequality
ψ ≤
1
t



| cosp ψ|p
√
t
t+
√
t
t
r(τ) dτ − Φ(cosp ψ) sinp ψ
+
| sinp ψ|p
(p − 1)
√
t
t+
√
t
t
s(τ) dτ +
P
t



(2.2.41)
2.2. PERIODIC COEFFICIENTS 44
for some P > 0 and ∈ (0, 1/3). Let t ≥ a be arbitrarily given. Let n ∈ N ∪ {0} be such
that nα ≤
√
t < (n + 1)α. Using the periodicity of function r and (2.2.2), we obtain
1
√
t
t+
√
t
t
r(τ) dτ − M(r) ≤
1
√
t
t+
√
t
t
r(τ) dτ −
1
√
t
t+nα
t
r(τ) dτ
+
1
√
t
t+nα
t
r(τ) dτ −
1
nα
t+nα
t
r(τ) dτ
≤
r+
α
√
t
+
1
nα
−
1
√
t
nαM(r) ≤
[r+
+ M(r)] α
√
t
.
(2.2.42)
Analogously, we can obtain
1
√
t
t+
√
t
t
s(τ) dτ − M(s) ≤
[s+
+ M(s)] β
√
t
. (2.2.43)
Obviously, inequalities (2.2.41), (2.2.42), and (2.2.43) give the statement of the lemma.
Next, we deal with a perturbed equation and we state the equation for its Pr¨ufer
angle. We also mention a consequence of Lemma 2.2.3, as the below given Lemma 2.2.6,
which will be essential in § 2.2.3.
Lemma 2.2.5. There exists ε > 0 such that the equation
1 +
ε
log2
t
−p
q
Φ (x ) +
Φ(x)
tp
q−p
+
ε
log2
t
= 0 (2.2.44)
is non-oscillatory.
Proof. The lemma follows from [16, Theorem 4.1] (see also [17]).
Considering (2.2.9), the equation for the Pr¨ufer angle η associated to Eq. (2.2.44) is
η =
1
t
1 +
ε
log2
t
| cosp η|p
− Φ(cosp η) sinp η
+ q−p
+
ε
log2
t
| sinp η|p
p − 1
.
(2.2.45)
Lemma 2.2.6. Let η be a solution of Eq. (2.2.45) on [a, ∞). Then, there exist P > 0
and ˆ > 0 such that the function ζ : Ra → R defined as
ζ(t) :=
t+
√
t
t
η(τ)
√
τ
dτ, t ≥ a,
2.2. PERIODIC COEFFICIENTS 45
satisfies the inequality
ζ ≥
1
t
| cosp ζ|p
1 +
ε
log2
t +
√
t
− Φ(cosp ζ) sinp ζ
+
| sinp ζ|p
p − 1
q−p
+
ε
log2
t +
√
t
−
P
tˆ
.
(2.2.46)
Proof. Since Eq. (2.2.44) is a special case of Eq. (2.2.1) for
r(t) = 1 +
ε
log2
t
, s(t) = q−p
+
ε
log2
t
,
we can use the above lemmata for ζ which corresponds to ψ.
Especially, from Lemma 2.2.3 (see (2.2.17)), we have
ζ ≥
1
t



| cosp ζ|p
√
t
t+
√
t
t
1 +
ε
log2
τ
dτ − Φ(cosp ζ) sinp ζ
+
| sinp ζ|p
(p − 1)
√
t
t+
√
t
t
q−p
+
ε
log2
τ
dτ −
P
t



≥
1
t
| cosp ζ|p
1 +
ε
log2
t +
√
t
− Φ(cosp ζ) sinp ζ
+
| sinp ζ|p
p − 1
q−p
+
ε
log2
t +
√
t
−
P
t
.
It means that it suffices to put P = P and ˆ = in (2.2.46).
§ 2.2.3 Main results and examples
Now, we can prove the announced result.
Theorem 2.2.1. If function r is α-periodic and has mean value M(r) = 1 and if function
s is β-periodic and has mean value M(s) = q−p
, then Eq. (2.2.1) is non-oscillatory.
Proof. Taking into account the half-linear Pythagorean identity (see (1.1.5)), we observe
max {| sinp y|p
, | cosp y|p
} ≥
1
2
, y ∈ R.
Hence, for ε > 0 from the statement of Lemma 2.2.5, there exists δ > 0 with the property
that
ε |cosp y|p
+
ε |sinp y|p
p − 1
> δ, y ∈ R;
2.2. PERIODIC COEFFICIENTS 46
i.e., it holds
ε |cosp y|p
log2
t +
√
t
+
ε |sinp y|p
(p − 1) log2
t +
√
t
>
D
t
, y ∈ R, (2.2.47)
for any constant D > 0 and > 0 and for all sufficiently large t.
Let ϕ be a solution of Eq. (2.2.9) which is associated to Eq. (2.2.1). Lemma 2.2.4 says
that the function ψ defined by (2.2.11) satisfies inequality (2.2.40). Thus, considering
(2.2.47), where D = P + P and = min{˜, ˆ}, we have
ψ ≤
1
t
| cosp ψ|p
− Φ(cosp ψ) sinp ψ + q−p | sinp ψ|p
p − 1
+
P
t˜
<
1
t
| cosp ψ|p
1 +
ε
log2
t +
√
t
− Φ(cosp ψ) sinp ψ
+
| sinp ψ|p
p − 1
q−p
+
ε
log2
t +
√
t
−
P
tˆ
(2.2.48)
for sufficiently large t. It is well-known that the non-oscillation of Eq. (2.2.1) is equivalent
to the boundedness from above of the Pr¨ufer angle ϕ (given by (2.2.9)). We can refer,
e.g., to [21, Section 1.1.3], [19], [71] (or consider directly (2.2.4) together with Eq. (2.2.9)
when sinp ϕ = 0). We remark that the space of all values of ϕ is unbounded if and only
if limt→∞ ϕ(t) = ∞. It follows from the periodicity of the half-linear sine function and
the right-hand side of Eq. (2.2.9) for values ϕ satisfying sinp ϕ = 0 (when the derivative
is positive).
Considering Lemma 2.2.5, we know that the Pr¨ufer angle η given by Eq. (2.2.45) is
bounded. Lemma 2.2.2 says that ϕ is bounded if and only if ψ is bounded. In particular,
ζ is bounded, because η, ζ are special cases of ϕ, ψ, respectively. Thus, Lemma 2.2.6
together with (2.2.48) guarantees that the considered solution ϕ (given by Eq. (2.2.9)) is
bounded, i.e., Eq. (2.2.1) is non-oscillatory. Indeed, it suffices to consider the solutions µ,
ν of the equations
µ =
1
t
| cosp µ|p
− Φ(cosp µ) sinp µ + q−p | sinp µ|p
p − 1
+
P
t˜
,
ν =
1
t
| cosp ν|p
1 +
ε
log2
t +
√
t
− Φ(cosp ν) sinp ν
+
| sinp ν|p
p − 1
q−p
+
ε
log2
t +
√
t
−
P
tˆ
determined by the same initial condition µ(T) = ν(T) = 0, where T is sufficiently large.
We have ν(t) ≥ µ(t), t ≥ T. Therefore (see again (2.2.12)),
lim sup
t→∞
ζ(t) = lim sup
t→∞
η(t) < ∞
2.2. PERIODIC COEFFICIENTS 47
gives
lim sup
t→∞
ϕ(t) = lim sup
t→∞
ψ(t) < ∞.
We formulate the following direct consequence of Theorem 2.1.4.
Corollary 2.2.1. Let r be an α-periodic function having mean value M(r) = 1 and let
s be a β-periodic function. Eq. (2.2.1) is oscillatory if M(s) > q−p
; and Eq. (2.2.1) is
non-oscillatory if M(s) < q−p
.
Remark 2.2.2. In fact, the non-oscillatory part of Corollary 2.2.1 is also a consequence of
our Theorem 2.2.1 and the half-linear Sturm comparison theorem 1.1.3.
Using Corollary 2.2.1, we can improve Theorem 2.2.1 in the next form common in the
literature.
Theorem 2.2.2. Let function f be α-periodic, positive, and continuous and let function
h be β-periodic and continuous for arbitrary α, β > 0. Consider the half-linear equation
[f(t)Φ (x )] +
h(t)
tp
Φ(x) = 0. (2.2.49)
Let
γ := q−p
M f1−q 1−p
= q−p

 1
α
α
0
f1−q
(τ) dτ


1−p
. (2.2.50)
(i) If M(h) > γ, then Eq. (2.2.49) is oscillatory.
(ii) If M(h) ≤ γ, then Eq. (2.2.49) is non-oscillatory.
Proof. We rewrite Eq. (2.2.49) as
f1−q
(t)
− p
q
Φ (x ) +
h(t)
tp
Φ(x) = 0,
i.e., it takes the form of Eq. (2.2.1) for
r(t) =
f1−q
(t)
M (f1−q)
, s(t) = M f1−q
p
q
h(t).
Theorem 2.2.1 and Corollary 2.2.1 give that Eq. (2.2.49) is non-oscillatory if and only if
M(s) = M f1−q
p
q
M(h) = M f1−q p−1
M(h) ≤ q−p
.
Using γ given in (2.2.50), we can reformulate this observation as follows. Eq. (2.2.49) is
non-oscillatory if and only if M(h) ≤ γ.
2.2. PERIODIC COEFFICIENTS 48
Remark 2.2.3. Let us consider the case when M(h) = γ. Note that it is not possible
to generalize the above obtained result (see Theorem 2.2.2 or directly Theorem 2.2.1)
for general function h having mean value. It follows, e.g., from the main result of [19].
We conjecture that such a generalization is not true even for limit periodic and almost
periodic functions on the position of h. Our conjecture is based on the constructions
mentioned in [76] (or see [75, Theorem 3.5] together with [11, Theorem 1.27]).
Immediately, Theorem 2.2.2 guarantees the conditional oscillation of general periodic
linear equations which is explicitly embodied in the below mentioned corollary.
Corollary 2.2.2. Let g1, g2 be periodic and continuous functions and let g1 be positive.
The equation
x
g1(t)
+
g2(t)
t2
x = 0
is oscillatory if and only if M (g1) M (g2) > 1/4.
Proof. It suffices to put p = 2 in Theorem 2.2.2.
Remark 2.2.4. If M (g1) M (g2) = 1/4 and if g2 is positive, then the statement of Corollary
2.2.2 follows from many known results.
To illustrate Theorem 2.2.2 and Corollary 2.2.2, we give the following two examples
which are not generally solvable using known oscillatory criteria. We recall that the most
general result concerning the conditional oscillation of Eq. (2.2.1) is proved in Section
2.1 (see [35]). There is analyzed the conditional oscillation of equations with coefficients
having mean values. The critical constant is found, but the critical case remains unsolved.
Remark 2.2.3 is devoted to the description of this problem.
On the other hand, the critical case is studied in papers [17, 19], where the coefficients
of the considered equations have the same period. The critical case with different periods
of coefficients has not been analyzed in the literature.
Example 2.2.1. Let α > 1/2, β1, β2 = 0, p = 3/2. The coefficients of the half-linear
equation
Φ (x )
α + cos [β1t] sin [β1t]
+
(cos [β2t] sin [β2t])2
t
3
2
Φ(x) = 0 (2.2.51)
satisfy the conditions of Theorem 2.2.2. Since
M (cos [β2t] sin [β2t])2
=
1
8
and (see (2.2.50))
γ = 3−3
2 M (α + cos [β1t] sin [β1t])2 − 1
2
=
1
27 α2 + 1
8
,
Eq. (2.2.51) is non-oscillatory if and only if 1 + 8α2
≤ (8/3)3
. We remark that this
equivalence is new for all β1, β2 = 0 satisfying β1/β2 /∈ Q, because, in this case, the
coefficient in the differential term and the coefficient in the potential of Eq. (2.2.51) do
not have any common period.
2.2. PERIODIC COEFFICIENTS 49
Example 2.2.2. Let σ(1), σ(2) > 1 be arbitrary. The linear equations
x
2 + sinσ(1) t
+
1 + sinσ(2) t
8t2
x = 0, (2.2.52)
x
2 + sinσ(1) t
+
1 + cosσ(2) t
8t2
x = 0, (2.2.53)
x
2 + cosσ(1) t
+
1 + sinσ(2) t
8t2
x = 0, (2.2.54)
x
2 + cosσ(1) t
+
1 + cosσ(2) t
8t2
x = 0 (2.2.55)
are in the so-called border case M (g1) M (g2) = 1/4 (see Corollary 2.2.2), because
M (c + d sinσ t) = M (c + d cosσ t) = c, c, d ∈ R, σ > 1.
Nevertheless, we actually know that these equations are non-oscillatory. This fact does
not follow from any previous result for, e.g., σ(1) = 2, σ(2) = 3. Indeed, in this case, the
coefficients in the differential terms of Eq. (2.2.52), (2.2.53), (2.2.54), and (2.2.55) have
the period 2π2 = 2π and the coefficients in the potentials have the period 2π3 = 8π
√
3/9
(see (1.1.3)). Since π3/π2 = 4
√
3/9 ∈ Q, the coefficients do not have any common period
for σ(1) = 2, σ(2) = 3.
Applying known comparison theorems, we can obtain several new results which follow
from Theorem 2.2.2. We mention at least one known comparison theorem and a new
result as Corollary 2.2.3 with the below given Example 2.2.3.
Theorem 2.2.3. Let r : Ra → R be a continuous positive function satisfying
∞
a
r1−q
(τ) dτ = ∞ (2.2.56)
and s1, s2 : Ra → R be continuous functions satisfying
∞
t
s2(τ) dτ ≥
∞
t
s1(τ) dτ , t ≥ T, (2.2.57)
for some T ≥ a, where the integrals
∞
T
s1(τ) dτ,
∞
T
s2(τ) dτ are convergent. Consider
the equations
[r(t)Φ (x )] + s1(t)Φ(x) = 0, (2.2.58)
[r(t)Φ (x )] + s2(t)Φ(x) = 0. (2.2.59)
If Eq. (2.2.59) is non-oscillatory, then Eq. (2.2.58) is non-oscillatory as well.
2.2. PERIODIC COEFFICIENTS 50
Proof. See [21, Theorem 2.3.1].
Corollary 2.2.3. Let function r be α-periodic, positive, and continuous and let function
s be β-periodic and continuous for arbitrary α, β > 0. Consider the equation
r−p
q (t)Φ (x ) + z(t)Φ(x) = 0, (2.2.60)
where z : Ra → R is a continuous function satisfying
∞
a
z(τ) dτ < ∞. (2.2.61)
If
M(s) =
1
β
β
0
s(τ) dτ ≤ q−p
[M (r)]1−p
= q−p

 1
α
α
0
r(τ) dτ


1−p
(2.2.62)
and if there exists t0 ≥ a for which
∞
t
s(τ)
τp
dτ ≥
∞
t
z(τ) dτ , t ≥ t0, (2.2.63)
then Eq. (2.2.60) is non-oscillatory.
Proof. The corollary follows from Theorem 2.2.2, (ii) and Theorem 2.2.3. At first, we
discuss the assumptions of Theorem 2.2.3. Putting s1(t) = z(t), s2(t) = s(t)/tp
for t ≥ a,
we consider Eq. (2.2.60) as Eq. (2.2.58) and the equation
r−p
q (t)Φ (x ) +
s(t)
tp
Φ(x) = 0 (2.2.64)
as Eq. (2.2.59), i.e., we replace function r by r−p
q . Since
∞
a
r−p
q (τ)
1−q
dτ =
∞
a
r(τ) dτ = lim
n→∞
n
a+α
a
r(τ) dτ = ∞,
condition (2.2.56) from Theorem 2.2.3 is fulfilled. The integral
∞
a
s1(τ) dτ is convergent
due to (2.2.61). The periodicity together with the continuity of function s implies its
boundedness. Therefore (consider that p > 1), we have
∞
a
s(τ)
τp
dτ ≤
∞
a
|s(τ)|
τp
dτ < ∞.
2.2. PERIODIC COEFFICIENTS 51
Hence, the integral
∞
a
s2(τ) dτ is convergent as well. Moreover, (2.2.63) gives (2.2.57).
To finish the proof, it suffices to show that Eq. (2.2.64) is non-oscillatory which implies
the non-oscillation of Eq. (2.2.60) (consider Theorem 2.2.3). Putting f(t) = r−p
q (t) and
h(t) = s(t) in Theorem 2.2.2, we can see that (2.2.62) ensures the validity of the inequality
in Theorem 2.2.2, (ii). Indeed, it holds
1
α
α
0
f1−q
(τ) dτ =
1
α
α
0
r−p
q (τ)
1−q
dτ =
1
α
α
0
r(τ) dτ.
Thus, Eq. (2.2.64) is non-oscillatory and, consequently, Eq. (2.2.60) is non-oscillatory as
well.
Example 2.2.3. Let a, b = 0 be arbitrarily given. We define
z(t) :=
π
4q
p
(|sin [bt]| + |cos [bt]| + ˜z(t)) , t ∈ R3,
where
˜z(t) :=



(t − 2n
)
n − 1
n
, t ∈ 2n
, 2n
+
1
4
, n ∈ N {1};
2n
+
1
2
− t
n − 1
n
, t ∈ 2n
+
1
4
, 2n
+
1
2
, n ∈ N {1};
−2 t − 2n
−
1
2
n − 1
n
, t ∈ 2n
+
1
2
, 2n
+
3
4
, n ∈ N {1};
−2 (2n
+ 1 − t)
n − 1
n
, t ∈ 2n
+
3
4
, 2n
+ 1 , n ∈ N {1};
0, t ∈ R3
n∈N {1}
[2n
, 2n
+ 1] .
We consider the equation
(|sin [at]| + |cos [at]|)−p
q Φ (x ) +
z(t)
tp
Φ(x) = 0 (2.2.65)
which is in the form of Eq. (2.2.60) for z(t) = z(t)/tp
. It is seen that
0 ≤
∞
t
z(τ) dτ =
∞
t
|z(τ)| dτ ≤
∞
t
H
τp
dτ < ∞, t ≥ 3, (2.2.66)
for some H > 0. We put
s(t) :=
π
4q
p
(|sin [bt]| + |cos [bt]|) , t ∈ R3.
2.3. SUMS OF PERIODIC COEFFICIENTS 52
Directly from limt→∞
t
t+1
p
= 1, we get
∞
t
˜z(τ)
τp
dτ < 0, i.e.,
∞
t
s(τ)
τp
dτ >
∞
t
z(τ) dτ,
for all sufficiently large t. Hence (see also (2.2.66)), we have (2.2.63). Since
M(s) =
π
4q
p
4
π
= q−p 4
π
1−p
= q−p
[M (|sin [at]| + |cos [at]|)]1−p
,
inequality (2.2.62) is satisfied as well. Finally, applying Corollary 2.2.3, we obtain the
non-oscillation of Eq. (2.2.65) which does not follow from any known theorem.
2.3 Perturbed equations with sums of periodic
coefficients
In this section, we continue in the study of Eq. (2.1.1). We are interested in perturbations
of both terms of this equation when both of the perturbations are the sums of periodic
functions. In contrast with the situation common in the literature, the functions in the
perturbations do not need to have any common period and can change sign. We prove
that all considered equations are non-oscillatory in the critical case. According to our
best knowledge, this result is new also in the linear case (i.e., for p = 2).
§ 2.3.1 Preliminaries
This paragraph is devoted to the description of the considered equations and to the
modified Pr¨ufer angle which is the main method in our processes.
Our main objective is to give a non-oscillation criterion for the half-linear differential
equations in the form
r0(t) +
r1(t)
log2
t
−p
q
Φ (x ) + s0(t) +
s1(t)
log2
t
Φ(x)
tp
= 0, (2.3.1)
where r0, r1, s0, s1 : Ra → R, a ≥ e, (e stands for the base of the natural logarithm log
and Ra := [a, ∞)) are continuous functions such that r0 is positive and α-periodic, s0 is
α-periodic, and
r1(t) =
n
i=1
Ri(t), s1(t) =
n
i=1
Si(t), t ∈ Ra, (2.3.2)
for arbitrarily given periodic continuous functions Ri and Si with periods αi and βi,
respectively. Of course, we can assume that all considered periods α, αi, βi are positive
and that some of functions Ri, Si are identically zero.
2.3. SUMS OF PERIODIC COEFFICIENTS 53
At this place, we recall that we use the definition of mean values 2.1.1 for continuous
functions as a tool that helps us to identify the critical case for studied equations. It is
seen that functions r1, s1 given in (2.3.2) have mean values
M(r1) =
n
i=1
M(Ri), M(s1) =
n
i=1
M(Si). (2.3.3)
Concerning the presented results, we will assume that M(r1), M(s1) ≥ 0.
In fact, we study oscillatory properties of the equation
r0(t) +
m
i=1 Ri(t)
log2
t
−p
q
Φ (x ) + s0(t) +
n
i=1 Si(t)
log2
t
Φ(x)
tp
= 0 (2.3.4)
with periodic coefficients R1, . . . , Rm, S1, . . . , Sn on Ra at infinity (i.e., value a is large
enough) when m
i=1 M(Ri) ≥ 0, n
i=1 M(Si) ≥ 0. For simplicity, we will consider
Eq. (2.3.1) only in the critical case (see the below given Theorem 2.3.3 and [16, 17, 22, 35])
given by
[M(r0)]
p
q M(s0) =
1
αp


a+α
a
r0(τ) dτ


p
q


a+α
a
s0(τ) dτ

 = q−p
, (2.3.5)
M(s1) [M(r0)]
p
q +
p
qp+1
M(r1) [M(r0)]−1
=
q1−p
2
, (2.3.6)
i.e.,
lim
t→∞





1
tp


a+t
a
s1(τ) dτ




a+t
a
r0(τ) dτ


p
q
+
p
a+t
a
r1(τ) dτ
qp+1
a+t
a
r0(τ) dτ





=
q1−p
2
.
Then (see the below given Theorem 2.3.4), we formulate the general result about the
oscillation and non-oscillation of Eq. (2.3.4).
To study Eq. (2.3.1), we will consider the equation
M(r0) +
M(r1)
log2
t
+
1
log4
t
−p
q
Φ (x )
+ M(s0) +
M(s1)
log2
t
+
1
log4
t
Φ(x)
tp
= 0
(2.3.7)
with constant coefficients and we will also use the notations
r+
0 := max {r0(t); t ∈ Ra}, r+
1 := sup {|r1(t)|; t ∈ Ra},
s+
0 := max {|s0(t)|; t ∈ Ra}, s+
1 := sup {|s1(t)|; t ∈ Ra},
(2.3.8)
and
R+
i := max {|Ri(t)|; t ∈ Ra}, S+
i := max {|Si(t)|; t ∈ Ra} (2.3.9)
2.3. SUMS OF PERIODIC COEFFICIENTS 54
for each i.
The main method used in this section is the analogy of the one used in the previous
section, i.e., the modified Pr¨ufer transformation (2.2.4) and the equation for the Pr¨ufer
angle (2.2.9). In particular, the Pr¨ufer angle ϕ associated to Eq. (2.3.1) via (2.2.4) satisfies
the equation
ϕ (t) =
1
t
r0(t) +
r1(t)
log2
t
| cosp ϕ(t)|p
− Φ(cosp ϕ(t)) sinp ϕ(t) + s0(t) +
s1(t)
log2
t
| sinp ϕ(t)|p
p − 1
.
(2.3.10)
The equation for the Pr¨ufer angle ϕ associated to Eq. (2.3.7) is (see Eq. (2.2.9))
ϕ (t) =
1
t
M(r0) +
M(r1)
log2
t
+
1
log4
t
| cosp ϕ(t)|p
− Φ(cosp ϕ(t)) sinp ϕ(t) + M(s0) +
M(s1)
log2
t
+
1
log4
t
| sinp ϕ(t)|p
p − 1
.
(2.3.11)
Similarly as in Lemma 2.2.2, for any solution ϕ of Eq. (2.2.9) on Ra, we define the
function ψ : Ra → R by the formula
ψ(t) :=
t+
√
t
t
ϕ(τ)
√
τ
dτ, t ≥ a. (2.3.12)
This auxiliary function ψ will play an important role in the rest of this section. Note that
Eq. (2.3.10) and (2.3.11) are special cases of Eq. (2.2.9). Thus, the above function ψ is
introduced also for solutions of Eq. (2.3.10) and (2.3.11).
§ 2.3.2 Lemmata
In this paragraph, we complete necessary statements which we will use to prove the main
result. We will need Lemma 2.2.2 from the previous section and a consequence of Lemma
2.2.3.
Lemma 2.3.1. Let ϕ be a solution of Eq. (2.2.9) on Ra. Then, there exist A, c > 0 such
that the function ψ : Ra → R defined in (2.3.12) satisfies the inequality
ψ (t) −
1
t



| cosp ψ(t)|p
√
t
t+
√
t
t
r(τ) dτ − Φ(cosp ψ(t)) sinp ψ(t)
+
| sinp ψ(t)|p
(p − 1)
√
t
t+
√
t
t
s(τ) dτ


 ≤
A
t1+c
for all t > a.
2.3. SUMS OF PERIODIC COEFFICIENTS 55
Proof. The lemma comes directly from Lemma 2.2.3.
Next, we will need the following results.
Lemma 2.3.2. Let ϕ be a solution of Eq. (2.3.11) on Ra. Then, the function ψ : Ra → R
defined in (2.3.12) satisfies the inequality
ψ (t) ≥
1
t
M(r0) +
M(r1)
log2
t
| cosp ψ(t)|p
− Φ(cosp ψ(t)) sinp ψ(t)
+ M(s0) +
M(s1)
log2
t
| sinp ψ(t)|p
p − 1
+
1
log5
t
(2.3.13)
for all sufficiently large t.
Proof. From Lemma 2.3.1, we have
ψ (t) ≥
1
t



| cosp ψ(t)|p
√
t
t+
√
t
t
M(r0) +
M(r1)
log2
τ
+
1
log4
τ
dτ − Φ(cosp ψ(t)) sinp ψ(t)
+
| sinp ψ(t)|p
(p − 1)
√
t
t+
√
t
t
M(s0) +
M(s1)
log2
τ
+
1
log4
τ
dτ −
A
tc



≥
1
t
| cosp ψ(t)|p
M(r0) +
M(r1)
log2
(t +
√
t )
+
1
log4
(t +
√
t )
− Φ(cosp ψ(t)) sinp ψ(t)
+
| sinp ψ(t)|p
p − 1
M(s0) +
M(s1)
log2
(t +
√
t )
+
1
log4
(t +
√
t )
−
A
tc
for all t > a. Via the mean value theorem, one can directly compute
0 ≤ lim sup
t→∞
log2
t log2
t +
√
t − log2
t ≤ lim
t→∞
log2
t
2 log t
t
√
t = 0. (2.3.14)
Thus, we have
M(r1)
log2
(t +
√
t )
−
M(r1)
log2
t
≤ M(r1)
log2
(t +
√
t ) − log2
t
log4
t
≤
1
log6
t
, (2.3.15)
M(s1)
log2
(t +
√
t )
−
M(s1)
log2
t
≤ M(s1)
log2
(t +
√
t ) − log2
t
log4
t
≤
p − 1
log6
t
(2.3.16)
for all large t. From (1.1.5), it is seen that
max {| sinp y|p
, | cosp y|p
} ≥
1
2
, y ∈ R.
2.3. SUMS OF PERIODIC COEFFICIENTS 56
Hence, for large t, it holds
|cosp y|p
log4
(t +
√
t )
+
|sinp y|p
(p − 1) log4
(t +
√
t )
>
2
log5
t
, y ∈ R. (2.3.17)
Altogether, using (2.3.15), (2.3.16), and (2.3.17), we obtain
ψ (t) ≥
1
t
M(r0) +
M(r1)
log2
t
| cosp ψ(t)|p
− Φ(cosp ψ(t)) sinp ψ(t)
+ M(s0) +
M(s1)
log2
t
| sinp ψ(t)|p
p − 1
+
2
log5
t
−
2
log6
t
−
A
tc
for large t which gives (2.3.13).
Lemma 2.3.3. Let ϕ be a solution of Eq. (2.3.10) on Ra. Then, there exists B > 0 such
that the function ψ : Ra → R defined by (2.3.12) satisfies the inequality
ψ (t) ≤
1
t
M(r0) +
M(r1)
log2
t
| cosp ψ(t)|p
− Φ(cosp ψ(t)) sinp ψ(t)
+ M(s0) +
M(s1)
log2
t
| sinp ψ(t)|p
p − 1
+
B
log6
t
for all sufficiently large t.
Proof. From Lemma 2.3.1, we know that the inequality
ψ (t) ≤
1
t



| cosp ψ(t)|p
√
t
t+
√
t
t
r0(τ) +
r1(τ)
log2
τ
dτ − Φ(cosp ψ(t)) sinp ψ(t)
+
| sinp ψ(t)|p
(p − 1)
√
t
t+
√
t
t
s0(τ) +
s1(τ)
log2
τ
dτ +
A
tc



holds for all t > a. It means that it suffices to prove
1
√
t
t+
√
t
t
r0(τ) dτ − M(r0) ≤
A0
√
t
,
1
√
t
t+
√
t
t
s0(τ) dτ − M(s0) ≤
B0
√
t
, (2.3.18)
1
√
t
t+
√
t
t
r1(τ)
log2
τ
dτ −
M(r1)
log2
t
≤
A1
log6
t
, (2.3.19)
2.3. SUMS OF PERIODIC COEFFICIENTS 57
and
1
√
t
t+
√
t
t
s1(τ)
log2
τ
dτ −
M(s1)
log2
t
≤
B1
log6
t
(2.3.20)
for some A0, B0, A1, B1 > 0 and for all large t.
Let f : R → R be an arbitrary continuous periodic function with period δ > 0. Let
a given number t be sufficiently large and l ∈ N be such that
√
t ∈ [lδ, (l + 1)δ). We have
1
√
t
t+
√
t
t
f(τ) dτ − M(f)
≤
1
√
t
t+
√
t
t
f(τ) dτ −
1
√
t
t+lδ
t
f(τ) dτ +
1
√
t
t+lδ
t
f(τ) dτ − M(f)
≤
1
√
t
t+
√
t
t+lδ
f(τ) dτ +
1
√
t
t+lδ
t
f(τ) dτ −
1
lδ
t+lδ
t
f(τ) dτ
≤
δ max
t∈[0,δ)
|f(t)|
√
t
+
1
lδ
−
1
√
t
lδM(f) ≤
δ max
t∈[0,δ)
|f(t)| + δM(f)
√
t
.
(2.3.21)
Thus, (2.3.18) is valid for (see (2.3.8))
A0 = α r+
0 + M(r0) , B0 = α s+
0 + M(s0) .
Since (2.3.21) is true for any periodic continuous function f, we obtain (see (2.3.2), (2.3.3),
(2.3.9))
1
√
t
t+
√
t
t
r1(τ) dτ − M(r1) =
1
√
t
t+
√
t
t
n
i=1
Ri(τ) dτ − M
n
i=1
Ri
=
n
i=1



1
√
t
t+
√
t
t
Ri(τ) dτ − M(Ri)



≤
n
i=1
1
√
t
t+
√
t
t
Ri(τ) dτ − M(Ri) ≤
n
i=1
αiR+
i + αiM(Ri)
√
t
.
(2.3.22)
Analogously,
1
√
t
t+
√
t
t
s1(τ) dτ − M(s1) ≤
n
i=1
βiS+
i + βiM(Si)
√
t
. (2.3.23)
2.3. SUMS OF PERIODIC COEFFICIENTS 58
Using (2.3.22), we have
1
√
t
t+
√
t
t
r1(τ)
log2
τ
dτ −
M(r1)
log2
t
≤
1
√
t
t+
√
t
t
r1(τ)
log2
τ
dτ −
1
√
t
t+
√
t
t
r1(τ)
log2
t
dτ +
1
√
t
t+
√
t
t
r1(τ)
log2
t
dτ −
M(r1)
log2
t
≤
r+
1
√
t
t+
√
t
t
1
log2
τ
−
1
log2
t
dτ +
1
log2
t
1
√
t
t+
√
t
t
r1(τ) dτ − M(r1)
≤ r+
1
log2
t +
√
t − log2
t
log4
t
+
1
log2
t
n
i=1
αiR+
i + αiM(Ri)
√
t
(2.3.24)
for large t. Considering (2.3.14), we obtain (2.3.19) from (2.3.24). Analogously, one can
obtain (2.3.20) applying (2.3.23). Hence, the proof is complete.
Lemma 2.3.4. Eq. (2.3.7) is non-oscillatory.
Proof. The non-oscillation of Eq. (2.3.7) follows from [16, Theorem 4.1] (see also [17])
and the Sturm half-linear comparison theorem 1.1.3. More precisely, from [16, Theorem
4.1] it follows that the equation
M(r0) +
M(r1)
log2
t
+
ε
[log t · log(log t)]2
−p
q
Φ (x )
+ M(s0) +
M(s1)
log2
t
+
ε
[log t · log(log t)]2
Φ(x)
tp
= 0
(2.3.25)
is non-oscillatory for any sufficiently small ε > 0 (it is described in [17]) and Eq. (2.3.25)
is a non-oscillatory majorant of Eq. (2.3.7).
Lemma 2.3.5. For a solution ϕ of Eq. (2.3.11) on Ra, it holds
lim sup
t→∞
ϕ(t) = lim sup
t→∞
ψ(t) < ∞, (2.3.26)
where ψ is introduced in (2.3.12).
Proof. Lemma 2.3.4 says that any considered solution ϕ is bounded from above. Indeed,
it suffices to consider (2.2.4) and Eq. (2.3.11) when sinp ϕ(t) = 0. For details, we can
refer, e.g., to [21, Section 1.1.3], [19], [39], [71]. Finally, the equality in (2.3.26) follows
from Lemma 2.2.2.
2.3. SUMS OF PERIODIC COEFFICIENTS 59
§ 2.3.3 Results and examples
Now, we can prove the announced result.
Theorem 2.3.1. Eq. (2.3.1) with (2.3.5) and (2.3.6) is non-oscillatory.
Proof. We recall that the non-oscillation of Eq. (2.3.1) is equivalent to the boundedness
of a solution ϕ of Eq. (2.3.10) on Ra (see again each one of papers [19], [39]) or [71]. In
addition, a solution ϕ of Eq. (2.3.10) on Ra is bounded if and only if lim supt→∞ ϕ(t) < ∞.
It is seen from the right-hand side of Eq. (2.3.10) when sinp ϕ(t) = 0.
Let sufficiently large T > a be given. Let us consider an arbitrary solution ϕ of
Eq. (2.3.10) on RT and the corresponding function ψ : RT → R given by (2.3.12).
Lemma 2.3.3 ensures
ψ (t) ≤
1
t
M(r0) +
M(r1)
log2
t
| cosp ψ(t)|p
− Φ(cosp ψ(t)) sinp ψ(t)
+ M(s0) +
M(s1)
log2
t
| sinp ψ(t)|p
p − 1
+
B
log6
t
, t > T.
Thus, we have
ψ (t) <
1
t
M(r0) +
M(r1)
log2
t
| cosp ψ(t)|p
− Φ(cosp ψ(t)) sinp ψ(t)
+ M(s0) +
M(s1)
log2
t
| sinp ψ(t)|p
p − 1
+
1
log5
t
, t > T,
(2.3.27)
because T can be chosen arbitrarily.
We consider the solution ˜ϕ of Eq. (2.3.11) given by the initial condition (see (1.1.3))
˜ϕ(T) = max ϕ T +
√
t ; t ∈ [0, T] + πp (2.3.28)
and the corresponding function ˜ψ given by (2.3.12). Considering the form of Eq. (2.3.11)
and (2.3.28), one can show that
ψ(T) < ˜ψ(T). (2.3.29)
Lemma 2.3.5 says that (2.3.26) is valid for ˜ϕ and ˜ψ, i.e., it holds
lim sup
t→∞
˜ϕ(t) = lim sup
t→∞
˜ψ(t) < ∞. (2.3.30)
Lemma 2.3.2 gives
˜ψ (t) ≥
1
t
M(r0) +
M(r1)
log2
t
cosp
˜ψ(t)
p
− Φ cosp
˜ψ(t) sinp
˜ψ(t)
+ M(s0) +
M(s1)
log2
t
sinp
˜ψ(t)
p
p − 1
+
1
log5
t
, t > T.
(2.3.31)
2.3. SUMS OF PERIODIC COEFFICIENTS 60
Considering (2.3.27), (2.3.29), (2.3.30), and (2.3.31), we obtain
lim sup
t→∞
ψ(t) ≤ lim sup
t→∞
˜ψ(t) < ∞.
Indeed, it suffices to consider the case when ψ(t0) = ˜ψ(t0) for any t0 > T. Using
Lemma 2.2.2, we know that ϕ is bounded from above which implies the non-oscillation of
Eq. (2.3.1).
To illustrate our results, we mention examples. We remark that all given examples
are not generally solvable using any previously known non-oscillation criteria.
Example 2.3.1. Immediately, Theorem 2.3.1 gives the non-oscillation of several equations.
For example, the equations

 1 +
sin t
p
+
q2
+ sin
√
2t
2p log2
t
−p
q
Φ (x )

 + q−p
+ sin (5t)
Φ(x)
tp
= 0,
(1 + arctan (sin3 t))−p
q Φ (x ) + q−p
1 +
πq |sin t|
4 log2
t
Φ(x)
tp
= 0
are non-oscillatory.
Theorem 2.3.1 implies new results in many special cases. We obtain a new result even
for linear equations with constant and periodic coefficients which is formulated as the
corollary below.
Corollary 2.3.1. Let f, g be periodic and continuous functions such that M(f), M(g) ≥ 0
and M(f) + M(g) = 1. The equation
1 +
f(t)
log2
t
−1
x +
1
4t2
1 +
g(t)
log2
t
x = 0 (2.3.32)
is non-oscillatory.
Example 2.3.2. Let a ∈ (0, 1) and , σ > 1 be arbitrary. For the linear equations
x
1 + (a + sin t)/log2
t
+
1 + (1 − a + sinσ t)/ log2
t
4t2
x = 0,
x
1 + (a + sin t)/log2
t
+
1 + (1 − a + cosσ t)/ log2
t
4t2
x = 0,
x
1 + (a + cos t)/log2
t
+
1 + (1 − a + sinσ t)/ log2
t
4t2
x = 0,
x
1 + (a + cos t)/log2
t
+
1 + (1 − a + cosσ t)/ log2
t
4t2
x = 0,
we can apply Corollary 2.3.1. Thus, the above equations are non-oscillatory.
2.3. SUMS OF PERIODIC COEFFICIENTS 61
To proceed further, we mention a direct consequence of a relevant result from the
literature.
Theorem 2.3.2. Let c1 be a positive α-periodic continuous function, let d1 be an α-periodic
continuous function, and let c2, d2 : Ra → R be arbitrary continuous functions for
which mean values M(c2), M(|c2|), M(d2), M(|d2|) exist. Let us consider the equation
c1(t) +
c2(t)
log2
t
−p
q
Φ(x ) + d1(t) +
d2(t)
log2
t
Φ(x)
tp
= 0 (2.3.33)
and denote
Γ := 2qp−1
M(d2) [M (c1)]
p
q + 2q−2
pM(c2) [M(c1)]−1
.
Let
c1(t) +
c2(t)
log2
t
> 0, t ≥ a, qp
M(d1) [M(c1)]
p
q = 1.
The following statements hold.
(i) Eq. (2.3.33) is oscillatory if Γ > 1.
(ii) Eq. (2.3.33) is non-oscillatory if Γ < 1.
Proof. See [22, Theorem 5.1], where it suffices to put n = 1.
Combining Theorems 2.1.4 and 2.3.2, we obtain the following one.
Theorem 2.3.3. The following statements hold.
(i) If [M(r0)]
p
q M(s0) > q−p
, then Eq. (2.3.1) is oscillatory.
(ii) If [M(r0)]
p
q M(s0) < q−p
, then Eq. (2.3.1) is non-oscillatory.
(iii) If [M(r0)]
p
q M(s0) = q−p
and
M(s1) [M(r0)]
p
q +
p
qp+1
M(r1) [M(r0)]−1
>
q1−p
2
,
then Eq. (2.3.1) is oscillatory.
(iv) If [M(r0)]
p
q M(s0) = q−p
and
M(s1) [M(r0)]
p
q +
p
qp+1
M(r1) [M(r0)]−1
<
q1−p
2
,
then Eq. (2.3.1) is non-oscillatory.
Proof. The theorem follows immediately from Theorem 2.1.4 (parts (i), (ii)) and Theorem
2.3.2 (parts (iii), (iv)). It suffices to consider the identities p−1 = p/q, (1−q)(−p/q) =
1.
2.3. SUMS OF PERIODIC COEFFICIENTS 62
Applying Theorem 2.3.3, we can improve Theorem 2.3.1 and Corollary 2.3.1 into the
following more convenient forms. We give illustrating examples as well.
Theorem 2.3.4. Eq. (2.3.4) is non-oscillatory if and only if
lim
t→∞



t
αp


a+α
a
r0(τ) dτ


p
q


a+α
a
s0(τ) dτ

 − q−p
t
+
1
tp


a+t
a
n
i=1
Si(τ) dτ




a+t
a
r0(τ) dτ


p
q
+
p
a+t
a
m
i=1
Ri(τ) dτ
qp+1
a+t
a
r0(τ) dτ





≤
q1−p
2
.
Proof. It suffices to consider Theorems 2.3.1 and 2.3.3.
Example 2.3.3. Let a, b, c, d > 0, a1, a2, a3, b1 = 0, p = 3/2. Let us consider the halflinear
equation

 1
a + c (cos (a1t) sin (a1t) + cos (a2t) + sin (a3t)) /log2
t
·
x
|x |


+ b + d
cos (b1t) sin (b1t)
log t
2
x
t3|x|
= 0.
(2.3.34)
Theorem 2.3.4 guarantees the oscillation of Eq. (2.3.34) if ab2
> 1/27; and its nonoscillation
if ab2
< 1/27. We put ab2
= 1/27. Since
M (cos (αt) sin (αt)) = M (cos (αt)) = M (sin (αt)) = 0, α = 0,
and
M [cos (αt) sin (αt)]2
=
1
8
, α = 0,
we obtain the oscillation of Eq. (2.3.34) for ad2
> 16/3 and the non-oscillation in the
opposite case ad2
≤ 16/3.
Corollary 2.3.2. Let f, g be periodic and continuous functions such that M(f), M(g) ≥ 0.
Eq. (2.3.32) is oscillatory if and only if M(f) + M(g) > 1.
2.3. SUMS OF PERIODIC COEFFICIENTS 63
Example 2.3.4. Using Corollary 2.3.2 (and Theorem 2.3.4), we can generalize Example
2.3.2. For any a1, a2, b1, b2 > 0 and , σ > 1, the linear equations
x
a1 + (b1 + sin t)/log2
t
+
a2 + (b2 + sinσ t)/ log2
t
4t2
x = 0,
x
a1 + (b1 + sin t)/log2
t
+
a2 + (b2 + cosσ t)/ log2
t
4t2
x = 0,
x
a1 + (b1 + cos t)/log2
t
+
a2 + (b2 + sinσ t)/ log2
t
4t2
x = 0,
x
a1 + (b1 + cos t)/log2
t
+
a2 + (b2 + cosσ t)/ log2
t
4t2
x = 0
are oscillatory for a1a2 > 1 and non-oscillatory for a1a2 < 1. In the limiting case a1a2 =
1, one can easily rewrite the considered equations in the form of Eq. (2.3.32), where
M(f) = b1/a1 and M(g) = a1b2. Therefore, in the case a1a2 = 1, the above equations are
oscillatory if and only if b1 > a1 (1 − a1b2).
If we know that an equation is conditionally oscillatory, then we can use it as a testing
equation for many other equations. For example, using the Sturm comparison theorem
1.1.3, we can proceed for perturbed Euler type half-linear equations as follows. Let us
consider
r
−p
q
0 (t)Φ (x ) + s0(t)
Φ(x)
tp
+ g(t)Φ(x) = 0 (2.3.35)
and
[r0(t) + f(t)]−p
q Φ (x ) + s0(t)
Φ(x)
tp
= 0, (2.3.36)
where f, g are arbitrary continuous functions and r0, s0 are α-periodic continuous functions
such that r0, f are positive and M(r0) = 1, M(s0) = q−p
.
Eq. (2.3.35) is non-oscillatory if there exist βi-periodic continuous functions Si, i ∈
{1, . . . , n}, such that
M
n
i=1
Si = 1,
n
i=1
Si(t) > 0, t ∈ R, (2.3.37)
and
lim sup
t→∞
g(t)tp
log2
t
n
i=1 Si(t)
<
q1−p
2
. (2.3.38)
Eq. (2.3.35) is oscillatory if the functions Si satisfy (2.3.37) and
lim inf
t→∞
g(t)tp
log2
t
n
i=1 Si(t)
>
q1−p
2
. (2.3.39)
2.4. MODIFIED EULER EQUATIONS 64
Indeed, from inequality (2.3.38), we obtain ε > 0 with the property that we have
g(t) <
q1−p
2
− ε
n
i=1 Si(t)
tp log2
t
for all sufficiently large t. Thus, it suffices to use Theorem 2.3.4 and Sturm comparison
theorem 1.1.3. Analogously, we get the statement concerning inequality (2.3.39).
Similarly, Eq. (2.3.36) is non-oscillatory if there exist αi-periodic continuous functions
Ri for i ∈ {1, . . . , m} such that
M
m
i=1
Ri = 1,
m
i=1
Ri(t) > 0, t ∈ R, (2.3.40)
and it holds
lim sup
t→∞
f(t) log2
t
m
i=1 Ri(t)
<
q2
2p
.
On the other hand, if the functions Ri satisfy (2.3.40) and
lim inf
t→∞
f(t) log2
t
m
i=1 Ri(t)
>
q2
2p
,
then Eq. (2.3.36) is oscillatory.
2.4 Modified Euler type equations
In this section, our aim is to extend the family of conditionally oscillatory equations.
More precisely, we identify the critical oscillation constant for the Euler type equations
in the form
r(t)tp−1
Φ(x ) +
s(t)
t logp
t
Φ(x) = 0, (2.4.1)
where r : Re → R+
and s : Re → R are continuous functions. We introduce another modification
of the half-linear Pr¨ufer angle in combination with the Riccati transformation.
The main motivation comes from Theorem 2.1.4 and the following theorem proved
in [49].
Theorem 2.4.1. Let r : R1 → R+
and s : R1 → R be continuous functions such that
there exist δ > 1 and ε ∈ (0, 1/2) for which
t+1
t
r(τ) dτ < δtε
,
t+1
t
|s(τ)| dτ < δtε
, t ∈ R1. (2.4.2)
Consider the equation
r1−p
(t) Φ (x ) +
s(t)
tp
Φ(x) = 0. (2.4.3)
2.4. MODIFIED EULER EQUATIONS 65
(i) If there exist α, R, S ∈ R+
satisfying
Rp−1
S >
p − 1
p
p
and
1
α
t+α
t
r(τ) dτ ≥ R,
1
α
t+α
t
s(τ) dτ ≥ S
for all sufficiently large t, then Eq. (2.4.3) is oscillatory.
(ii) If there exist α, R, S ∈ R+
satisfying
Rp−1
S <
p − 1
p
p
and
1
α
t+α
t
r(τ) dτ ≤ R,
1
α
t+α
t
s(τ) dτ ≤ S
for all sufficiently large t, then Eq. (2.4.3) is non-oscillatory.
We remark that the results proven in this section are stronger than the ones known
for Eq. (2.4.3) (compare the below given assumptions (2.4.13) and (2.4.14) with (2.4.2)).
To the best of our knowledge, the presented results are new even for Eq. (2.4.1) with
periodic coefficients r, s.
The section is organized as follows. The notion of the modified Pr¨ufer angle is mentioned
in the next paragraph. Auxiliary results are collected in § 2.4.2. The content of
§ 2.4.1 and § 2.4.2 gives the description of our method which is used in the proofs of the
main results in § 2.4.3 which is finished by corollaries and examples.
§ 2.4.1 Equation for Pr¨ufer angle
In this paragraph, we derive the equation for the modified half-linear Pr¨ufer angle which
will be fundamental for our investigation. To do this, we have to use different transformations
than in the previous sections and combine it with the Riccati equation (1.1.13)
corresponding to the general half-linear equation (1.1.1) via (1.1.12).
We introduce the modified Pr¨ufer transformation in the form
x(t) = ρ(t) sinp ϕ(t), rq−1
(t)x (t) =
ρ(t)
log t
cosp ϕ(t) (2.4.4)
together with the substitution
v(t) = (log t)
p
q w(t), t ∈ Re. (2.4.5)
2.4. MODIFIED EULER EQUATIONS 66
Note that substitutions similar to (2.4.5) can be used also in the Riccati equation (1.1.13).
This approach leads to the so-called adapted (or weighted) Riccati equation. Nevertheless,
we use this process only partially (see below) and we have to take into consideration the
modified Pr¨ufer transformation (2.4.4) as well.
Using (2.4.5), (1.1.12), (2.4.4), and (1.1.4) successively, we obtain
v(t) = (log t)
p
q w(t) = (log t)
p
q r(t)Φ
x (t)
x(t)
= (log t)
p
q r(t)
Φ (r1−q
(t) cosp ϕ(t))
Φ (log t sinp ϕ(t))
= Φ
cosp ϕ(t)
sinp ϕ(t)
.
(2.4.6)
One can easily verify that
ˆv(t) := Φ
cosp t
sinp t
(2.4.7)
solves the equation
ˆv (t) + p − 1 + (p − 1)|ˆv(t)|q
= 0. (2.4.8)
Eq. (2.4.8) is the Riccati equation associated to the equation in (1.1.2). Hence, due to
(1.1.5), (1.1.4), (2.4.7), and (2.4.8), we have
v (t) = [ˆv(ϕ(t))] = [−p + 1 − (p − 1)|ˆv(ϕ(t))|q
] ϕ (t)
= (1 − p) 1 +
cosp ϕ(t)
sinp ϕ(t)
p
ϕ (t) =
1 − p
|sinp ϕ(t)|p ϕ (t).
(2.4.9)
On the other side, considering (2.4.5) together with (1.1.13), we have
v (t) =
p
q
(log t)
p
q
−1 w(t)
t
+ (log t)
p
q w (t)
=
p
q
·
v(t)
t log t
+ (log t)
p
q −z(t) − (p − 1)r1−q
(t)|w(t)|q
=
p
q
·
v(t)
t log t
− (log t)
p
q z(t) − (p − 1)r1−q
(t)
|v(t)|q
log t
.
(2.4.10)
We combine (2.4.9) and (2.4.10). This leads to
1 − p
|sinp ϕ(t)|p ϕ (t) =
p
q
·
v(t)
t log t
− (log t)
p
q z(t) − (p − 1)r1−q
(t)
|v(t)|q
log t
.
Taking into account (2.4.6), we obtain
(1 − p) ϕ (t) =
p
q
·
1
t log t
Φ (cosp ϕ(t)) sinp ϕ(t)
− (log t)
p
q z(t) |sinp ϕ(t)|p
− (p − 1)r1−q
(t)
| cosp ϕ(t)|p
log t
2.4. MODIFIED EULER EQUATIONS 67
which gives us the desired equation for the Pr¨ufer angle as
ϕ (t) = r1−q
(t)
| cosp ϕ(t)|p
log t
−
1
t log t
Φ (cosp ϕ(t)) sinp ϕ(t) +
(log t)
p
q
p − 1
z(t) |sinp ϕ(t)|p
.
(2.4.11)
In this section, we apply Eq. (2.4.11) to the study of Eq. (2.4.1), i.e., we have the
equation for the Pr¨ufer angle in the form
ϕ (t) =
1
t log t
r1−q
(t)| cosp ϕ(t)|p
− Φ (cosp ϕ(t)) sinp ϕ(t)
+ s(t)
|sinp ϕ(t)|p
p − 1
,
(2.4.12)
which can be simply verified.
§ 2.4.2 Pr¨ufer angle of average function
In this paragraph, we consider that the coefficients r : Re → R+
and s : Re → R in
Eq. (2.4.1) are such that
lim
t→∞
t+α
t
r1−q
(τ) dτ
√
t log t
= 0, (2.4.13)
lim
t→∞
t+α
t
|s(τ)| dτ
√
t log t
= 0 (2.4.14)
hold for some α ∈ R+
. For this number α, we define the function ψ which determines the
average value of an arbitrarily given solution ϕ of Eq. (2.4.12) over intervals of the length
α, i.e., we put
ψ(t) :=
1
α
t+α
t
ϕ(τ) dτ, t ∈ Re,
where ϕ is a solution of Eq. (2.4.12) on Re. We formulate and prove auxiliary results
concerning the function ψ.
Lemma 2.4.1. It holds
lim
t→∞
t log t |ϕ(s) − ψ(t)| = 0 (2.4.15)
uniformly with respect to s ∈ [t, t + α].
2.4. MODIFIED EULER EQUATIONS 68
Proof. For s ∈ [t, t + α], it is seen that
lim sup
t→∞
t log t |ϕ(s) − ψ(t)|
≤ lim sup
t→∞
t log t
t+α
t
|ϕ (τ)| dτ
= lim sup
t→∞
t log t
t+α
t
1
τ log τ
r1−q
(τ)| cosp ϕ(τ)|p
− Φ(cosp ϕ(τ)) sinp ϕ(τ) + s(τ)
| sinp ϕ(τ)|p
p − 1
dτ
≤ lim sup
t→∞
t log t
t+α
t
1
t log t
r1−q
(τ)| cosp ϕ(τ)|p
+ |Φ(cosp ϕ(τ)) sinp ϕ(τ)| + |s(τ)|
| sinp ϕ(τ)|p
p − 1
dτ.
Since (see directly (1.1.5))
| sinp x|p
≤ 1, | cosp x|p
≤ 1, x ∈ R, (2.4.16)
and, consequently,
|Φ(cosp x) sinp x| = | cosp x|p−1
| sinp x| ≤ 1, x ∈ R, (2.4.17)
we have
0 ≤ lim sup
t→∞
t log t |ϕ(s) − ψ(t)|
≤ lim sup
t→∞
1
√
t log t
t+α
t
r1−q
(τ) + 1 +
|s(τ)|
p − 1
dτ, s ∈ [t, t + α].
Using (2.4.13) and (2.4.14), we obtain
0 ≤ lim inf
t→∞
t log t |ϕ(s) − ψ(t)| ≤ lim sup
t→∞
t log t |ϕ(s) − ψ(t)| = 0
uniformly with respect to s ∈ [t, t + α].
Lemma 2.4.2. It holds
ψ (t) =
1
t log t

|cosp ψ(t)|p
α
t+α
t
r1−q
(τ) dτ − Φ (cosp ψ(t)) sinp ψ(t)
+
|sinp ψ(t)|p
(p − 1)α
t+α
t
s(τ) dτ + Ψ(t)


2.4. MODIFIED EULER EQUATIONS 69
for all t > e, where Ψ : Re → R is a continuous function such that limt→∞ Ψ(t) = 0.
Proof. For any t > e, we have
ψ (t) =
1
α
t+α
t
1
τ log τ
r1−q
(τ)| cosp ϕ(τ)|p
− Φ(cosp ϕ(τ)) sinp ϕ(τ) + s(τ)
| sinp ϕ(τ)|p
p − 1
dτ.
We can replace ψ (t) by
1
α
t+α
t
1
t log t
r1−q
(τ)| cosp ϕ(τ)|p
− Φ(cosp ϕ(τ)) sinp ϕ(τ) + s(τ)
| sinp ϕ(τ)|p
p − 1
dτ,
because we can easily estimate (see (2.4.13), (2.4.14), (2.4.16), (2.4.17))
lim sup
t→∞
t log t
α
t+α
t
1
τ log τ
r1−q
(τ)| cosp ϕ(τ)|p
− Φ(cosp ϕ(τ)) sinp ϕ(τ) + s(τ)
| sinp ϕ(τ)|p
p − 1
dτ
−
t+α
t
1
t log t
r1−q
(τ)| cosp ϕ(τ)|p
− Φ(cosp ϕ(τ)) sinp ϕ(τ) + s(τ)
| sinp ϕ(τ)|p
p − 1
dτ
≤ lim sup
t→∞
t log t
α
t+α
t
1
t log t
−
1
τ log τ
r1−q
(τ) + 1 + |s(τ)|
1
p − 1
dτ
2.4. MODIFIED EULER EQUATIONS 70
≤ lim sup
t→∞
t log t
α
t+α
t
1
t log t
−
1
(t + α) log(t + α)
×
× r1−q
(τ) + 1 + |s(τ)|
1
p − 1
dτ
≤ lim sup
t→∞
t log t
α
·
α
t (t + α) log t
t+α
t
r1−q
(τ) + 1 + |s(τ)|
1
p − 1
dτ
≤ lim sup
t→∞
1
t + α
t+α
t
r1−q
(τ) + 1 + |s(τ)|
1
p − 1
dτ = 0.
Applying the uniform continuity of the half-linear trigonometric functions and (2.4.15)
in Lemma 2.4.1, we see that
lim
t→∞
Φ (cosp ψ(t)) sinp ψ(t)
−
1
α
t+α
t
Φ(cosp ϕ(τ)) sinp ϕ(τ) dτ = 0.
(2.4.18)
In addition, the half-linear trigonometric functions are continuously differentiable and
periodic (see, e.g., [21, Section 1.1.2]). Hence, they have the Lipschitz property on R and,
consequently, there exists a constant L ∈ R+
for which
|cosp x|p
− |cosp y|p
≤ L|x − y|, x, y ∈ R, (2.4.19)
|sinp x|p
− |sinp y|p
≤ L|x − y|, x, y ∈ R. (2.4.20)
Therefore, from (2.4.13), Lemma 2.4.1, and (2.4.19), it follows
lim sup
t→∞
|cosp ψ(t)|p
α
t+α
t
r1−q
(τ) dτ
−
1
α
t+α
t
r1−q
(τ) |cosp ϕ(τ)|p
dτ
≤
1
α
lim sup
t→∞
t+α
t
r1−q
(τ) |cosp ψ(t)|p
− |cosp ϕ(τ)|p
dτ
≤
1
α
lim sup
t→∞
√
t log t
√
t log t
t+α
t
r1−q
(τ)L |ψ(t) − ϕ(τ)| dτ = 0.
(2.4.21)
2.4. MODIFIED EULER EQUATIONS 71
Analogously, from (2.4.14), Lemma 2.4.1, and (2.4.20), we have
lim sup
t→∞
|sinp ψ(t)|p
(p − 1)α
t+α
t
s(τ) dτ −
1
α
t+α
t
s(τ)
|sinp ϕ(τ)|p
p − 1
dτ
≤
1
(p − 1)α
lim sup
t→∞
t+α
t
|s(τ)| · |sinp ψ(t)|p
− |sinp ϕ(τ)|p
dτ
≤
1
(p − 1)α
lim sup
t→∞
√
t log t
√
t log t
t+α
t
|s(τ)| L |ψ(t) − ϕ(τ)| dτ = 0.
(2.4.22)
Finally, the statement of the lemma directly comes from the combination of (2.4.18),
(2.4.21), and (2.4.22). The continuity of Ψ is obvious.
§ 2.4.3 Oscillation constant
At first, we recall known results concerning the studied equations with constant coeffi-
cients.
Theorem 2.4.2. If A, B ∈ R+
satisfy B/A > q−p
, then the equation
Atp−1
Φ(x ) +
B
t logp
t
Φ(x) = 0
is oscillatory.
Proof. See, e.g., [21, Theorem 1.4.4] (or directly [27] and [26]).
Theorem 2.4.3. If C, D ∈ R+
satisfy D/C < q−p
, then the equation
Ctp−1
Φ(x ) +
D
t logp
t
Φ(x) = 0
is non-oscillatory.
Proof. See again [21, Theorem 1.4.4] (or [27, 26]).
Now, we can prove the announced results which identify the critical oscillation constant
for the analyzed equations with very general coefficients.
Theorem 2.4.4. Let α, R, S ∈ R+
be such that (2.4.14) is valid and Rp−1
S > q−p
. If
there exists T > e with the property that
1
α
t+α
t
r1−q
(τ) dτ ≥ R,
1
α
t+α
t
s(τ) dτ ≥ S, t ∈ RT ,
then Eq. (2.4.1) is oscillatory.
2.4. MODIFIED EULER EQUATIONS 72
Proof. We will deal with the equation for the Pr¨ufer angle corresponding to the average
function ψ. It is well-known that the non-oscillation of solutions of Eq. (2.4.1) is equivalent
to the boundedness from above of the Pr¨ufer angle ϕ given by Eq. (2.4.12). We
refer, e.g., to [17, 19, 39, 71]. It also suffices to consider directly (2.4.4) and Eq. (2.4.12)
when sinp ϕ(t) = 0. Based on Lemma 2.4.1, we know that the boundedness (from above)
of ϕ is equivalent to the boundedness (from above) of ψ. Hence, we will show that ψ is
unbounded from above. At first, we assume that (2.4.13) is true.
Taking into account Lemma 2.4.2, we have
ψ (t) =
1
t log t

|cosp ψ(t)|p
α
t+α
t
r1−q
(τ) dτ − Φ (cosp ψ(t)) sinp ψ(t)
+
|sinp ψ(t)|p
(p − 1)α
t+α
t
s(τ) dτ + Ψ(t)


≥
1
t log t
R |cosp ψ(t)|p
− Φ (cosp ψ(t)) sinp ψ(t)
+ S
|sinp ψ(t)|p
p − 1
+ Ψ(t)
for all t ∈ RT and for some continuous function Ψ : RT → R satisfying
lim
t→∞
Ψ(t) = 0. (2.4.23)
Let ε > 0 be arbitrary. From (1.1.5) and (2.4.23), we have
ε | cosp x|p
+
| sinp x|p
p − 1
> |Ψ(t)| (2.4.24)
for all x ∈ R and for all large t ∈ RT . Thus,
ψ (t) >
1
t log t
(R − ε) |cosp ψ(t)|p
− Φ (cosp ψ(t)) sinp ψ(t)
+ (S − ε)
|sinp ψ(t)|p
p − 1
(2.4.25)
for all large t ∈ RT . Let ε be so small that (R − ε)p−1
(S − ε) > q−p
and R−ε > 0. Using
Theorem 2.4.2 for A = (R − ε)
1
1−q and B = S − ε, we know that the equation
(R − ε)
1
1−q tp−1
Φ(x ) +
S − ε
t logp
t
Φ(x) = 0
2.4. MODIFIED EULER EQUATIONS 73
is oscillatory, i.e., any solution ˆϕ : RT → R of the equation
ˆϕ (t) =
1
t log t
(R − ε) | cosp ˆϕ(t)|p
− Φ(cosp ˆϕ(t)) sinp ˆϕ(t)
+ (S − ε)
| sinp ˆϕ(t)|p
p − 1
(2.4.26)
has the property that lim supt→∞ ˆϕ(t) = ∞. Indeed, one can simply compute that
B/A = (R − ε)p−1
(S − ε) > q−p
.
Considering (2.4.25) with (2.4.26) and the 2πp-periodicity of the functions sinp and cosp,
we see that lim supt→∞ ˆϕ(t) = ∞ implies lim supt→∞ ψ(t) = ∞.
To finish the proof, we have to consider the case when (2.4.13) is not valid. Evidently,
there exists a continuous function ˜r : Re → R+
with the properties
r1−q
(t) ≥ ˜r1−q
(t),
1
α
t+α
t
˜r1−q
(τ) dτ ≥ R, t ∈ RT ,
and
lim
t→∞
t+α
t
˜r1−q
(τ) dτ
√
t log t
= 0. (2.4.27)
We actually know that the equation
˜r(t)tp−1
Φ(x ) +
s(t)
t logp
t
Φ(x) = 0
is oscillatory (cf. (2.4.13) and (2.4.27)). Since r(t) ≤ ˜r(t) for all t ∈ RT , the Sturm
half-linear comparison theorem 1.1.3 gives the oscillation of Eq. (2.4.1).
Theorem 2.4.5. Let α, R, S ∈ R+
be such that (2.4.14) is valid and Rp−1
S < q−p
. If
there exists T > e with the property that
1
α
t+α
t
r1−q
(τ) dτ ≤ R,
1
α
t+α
t
s(τ) dτ ≤ S, t ∈ RT , (2.4.28)
then Eq. (2.4.1) is non-oscillatory.
Proof. We will proceed analogously as in the proof of Theorem 2.4.4. In this proof, we will
show that ψ is bounded from above. Note that (2.4.13) is valid (see the first inequality
in (2.4.28)).
2.4. MODIFIED EULER EQUATIONS 74
From Lemma 2.4.2, we obtain
ψ (t) ≤
1
t log t
R |cosp ψ(t)|p
− Φ (cosp ψ(t)) sinp ψ(t)
+ S
|sinp ψ(t)|p
p − 1
+ Ψ(t)
for any t ∈ RT and for a continuous function Ψ : RT → R satisfying (2.4.23). In addition,
using (2.4.24), we obtain
ψ (t) <
1
t log t
(R + ε) |cosp ψ(t)|p
− Φ (cosp ψ(t)) sinp ψ(t)
+ (S + ε)
|sinp ψ(t)|p
p − 1
(2.4.29)
for any ε > 0 and all sufficiently large t ∈ RT . We choose ε in such a way that
(R + ε)p−1
(S + ε) < q−p
.
We put C = (R + ε)
1
1−q and D = S + ε in Theorem 2.4.3. Since
D/C = (R + ε)p−1
(S + ε) < q−p
,
we know that the equation
(R + ε)
1
1−q tp−1
Φ(x ) +
S + ε
t logp
t
Φ(x) = 0
is non-oscillatory. This fact means that any solution ˇϕ : RT → R of the equation
ˇϕ (t) =
1
t log t
(R + ε) | cosp ˇϕ(t)|p
− Φ(cosp ˇϕ(t)) sinp ˇϕ(t)
+ (S + ε)
| sinp ˇϕ(t)|p
p − 1
(2.4.30)
has the property that lim supt→∞ ˇϕ(t) < ∞. Finally, considering (2.4.29) together with
(2.4.30) and considering the 2πp-periodicity of the generalized trigonometric functions, we
have the inequality
lim sup
t→∞
ψ(t) < ∞.
Therefore, the statement of the theorem is proven.
Now, we mention definitions which enable us to formulate the below given Corollaries
2.4.1 and 2.4.2 (and which we use later as well).
2.4. MODIFIED EULER EQUATIONS 75
Definition 2.4.1. A continuous function f : R → R is called almost periodic if, for all
ε > 0, there exists l(ε) > 0 such that any interval of length l(ε) of the real line contains
at least one point s for which
|f(t + s) − f(t)| < ε, t ∈ R.
It is well-known that there exist different (equivalent) ways to define almost periodic
functions. The above given definition is the so-called Bohr definition. Another way is
given by the Bochner definition which follows.
Definition 2.4.2. Let f : R → R be a continuous function. We say that f is almost
periodic if, from any sequence of the form {f(t + sn)}n∈N, where sn are real numbers, one
can extract a subsequence which converges uniformly with respect to t ∈ R.
We remark that the equivalence of Definitions 2.4.1 and 2.4.2 is shown, e.g., in [29,
Theorem 1.14]. As a direct generalization of the almost periodicity, we consider the notion
of the so-called asymptotic almost periodicity.
Definition 2.4.3. We say that a continuous function f : Re → R is asymptotically almost
periodic if f can be expressed in the form f(t) = f1(t) + f2(t), t ∈ Re, where f1 is almost
periodic and f2 has the property that limt→∞ f2(t) = 0.
Concerning coefficients with mean values, we obtain a new result which reads as fol-
lows.
Corollary 2.4.1. Let continuous functions r : Re → R+
and s : Re → R be such that the
mean values M(r1−q
) ∈ R+
, M(s) ∈ R exist and let (2.4.14) be valid for some α ∈ R+
.
Let [M(r1−q
)]
p−1
M(s) = q−p
. Eq. (2.4.1) is oscillatory if and only if [M(r1−q
)]
p−1
M(s) >
q−p
.
Proof. The corollary follows from Theorems 2.4.4 and 2.4.5. Let ε > 0 be arbitrary. The
existence of M(r1−q
) and M(s) implies the existence of n ∈ N such that
1
nα
t+nα
t
r1−q
(τ) dτ − M(r1−q
) < ε,
1
nα
t+nα
t
s(τ) dτ − M(s) < ε
for all t ∈ Re. If [M(r1−q
)]
p−1
M(s) > q−p
, then it suffices to choose ε so that
M(r1−q
) − ε > 0, M(r1−q
) − ε
p−1
[M(s) − ε] > q−p
,
to put R = M(r1−q
) − ε and S = M(s) − ε, and to replace α by nα in Theorem 2.4.4.
Obviously, (2.4.14) is true also for nα. If [M(r1−q
)]
p−1
M(s) < q−p
, then we choose ε so
that
M(s) + ε > 0, M(r1−q
) + ε
p−1
[M(s) + ε] < q−p
,
we consider R = M(r1−q
)+ε and S = M(s)+ε, and we replace α by nα in Theorem 2.4.5.
2.4. MODIFIED EULER EQUATIONS 76
Remark 2.4.1. We point out that the requirement about the validity of (2.4.14) for some
α ∈ R+
cannot be omitted in the statement of Corollary 2.4.1. Indeed, the existence of
M(s) does not imply (2.4.14). We remark that, from the same reason, Theorem 2.1.4
does not follow from Theorem 2.4.1.
For asymptotically almost periodic coefficients, we get a new result as well. Again, we
formulate it explicitly.
Corollary 2.4.2. Let functions r1−q
: Re → R+
and s : Re → R be asymptotically almost
periodic and let [M(r1−q
)]
p−1
M(s) ∈ R+
{q−p
}. Then, Eq. (2.4.1) is oscillatory if and
only if [M(r1−q
)]
p−1
M(s) > q−p
.
Proof. Since any asymptotically almost periodic function has mean value and it is bounded
(see, e.g., [12, 29]), this corollary is a consequence of Corollary 2.4.1.
Remark 2.4.2. Let us pay our attention to Eq. (2.1.65). We repeat that the result about
Eq. (2.1.65), which corresponds to Corollary 2.4.1, is proven in Section 2.1 and that the
one, which corresponds to Corollary 2.4.2, is proven in [38].
Remark 2.4.3. In Corollaries 2.4.1 and 2.4.2, the case [M(r1−q
)]
p−1
M(s) = q−p
cannot be
solved as oscillatory or non-oscillatory for general coefficients (which have mean values
or which are asymptotically almost periodic). We conjecture that this case is not possible
to solve even for general almost periodic coefficients. Our conjecture is based on
constructions of almost periodic functions mentioned in [76] (see also [75, 77]).
At the end, we give some examples to illustrate the proven results.
Example 2.4.1. Let us consider constants u > 0, v ∈ R, and w = 0 and the function
h : R1 → R given by the formula
h(t) :=



v + nw2n
(t − n) , t ∈ n, n +
1
2n
, n ∈ N;
v + nw2n
n +
2
2n
− t , t ∈ n +
1
2n
, n +
2
2n
, n ∈ N;
v, t ∈ n +
2
2n
, n + 1 , n ∈ N.
We analyze the equation
(tx )3
u
+
h(t)
t log4
t
x3
= 0. (2.4.31)
Hence, we deal with Eq. (2.4.1), where p = 4 and
r(t) =
1
u
, s(t) = h(t), t ∈ Re.
One can verify that
M(h) = v, M r1−q
= M r−1
3 = M u
1
3 = u
1
3 > 0.
2.4. MODIFIED EULER EQUATIONS 77
Therefore, applying Corollary 2.4.1 (condition (2.4.14) is trivially valid for all α ∈ R+
), we
know that Eq. (2.4.31) is oscillatory when 44
uv > 34
and non-oscillatory when 44
uv < 34
.
Note that the second coefficient h has mean value, but it is not asymptotically almost
periodic (it suffices to consider that lim supt→∞ |h(t)| = ∞).
Example 2.4.2. For a, b, c > 0, we consider the equation
t2
x
1 + t [1 + sin t cos t]
5
+
|sin(at)| + arctan[sin(bt) + cos(bt)]
c t +
√
t log6
t
x5
= 0
(2.4.32)
which is in the form of Eq. (2.4.1) with
r(t) =
t
1 + t [1 + sin t cos t]
5
, t ∈ Re,
s(t) =
t (|sin(at)| + arctan[sin(bt) + cos(bt)])
c t +
√
t
, t ∈ Re,
and p = 6. Since r1−q
and s are asymptotically almost periodic functions (see, e.g.,
[12, 29]), we can use directly Corollary 2.4.2. We have
M r1−q
= M r−1
5 = M
1
t
+ 1 + sin t cos t = 1,
cM(s) = M (|sin(at)| + arctan[sin(bt) + cos(bt)]) = M (|sin(at)|) =
2
π
.
Therefore, Eq. (2.4.32) is oscillatory for
c < Γ :=
6
5
6
2
π
and non-oscillatory for c > Γ.
Example 2.4.3. Let p = 3/2 and let f, g : [−1, 1] → R+
be continuous functions. We
find the oscillation constant for the equation
tf (sin t) Φ(x ) +
g (sin t)
t logp
t
Φ(x) = 0. (2.4.33)
Evidently, the functions r(t) = f (sin t) and s(t) = g (sin t) are periodic and
M r−2
=
1
2π
π
−π
dt
f (sin t)
, M(s) =
1
2π
π
−π
g (sin t) dt.
2.5. CRITICAL CASE 78
We can apply, e.g., Corollary 2.4.2. If
Γ :=
1
2π
π
−π
g (sin t) dt
1
2π
π
−π
dt
f (sin t)
> 3−3/2
,
then Eq. (2.4.33) is oscillatory. If Γ < 3−3/2
, then Eq. (2.4.33) is non-oscillatory.
2.5 Modified Euler type equations in critical case
In this section, we study the oscillation behavior of the equation
r−p
q (t) tp−1
Φ(x ) +
s(t)
t logp
t
Φ(x) = 0, (2.5.1)
where r > 0 and s are continuous functions. The motivation comes from Section 2.4,
where the equation
r(t) tp−1
Φ(x ) +
s(t)
t logp
t
Φ(x) = 0 (2.5.2)
is proved to be conditionally oscillatory. Note that, in Section 2.4, Eq. (2.5.2) is considered
without the power −p/q in the first term. Nevertheless, since function r is positive, it
does not have any impact. We consider Eq. (2.5.2) in the presented form only due to
technical reasons, i.e., the technical parts of our processes are more transparent.
Since the critical case when the coefficients indicate exactly the critical value is open,
the aim of this section is to fill this gap. We will consider Eq. (2.5.1) with periodic
continuous coefficients. We will not require any common period for coefficients r and s.
§ 2.5.1 Preliminaries
In this paragraph, we describe the equation for the modified half-linear Pr¨ufer angle given
by the studied type of equations and we prove the auxiliary lemmata.
Now, let us turn our attention to the half-linear equation
r−p
q (t) tp−1
Φ(x ) +
s(t)
t logp
t
Φ(x) = 0 (2.5.3)
and the corresponding equation for the Pr¨ufer angle
ϕ (t) =
1
t log t
r(t)| cosp ϕ(t)|p
− Φ (cosp ϕ(t)) sinp ϕ(t)
+ s(t)
|sinp ϕ(t)|p
p − 1
,
(2.5.4)
where r : R → R is a continuous, positive, and α-periodic function and s : R → R is
a continuous and β-periodic function.
2.5. CRITICAL CASE 79
Since Eq. (2.5.4) for the Pr¨ufer angle of Eq. (2.5.3) was obtained analogously as in
§ 2.4.1, we mention only main steps. We used the Riccati type transformation
w(t) = r−p
q (t) tp−1
Φ
x (t)
x(t)
to Eq. (2.5.3). This lead to the equation
w (t) +
s(t)
t logp
t
+ (p − 1) r−p
q (t) tp−1
1
1−p
|w(t)|
p
p−1 = 0. (2.5.5)
Then, using the substitution
v(t) = (log t)
p
q w(t), t ∈ (e, ∞),
in Eq. (2.5.5) and taking into account the modified Pr¨ufer transformation
x(t) = ρ(t) sinp ϕ(t), r−p
q (t) tp−1
q−1
x (t) =
ρ(t)
log t
cosp ϕ(t),
we obtained Eq. (2.5.4).
Further, let us mention the definition of the mean value of an arbitrary periodic
function which is essential for our results. (See also the more general Definition 2.1.1.)
Definition 2.5.1. The mean value M(f) of a periodic function f : R → R with period
P > 0 is defined as
M(f) :=
1
P
P
0
f(τ) dτ.
Finally, for the upcoming use, we put
˜r := sup {r(t); t > e}, ˜s := sup {|s(t)|; t > e}, (2.5.6)
we denote 2 := min {p−1, 1}, and we mention that the half-linear trigonometric functions
satisfy
|cosp a| ≤ 1, |sinp a| ≤ 1, a ∈ R. (2.5.7)
Further, for ϑ > 0 be arbitrary, we define
ψ(t) :=
1
√
t
t+
√
t
t
ϕ(τ) dτ, t ≥ e + ϑ, (2.5.8)
where ϕ is a solution of Eq. (2.5.4) on [e + ϑ, ∞). Now, we formulate and prove auxiliary
results concerning this function ψ.
2.5. CRITICAL CASE 80
Lemma 2.5.1. If ϕ is a solution of Eq. (2.5.4) on [e + ϑ, ∞), then the function
ψ : [e + ϑ, ∞) → R defined by (2.5.8) satisfies
|ϕ(τ) − ψ(t)| ≤
C
√
t log t
, t ≥ e + ϑ, τ ∈ t, t +
√
t , (2.5.9)
for some constant C > 0.
Proof. The continuity of function ϕ implies that, for any t ≥ e + ϑ, there exists
˜t ∈ t, t +
√
t such that ψ(t) = ϕ ˜t . Hence, for all t ≥ e + ϑ, τ ∈ t, t +
√
t , we
get
|ϕ(τ) − ψ(t)| = ϕ(τ) − ϕ ˜t ≤
t+
√
t
t
|ϕ (τ)| dτ
≤
1
t log t



t+
√
t
t
r(τ)| cosp ϕ(τ)|p
+ |Φ(cosp ϕ(τ)) sinp ϕ(τ)| dτ
+
t+
√
t
t
| sinp ϕ(τ)|p
p − 1
|s(τ)| dτ


 ,
i.e., we obtain (see (2.5.7), (2.5.6))
|ϕ(τ) − ψ(t)| ≤
1
t log t
t+
√
t
t
˜r + 1 +
˜s
p − 1
dτ ≤
C
√
t log t
,
where
C := ˜r + 1 +
˜s
p − 1
. (2.5.10)
Lemma 2.5.2. The inequality
ψ (t) −
1
t log t



|cosp ψ(t)|p
√
t
t+
√
t
t
r(τ) dτ − Φ (cosp ψ(t)) sinp ψ(t)
+
|sinp ψ(t)|p
(p − 1)
√
t
t+
√
t
t
s(τ) dτ


 <
D
t1+ log t
holds for some D > 0 and for all t > e + ϑ.
2.5. CRITICAL CASE 81
Proof. For all t > e + ϑ, we have
ψ (t) = 1 +
1
2
√
t
ϕ t +
√
t
√
t
−
ϕ(t)
√
t
−
1
2
√
t3
t+
√
t
t
ϕ(τ) dτ
=
1
√
t
t+
√
t
t
ϕ (τ) dτ +
1
2t
ϕ t +
√
t −
1
2
√
t3
t+
√
t
t
ϕ(τ) dτ
=
1
√
t
t+
√
t
t
1
τ log τ
r(τ)| cosp ϕ(τ)|p
− Φ (cosp ϕ(τ)) sinp ϕ(τ)
+ s(τ)
|sinp ϕ(τ)|p
p − 1
dτ +
1
2
√
t3
t+
√
t
t
ϕ t +
√
t − ϕ(τ) dτ.
Since (see also (2.5.7), (2.5.6), and (2.5.10))
1
2
√
t3
t+
√
t
t
ϕ t +
√
t − ϕ(τ) dτ ≤
1
2
√
t3
t+
√
t
t
t+
√
t
τ
|ϕ (σ)| dσ dτ
≤
1
2
√
t3
t+
√
t
t
t+
√
t
τ
1
σ log σ
r(σ)| cosp ϕ(σ)|p
− Φ (cosp ϕ(σ)) sinp ϕ(σ)
+ s(σ)
|sinp ϕ(σ)|p
p − 1
dσ dτ
≤
1
2
√
t5 log t
t+
√
t
t
t+
√
t
t
˜r + 1 +
˜s
p − 1
dσ dτ ≤
C
2
√
t3 log t
,
it suffices to consider
1
√
t
t+
√
t
t
1
τ log τ
r(τ)| cosp ϕ(τ)|p
− Φ (cosp ϕ(τ)) sinp ϕ(τ)
+ s(τ)
|sinp ϕ(τ)|p
p − 1
dτ.
In fact, we will consider
1
√
t3 log t
t+
√
t
t
r(τ)| cosp ϕ(τ)|p
− Φ (cosp ϕ(τ)) sinp ϕ(τ)
+ s(τ)
|sinp ϕ(τ)|p
p − 1
dτ,
(2.5.11)
2.5. CRITICAL CASE 82
because
t+
√
t
t
1
τ log τ
[r(τ)| cosp ϕ(τ)|p
− Φ(cosp ϕ(τ)) sinp ϕ(τ)] dτ
+
t+
√
t
t
1
τ log τ
| sinp ϕ(τ)|p
p − 1
s(τ) dτ
−
t+
√
t
t
1
t log t
[r(τ)| cosp ϕ(τ)|p
− Φ(cosp ϕ(τ)) sinp ϕ(τ)] dτ
−
t+
√
t
t
1
t log t
| sinp ϕ(τ)|p
p − 1
s(τ) dτ
≤
t+
√
t
t
˜r + 1 +
˜s
p − 1
1
t log t
−
1
τ log τ
dτ
≤ C
√
t
t +
√
t log t +
√
t − t log t
t t +
√
t log t +
√
t log t
≤
KC
t log t
for all t ≥ e + ϑ, where K > 0 is such a constant that
t +
√
t log t +
√
t − t log t
log t +
√
t
≤ K
√
t, t ≥ e + ϑ.
Considering the form of (2.5.11), to finish the proof, it suffices to prove the following
inequalities
| cosp ψ(t)|p
√
t
t+
√
t
t
r(τ) dτ
−
1
√
t
t+
√
t
t
r(τ)| cosp ϕ(τ)|p
dτ ≤
E1
√
t log t
,
(2.5.12)
1
√
t
t+
√
t
t
Φ(cosp ψ(t)) sinp ψ(t) dτ
−
1
√
t
t+
√
t
t
Φ(cosp ϕ(τ)) sinp ϕ(τ) dτ ≤
E2
t log2
t
,
(2.5.13)
2.5. CRITICAL CASE 83
| sinp ψ(t)|p
√
t
t+
√
t
t
s(τ) dτ −
1
√
t
t+
√
t
t
s(τ)| sinp ϕ(τ)|p
dτ ≤
E3
√
t log t
(2.5.14)
for some constants E1, E2, E3 > 0 and for all t ≥ e + ϑ.
The half-linear trigonometric functions sinp and cosp are continuously differentiable
with period 2πp. Thus, there exists A > 0 for which
| cosp a|p
− | cosp b|p
≤ A|a − b|, a, b ∈ R, (2.5.15)
cosp a − cosp b ≤ A|a − b|, a, b ∈ R, (2.5.16)
| sinp a|p
− | sinp b|p
≤ A|a − b|, a, b ∈ R, (2.5.17)
sinp a − sinp b ≤ A|a − b|, a, b ∈ R. (2.5.18)
In addition, directly from the definition of Φ, it follows the existence of B > 0 such that
|Φ(a) − Φ(b)| ≤ [B|a − b|]min{1,p−1}
, a, b ∈ [−1, 1]. (2.5.19)
At first, we consider inequality (2.5.12) which comes from (see also (2.5.6), (2.5.9),
and (2.5.15))
1
√
t
t+
√
t
t
r(τ) (| cosp ψ(t)|p
− | cosp ϕ(τ)|p
) dτ
≤
1
√
t
t+
√
t
t
r(τ)A|ψ(t) − ϕ(τ)| dτ ≤
˜rAC
√
t log t
, t ≥ e + ϑ.
Similarly, we can obtain (2.5.14) from (see (2.5.6), (2.5.9), and (2.5.17))
1
√
t
t+
√
t
t
s(τ) (| sinp ψ(t)|p
− | sinp ϕ(τ)|p
) dτ
≤
1
√
t
t+
√
t
t
|s(τ)| A |ψ(t) − ϕ(τ)| dτ ≤
˜sAC
√
t log t
, t ≥ e + ϑ.
2.5. CRITICAL CASE 84
It remains to show (2.5.13). We have (see (2.5.7))
1
√
t
t+
√
t
t
[Φ(cosp ψ(t)) sinp ψ(t) − Φ(cosp ϕ(τ)) sinp ϕ(τ)] dτ
≤
1
√
t
t+
√
t
t
|Φ(cosp ψ(t)) sinp ψ(t) − Φ(cosp ψ(t)) sinp ϕ(τ)| dτ
+
1
√
t
t+
√
t
t
|Φ(cosp ψ(t)) sinp ϕ(τ) − Φ(cosp ϕ(τ)) sinp ϕ(τ)| dτ
≤
1
√
t
t+
√
t
t
|sinp ψ(t) − sinp ϕ(τ)| dτ
+
1
√
t
t+
√
t
t
|Φ(cosp ψ(t)) − Φ(cosp ϕ(τ))| dτ
for all t ≥ e + ϑ and, using (2.5.9), (2.5.16), (2.5.18), and (2.5.19), we obtain
1
√
t
t+
√
t
t
[Φ(cosp ψ(t)) sinp ψ(t) − Φ(cosp ϕ(τ)) sinp ϕ(τ)] dτ
≤
1
√
t
t+
√
t
t
A |ψ(t) − ϕ(τ)| dτ
+
1
√
t
t+
√
t
t
[B |cosp ψ(t) − cosp ϕ(τ)|]min{1,p−1}
dτ
≤
1
√
t
t+
√
t
t
AC
√
t log t
+ [AB |ψ(t) − ϕ(τ)|]min{1,p−1}
dτ
≤
AC
√
t log t
+
ABC
√
t log t
min{1,p−1}
for all t ≥ e + ϑ, i.e., (2.5.13) is valid for
E2 := AC + [ABC]min{1,p−1}
.
The proof is complete.
2.5. CRITICAL CASE 85
Now, we recall one known result and we provide its direct consequence which we will
use in the proof of Theorem 2.5.2 in the next paragraph.
Theorem 2.5.1. If M, N > 0 are such that Mp−1
N = q−p
, then the equation
M +
1
t
−p
q
Φ(x ) +
1
tp
N +
1
t
Φ(x) = 0 (2.5.20)
is non-oscillatory.
Proof. See, e.g., [17].
Corollary 2.5.1. If M, N > 0 are such that Mp−1
N = q−p
, then the equation
M +
1
log t
−p
q
tp−1
Φ(x ) +
N + 1
log t
t logp
t
Φ(x) = 0 (2.5.21)
is non-oscillatory.
Proof. Let us consider Eq. (2.5.21), where x = x(t) and (·) = d
dt
. Using the transformation
of the independent variable s = log t when x(t) = y(s), we have
1
t
d
ds
M +
1
s
−p
q
tp−1
Φ
1
t
dy
ds
+
1
tsp
N +
1
s
Φ(y) = 0.
This equation can be easily simplified into the form
M +
1
s
−p
q
Φ(y ) +
1
sp
N +
1
s
Φ(y) = 0. (2.5.22)
Hence (cf. Eq. (2.5.20) and Eq. (2.5.22)), it suffices to apply Theorem 2.5.1.
§ 2.5.2 Results and examples
Applying Lemma 2.5.2 and Corollary 2.5.1, we prove the following theorem.
Theorem 2.5.2. Let α, β > 0. If r : R → R is α-periodic and s : R → R is β-periodic
such that 
 1
α
α
0
r(τ) dτ


p−1
1
β
β
0
s(τ) dτ = [M(r)]p−1
M(s) = q−p
, (2.5.23)
then Eq. (2.5.3) is non-oscillatory.
2.5. CRITICAL CASE 86
Proof. In this proof, we consider the equation for the Pr¨ufer angle ϕ and the corresponding
equation for ψ. The used method is based on the fact that the non-oscillation of solutions
of Eq. (2.5.3) is equivalent to the boundedness from above of a solution ϕ of Eq. (2.5.4).
We can refer to [42] or also to each one of papers [17, 19, 54, 71, 39]. In addition,
Lemma 2.5.1 implies that a solution ϕ : [e + ϑ, ∞) → R of Eq. (2.5.4) is bounded from
above if and only if ψ given by (2.5.8) is bounded from above.
From Lemma 2.5.2, we have
ψ (t) <
1
t log t



|cosp ψ(t)|p
√
t
t+
√
t
t
r(τ) dτ − Φ (cosp ψ(t)) sinp ψ(t)
+
|sinp ψ(t)|p
(p − 1)
√
t
t+
√
t
t
s(τ) dτ +
D
t



for all t > e + ϑ and for some D. Especially,
ψ (t) <
1
t log t



|cosp ψ(t)|p
√
t
t+
√
t
t
r(τ) dτ − Φ (cosp ψ(t)) sinp ψ(t)
+
|sinp ψ(t)|p
(p − 1)
√
t
t+
√
t
t
s(τ) dτ +
D
log2
t



(2.5.24)
for all t > e+ϑ. Then, using the periodicity of coefficients r, s, we obtain (see (2.5.6) and
(2.5.24))
ψ (t) <
1
t log t
|cosp ψ(t)|p
M(r) +
˜rα
√
t
− Φ (cosp ψ(t)) sinp ψ(t)
+
|sinp ψ(t)|p
p − 1
M(s) +
˜sβ
√
t
+
D
log2
t
(2.5.25)
for all t > e + ϑ. Indeed, for any periodic continuous function f with period P > 0 and
positive mean value M(f), we have
1
√
t
t+
√
t
t
f(τ) dτ =
1
√
t



t+Pn
t
f(τ) dτ +
t+
√
t
t+Pn
f(τ) dτ



≤
1
Pn
t+Pn
t
f(τ) dτ +
1
√
t
t+P(n+1)
t+Pn
|f(τ)| dτ ≤ M(f) +
˜fP
√
t
,
2.5. CRITICAL CASE 87
where ˜f := max {|f(t)|; t ∈ [0, P]} and n ∈ N ∪ {0} is such that Pn ≤
√
t and that
P(n + 1) >
√
t.
For R := max {1, p − 1}, the Pythagorean identity (1.1.5) gives
R | cosp a|p
+
| sinp a|p
p − 1
≥ 1, a ∈ R. (2.5.26)
Considering (2.5.25) and (2.5.26), we have
ψ (t) <
1
t log t
|cosp ψ(t)|p
M(r) +
˜rα
√
t
+
RD
log2
t
− Φ (cosp ψ(t)) sinp ψ(t)
+
|sinp ψ(t)|p
p − 1
M(s) +
˜sβ
√
t
+
RD
log2
t
for all t > e + ϑ and, consequently, we have
ψ (t) <
1
t log t
|cosp ψ(t)|p
M(r) +
1
log t
− Φ (cosp ψ(t)) sinp ψ(t)
+
|sinp ψ(t)|p
p − 1
M(s) +
1
log t
(2.5.27)
for all large t.
The equation
ϕ (t) =
1
t log t
|cosp ϕ(t)|p
M(r) +
1
log t
− Φ (cosp ϕ(t)) sinp ϕ(t)
+
|sinp ϕ(t)|p
p − 1
M(s) +
1
log t
(2.5.28)
has the form of the equation for the Pr¨ufer angle ϕ which corresponds to Eq. (2.5.21),
where M = M(r) and N = M(s). Therefore (see (2.5.23)), Corollary 2.5.1 guarantees
that any solution ϕ : [e + ϑ, ∞) → R of Eq. (2.5.28) is bounded from above. Comparing
(2.5.27) with Eq. (2.5.28) and considering the 2πp-periodicity of the half-linear trigonometric
functions, we know that the considered function ψ is bounded from above. This
means that any non-zero solution of Eq. (2.5.3) is non-oscillatory.
Now, we explicitly mention a corollary of the main results of the previous section that
we will combine with Theorem 2.5.2.
2.5. CRITICAL CASE 88
Theorem 2.5.3. Let r, s : R → R be periodic.
(i) If [M(r)]p−1
M(s) > q−p
, then Eq. (2.5.3) is oscillatory.
(ii) If [M(r)]p−1
M(s) < q−p
, then Eq. (2.5.3) is non-oscillatory.
Proof. The statements of the theorem can be obtained immediately from Theorem 2.4.4
and 2.4.5.
Using Theorem 2.5.3, we can generalize Theorem 2.5.2 as follows.
Theorem 2.5.4. Let r, s : R → R be periodic. Eq. (2.5.3) is oscillatory if and only if
[M(r)]p−1
M(s) > q−p
.
We get a new result even for linear equations. Thus, we formulate the corollary below.
Corollary 2.5.2. Let r : R → R be continuous, positive, and periodic function and let
s : R → R be continuous and periodic function. The equation
t
r(t)
x +
s(t)
t log2
t
x = 0 (2.5.29)
is oscillatory if and only if 4M(r)M(s) > 1.
To illustrate the presented results, we give some examples of equations whose oscillation
properties do not follow from previously known oscillation criteria. At first, we
mention an example to illustrate Theorem 2.5.2.
Example 2.5.1. For any p > 1, the equation


2 + sin
√
qt
2q
−p
q
tp−1
Φ (x )

 +
p − 1 + cos (pt)
pt logp
t
Φ(x) = 0 (2.5.30)
is in the critical case because
M(r) = M
2 + sin
√
qt
2q
=
1
q
= M
p − 1 + cos (pt)
p
= M(s).
Hence, [M(r)]p−1
M(s) = q−p
and Eq. (2.5.30) is non-oscillatory due to Theorem 2.5.2.
Of course, the oscillation behavior of Eq. (2.5.30) is solvable in many slightly modified
situations as well. For example, its coefficients may involve parameters. Thus, we can
apply Theorem 2.5.4 as follows.
2.6. RIEMANN–WEBER EQUATIONS 89
Example 2.5.2. Let a > 1 and b, c, d = 0 be real parameters. We consider the equation
a + sin (ct)
q
−p
q
tp−1
Φ (x ) +
p − 1 + cos (dt)
bt logp
t
Φ(x) = 0 (2.5.31)
with
M(r) = M
a + sin (ct)
q
=
a
q
and
M(s) = M
p − 1 + cos (dt)
b
=
p − 1
b
.
Therefore, by Theorem 2.5.4, Eq. (2.5.31) is oscillatory for ap−1
p/b > 1 and non-oscillatory
otherwise.
Finally, we mention the following simple example of linear equations whose oscillation
properties are solvable by Corollary 2.5.2.
Example 2.5.3. Consider the equation
t
a1 + b1 sin(c1t) + d1 cos(c1t)
x
+
a2 + b2 sin(c2t) cos(c2t) + d2 arcsin[cos(c2t)]
t log2
t
x = 0,
(2.5.32)
where ai, bi, ci, di ∈ R, ci = 0, i ∈ {1, 2}, a1 > |b1| + |d1|. It is seen that M(r) = a1
and M(s) = a2 (cf. Eq. (2.5.29) and Eq. (2.5.32)). Hence, Eq. (2.5.32) is oscillatory
for a1a2 > 1/4 and non-oscillatory for a1a2 ≤ 1/4. We emphasize that this conclusion
remains valid even for, e.g., c1 = 1 and c2 = π or c2 =
√
2, when r and s do not possess
any common period.
2.6 Riemann–Weber type equations
This section is devoted to the study of the half-linear differential equations
r(t)tp−1
Φ(x ) +
s(t)
t logp
t
Φ(x) = 0, Φ(x) = |x|p−1
sgn x (2.6.1)
and
r1(t) +
r2(t)
[log (log t)]2
−p
q
tp−1
Φ(x )
+
1
t logp
t
s1(t) +
s2(t)
[log (log t)]2 Φ(x) = 0
(2.6.2)
2.6. RIEMANN–WEBER EQUATIONS 90
with continuous coefficients r > 0, s, r1 > 0, r2, s1, s2.
In this section, we fully solve the critical case of Eq. (2.6.1) with periodic coefficients
r and s, i.e., we solve the oscillation behavior of this equation in full. At the same time,
we turn our attention to the perturbed equation (2.6.2), where the coefficients r1, s1 are
periodic and the coefficients r2, s2 in the perturbations are very general and they can
change their signs.
Now, we describe in detail related results and the main motivation for the research
presented in this section. At first, we mention paper [19]. Its main result deals with the
Euler type equation
[r(t)Φ(x )] +
γc(t)
tp
Φ(x) = 0 (2.6.3)
and with the Riemann–Weber type equation
[r(t)Φ(x )] +
1
tp
γc(t) +
µd(t)
log2
t
Φ(x) = 0, (2.6.4)
where r, c, and d are periodic positive functions with the same period. Since [19] is one of
the main motivations for our research, we reformulate its main result in full. We should
recall that the mean value of a periodic function f over its period, say T > 0, is the
number
M(f) =
1
T
a+T
a
f(x) dx,
where a ∈ R is arbitrary. We can also refer to Definition 2.1.1. Further, we will use the
notation
γp =
p − 1
p
p
, µp =
1
2
p − 1
p
p−1
.
Theorem 2.6.1 ([19]). Eq. (2.6.3) is non-oscillatory if and only if
γ ≤ γrc := γp M r1−q 1−p
[M(c)]−1
.
In the limiting case γ = γrc, Eq. (2.6.4) is non-oscillatory if
µ < µrd := µp M r1−q
[M(d)]−1
,
and it is oscillatory if µ > µrd.
The next motivation comes from papers [14, 17, 16, 18]. At this place, we state a result
concerning the equation
α1 +
α2
log2
t
−p
q
Φ(x ) +
1
tp
β1 +
β2
log2
t
Φ(x) = 0, (2.6.5)
where α1, α2, β1, β2 are constants and α1 > 0. Note that, due to the exponent in the first
term of Eq. (2.6.5), the formulations of results are technically easier and the exponent does
2.6. RIEMANN–WEBER EQUATIONS 91
not mean any restriction and can be removed. The following theorem can be obtained,
e.g., as a direct corollary of the main result of [17] (or deduced from [14, 16, 18]). We will
also use this theorem in the proof of Lemma 2.6.2 below which is essential to prove our
main result.
Theorem 2.6.2 ([17]). The following statements hold.
(i) Eq. (2.6.5) is oscillatory if β1αp−1
1 > γp, and non-oscillatory if β1αp−1
1 < γp.
(ii) Let β1αp−1
1 = γp. Eq. (2.6.5) is oscillatory if
β2αp−1
1 + (p − 1)γpα2α−1
1 > µp,
and non-oscillatory if
β2αp−1
1 + (p − 1)γpα2α−1
1 < µp.
As the third result which is strongly connected to the presented one, we mention
Theorem 2.5.3 which comes from the results of Section § 2.4.3.
In this section, we generalize Theorem 2.5.3 into a very general situation. Especially,
we solve the critical case [M(r)]p−1
M(s) = γp. Our aim is to obtain a result similar
to Theorem 2.6.1 which covers also non-periodic coefficients. To make this, we apply
Theorem 2.6.2 and the method based on the combination of the modified half-linear
Pr¨ufer angle and the Riccati equation.
This section is organized as follows. In the following paragraph, we derive the equation
for the modified Pr¨ufer angle, which will be an important tool in the rest of this chapter.
Then, we study the behavior of the Pr¨ufer angle. This leads to the proof of the main
result in § 2.6.2. The section is finished by corollaries and examples in § 2.6.3.
§ 2.6.1 Modified Pr¨ufer angle and average function
At this place, we provide some background calculations which lead to auxiliary equations
that are necessary for our approach. Throughout this section, we will use the notation
R+
a := (a, ∞) for a ∈ R. In our main result (see Theorem 2.6.3 below), we will consider
the equation
r1(t) +
r2(t)
[log(log t)]2
−p
q
tp−1
Φ(x )
+
1
t logp
t
s1(t) +
s2(t)
[log(log t)]2 Φ(x) = 0,
(2.6.6)
where r1 : R → R+
and s1 : R → R are α-periodic continuous functions for some α ∈ R+
and where r2, s2 : R+
e → R are continuous functions such that
r1(t) +
r2(t)
[log(log t)]2 > 0, t ∈ R+
e , (2.6.7)
2.6. RIEMANN–WEBER EQUATIONS 92
lim
t→∞
1
√
t log t
t+α
t
|r2(u)| du = 0, (2.6.8)
and
lim
t→∞
1
√
t log t
t+α
t
|s2(u)| du = 0. (2.6.9)
For future use, we put
r+
1 := max
t∈[0,α]
r1(t), s+
1 := max
t∈[0,α]
|s1(t)|. (2.6.10)
For our investigation of Eq. (2.6.6), we need to express the half-linear Pr¨ufer angle in
a very special form. Let us briefly describe its derivation. At first, we apply the Riccati
transformation
w(t) = r1(t) +
r2(t)
[log(log t)]2
−p
q
tp−1
Φ
x (t)
x(t)
, (2.6.11)
where x is a nontrivial solution of Eq. (2.6.6). The obtained function w satisfies the
Riccati equation
w (t) +
1
t logp
t
s1(t) +
s2(t)
[log(log t)]2
+
p − 1
t
r1(t) +
r2(t)
[log(log t)]2 |w(t)|q
= 0
(2.6.12)
associated to Eq. (2.6.6) whenever x(t) = 0. For details about the Riccati transformation
and equation, see Section 1.1 (we also refer to [21, Section 1.1.4]).
Now, we use the transformation
v(t) = (log t)
p
q w(t), t ∈ R+
e , (2.6.13)
in Eq. (2.6.12) which brings us to the adapted (or weighted) Riccati type equation
v (t) =
p
q
(log t)
p
q
−1 w(t)
t
+ (log t)
p
q w (t)
=
p
q
v(t)
t log t
−
1
t log t
s1(t) +
s2(t)
[log(log t)]2
−
p − 1
t
r1(t) +
r2(t)
[log(log t)]2
|v(t)|q
log t
.
(2.6.14)
Thus, in one hand, we keep the adapted Riccati equation (2.6.14). In the other hand,
we have the modified half-linear Pr¨ufer transformation
x(t) = ρ(t) sinp ϕ(t), r1(t) +
r2(t)
[log(log t)]2
−1
tx (t) =
ρ(t)
log t
cosp ϕ(t), (2.6.15)
2.6. RIEMANN–WEBER EQUATIONS 93
where sinp and cosp stands for the half-linear sine and cosine function, respectively. For
fundamental properties of the half-linear trigonometric functions sinp and cosp, see Section
1.1 (or [21, Section 1.1.2]). In this section, we have to mention only that the half-linear
sine and cosine functions are periodic and continuously differentiable and that they satisfy
the half-linear Pythagorean identity (1.1.5). Especially,
| sinp x| ≤ 1, | cosp x| ≤ 1, |Φ (cosp x)| ≤ 1, x ∈ R. (2.6.16)
We combine the adapted Riccati equation (2.6.14) with the Pr¨ufer transformation
(2.6.15). We begin with the observations that the function
y(t) = Φ
cosp t
sinp t
solves the equation
y (t) + p − 1 + (p − 1)|y(t)|q
= 0
and that (see (2.6.11), (2.6.13), and (2.6.15))
v(t) = (log t)
p
q r1(t) +
r2(t)
[log(log t)]2
−p
q
tp−1
Φ
x (t)
x(t)
= Φ
cosp ϕ(t)
sinp ϕ(t)
. (2.6.17)
Using (1.1.5), these two observations lead to the second expression (the first one is the
adapted Riccati equation (2.6.14) itself)
v (t) = [y(ϕ(t))] = [1 − p + (1 − p)|y(ϕ(t))|q
] ϕ (t)
= (1 − p) 1 + Φ
cosp ϕ(t)
sinp ϕ(t)
q
ϕ (t)
= (1 − p) 1 +
cosp ϕ(t)
sinp ϕ(t)
p
ϕ (t) =
1 − p
|sinp ϕ(t)|p ϕ (t).
(2.6.18)
Finally, we compare both of the expressions for v (t), namely (2.6.14) and (2.6.18).
Hence, we have
1 − p
|sinp ϕ(t)|p ϕ (t)
=
p
q
v(t)
t log t
−
1
t log t
s1(t) +
s2(t)
[log(log t)]2 −
p − 1
t
r1(t) +
r2(t)
[log(log t)]2
|v(t)|q
log t
,
from where we immediately express the derivative of the modified Pr¨ufer angle (we are
aware of (2.6.17))
ϕ (t) =
1
t log t
r1(t) +
r2(t)
[log(log t)]2 | cosp ϕ(t)|p
− Φ (cosp ϕ(t)) sinp ϕ(t)
+ s1(t) +
s2(t)
[log(log t)]2
|sinp ϕ(t)|p
p − 1
.
(2.6.19)
2.6. RIEMANN–WEBER EQUATIONS 94
We will use Eq. (2.6.19) to the study of oscillatory properties of Eq. (2.6.6).
For the period α of the functions r1, s1, we define the function ψ which determines the
average value of an arbitrarily given solution ϕ : R+
e → R of Eq. (2.6.19) over intervals of
the length α, i.e., we put
ψ(t) :=
1
α
t+α
t
ϕ(u) du, t ∈ R+
e , (2.6.20)
where ϕ is a solution of Eq. (2.6.19) on R+
e .
We prove an auxiliary result concerning the function ψ.
Lemma 2.6.1. The limit
lim
t→∞
t log t |ϕ(s) − ψ(t)| = 0 (2.6.21)
exists uniformly with respect to s ∈ [t, t + α].
Proof. For s ∈ [t, t + α], we have
0 ≤ lim inf
t→∞
t log t |ϕ(s) − ψ(t)| ≤ lim sup
t→∞
t log t |ϕ(s) − ψ(t)|
= lim sup
t→∞
t log t ϕ(s) −
1
α
t+α
t
ϕ(u) du = lim sup
t→∞
t log t
1
α
t+α
t
ϕ(s) − ϕ(u) du
≤ lim sup
t→∞
t log t max
s1,s2∈[t,t+α]
|ϕ(s1) − ϕ(s2)| = lim sup
t→∞
t log t max
s1,s2∈[t,t+α]
s2
s1
ϕ (u) du
= lim sup
t→∞
t log t max
s1,s2∈[t,t+α]
s2
s1
1
u log u
r1(u) +
r2(u)
[log(log u)]2 | cosp ϕ(u)|p
− Φ(cosp ϕ(u)) sinp ϕ(u) + s1(u) +
s2(u)
[log(log u)]2
| sinp ϕ(u)|p
p − 1
du
2.6. RIEMANN–WEBER EQUATIONS 95
= lim sup
t→∞
t log t max
s1,s2∈[t,t+α]
1
s1 log s1
s3
s1
r1(u) +
r2(u)
[log(log u)]2 | cosp ϕ(u)|p
− Φ(cosp ϕ(u)) sinp ϕ(u) + s1(u) +
s2(u)
[log(log u)]2
| sinp ϕ(u)|p
p − 1
du
+
1
s2 log s2
s2
s3
r1(u) +
r2(u)
[log(log u)]2 | cosp ϕ(u)|p
− Φ(cosp ϕ(u)) sinp ϕ(u) + s1(u) +
s2(u)
[log(log u)]2
| sinp ϕ(u)|p
p − 1
du
≤ lim sup
t→∞
t log t max
s1∈[t,t+α]
1
s1 log s1
·
s3
s1
r1(u) +
r2(u)
[log(log u)]2 | cosp ϕ(u)|p
− Φ(cosp ϕ(u)) sinp ϕ(u) + s1(u) +
s2(u)
[log(log u)]2
| sinp ϕ(u)|p
p − 1
du
+ max
s2∈[t,t+α]
1
s2 log s2
·
s2
s3
r1(u) +
r2(u)
[log(log u)]2 | cosp ϕ(u)|p
− Φ(cosp ϕ(u)) sinp ϕ(u) + s1(u) +
s2(u)
[log(log u)]2
| sinp ϕ(u)|p
p − 1
du
≤ lim sup
t→∞
t log t
1
t log t
max
s1,s2∈[t,t+α]
s2
s1
r1(u) +
r2(u)
[log(log u)]2 | cosp ϕ(u)|p
− Φ(cosp ϕ(u)) sinp ϕ(u) + s1(u) +
s2(u)
[log(log u)]2
| sinp ϕ(u)|p
p − 1
du
+
1
t log t
max
s1,s2∈[t,t+α]
s2
s1
r1(u) +
r2(u)
[log(log u)]2 | cosp ϕ(u)|p
− Φ(cosp ϕ(u)) sinp ϕ(u) + s1(u) +
s2(u)
[log(log u)]2
| sinp ϕ(u)|p
p − 1
du
≤ lim sup
t→∞
2
√
t log t
max
s1,s2∈[t,t+α]
s2
s1
r1(u) +
r2(u)
[log(log u)]2 | cosp ϕ(u)|p
− Φ(cosp ϕ(u)) sinp ϕ(u) + s1(u) +
s2(u)
[log(log u)]2
| sinp ϕ(u)|p
p − 1
du
2.6. RIEMANN–WEBER EQUATIONS 96
≤ 2 lim sup
t→∞
1
√
t log t
max
s1,s2∈[t,t+α]
s2
s1
r+
1 +
|r2(u)|
[log(log u)]2
+1 +
1
p − 1
s+
1 +
|s2(u)|
[log(log u)]2 du
≤ 2 lim sup
t→∞
1
√
t log t
t+α
t
r+
1 +
|r2(u)|
[log(log t)]2 + 1 +
1
p − 1
s+
1 +
|s2(u)|
[log(log t)]2 du = 0,
where (2.6.8), (2.6.9), (2.6.10), and (2.6.16) are used.
§ 2.6.2 Preliminaries and results
At first, we discuss the oscillation behavior of the equation
α1 +
α2
[log (log t)]2
−p
q
tp−1
Φ(x ) +
1
t logp
t
β1 +
β2
[log (log t)]2 Φ(x) = 0 (2.6.22)
with constant coefficients α1 ∈ R+
, α2, β1, β2 ∈ R. Applying a simple transformation, one
can get the following lemma.
Lemma 2.6.2. Eq. (2.6.22) is oscillatory for αp−1
1 β1 > q−p
and non-oscillatory for
αp−1
1 β1 < q−p
. In the limiting case αp−1
1 β1 = q−p
, Eq. (2.6.22) is oscillatory if
β2αp−1
1 +
p − 1
qp
α2
α1
>
q1−p
2
, (2.6.23)
and non-oscillatory if
β2αp−1
1 +
p − 1
qp
α2
α1
<
q1−p
2
. (2.6.24)
Proof. In Eq. (2.6.22), we have x = x(t) and (·) = d/dt. Using the transformation of the
independent variable
s = log t, i.e.,
d
dt
=
1
t
d
ds
,
we obtain (we put x(t) = y(s))
1
t
d
ds
α1 +
α2
log2
s
−p
q
tp−1
Φ
1
t
dy
ds
+
1
tsp
β1 +
β2
log2
s
Φ(y) = 0.
This leads to the equation
α1 +
α2
log2
s
−p
q
Φ(y ) +
1
sp
β1 +
β2
log2
s
Φ(y) = 0,
where y = y(s) and (·) = d/ds. Now, it suffices to use Theorem 2.6.2.
2.6. RIEMANN–WEBER EQUATIONS 97
From Lemma 2.6.2, we get the lemma below which closes the preliminary results.
Lemma 2.6.3. Let M(r1), M(s1) ∈ R+
be such that [M(r1)]p−1
M(s1) = q−p
.
(i) If X, Y ∈ R satisfy
[M(r1)]p−1
Y +
p − 1
qp
X
M(r1)
>
q1−p
2
, (2.6.25)
then any solution θ : R+
e → R of the equation
θ (t) =
1
t log t
M(r1) +
X
[log(log t)]2 | cosp θ(t)|p
−Φ (cosp θ(t)) sinp θ(t) + M(s1) +
Y
[log(log t)]2
|sinp θ(t)|p
p − 1
(2.6.26)
is unbounded from above.
(ii) If V, W ∈ R satisfy
[M(r1)]p−1
W +
p − 1
qp
V
M(r1)
<
q1−p
2
, (2.6.27)
then any solution ξ : R+
e → R of the equation
ξ (t) =
1
t log t
M(r1) +
V
[log(log t)]2 | cosp ξ(t)|p
−Φ (cosp ξ(t)) sinp ξ(t) + M(s1) +
W
[log(log t)]2
|sinp ξ(t)|p
p − 1
(2.6.28)
is bounded from above.
Proof. Comparing Eq. (2.6.26) and Eq. (2.6.28) with Eq. (2.6.19), one can see that
Eq. (2.6.26) and Eq. (2.6.28) is the equation of the Pr¨ufer angle for
M(r1) +
X
[log(log t)]2
−p
q
tp−1
Φ(x )
+
1
t logp
t
M(s1) +
Y
[log(log t)]2 Φ(x) = 0
(2.6.29)
and
M(r1) +
V
[log(log t)]2
−p
q
tp−1
Φ(x )
+
1
t logp
t
M(s1) +
W
[log(log t)]2 Φ(x) = 0,
(2.6.30)
2.6. RIEMANN–WEBER EQUATIONS 98
respectively.
Let us focus on the first case. The assumption [M(r1)]p−1
M(s1) = q−p
and (2.6.25)
give that Eq. (2.6.29) is oscillatory (see (2.6.23) in Lemma 2.6.2). Now, it suffices to
consider directly the Pr¨ufer transformation (2.6.15) and take into account the form of
Eq. (2.6.26), where sinp θ(t) = 0 implies θ (t) > 0 for all large t. Therefore, Eq. (2.6.29) is
oscillatory if and only if its Pr¨ufer angle θ is unbounded from above. Part (i) is proved.
Considering (2.6.24) and (2.6.27), the case (ii) is analogous (Eq. (2.6.30) is nonoscillatory
if and only if the Pr¨ufer angle ξ is bounded from above).
Now, we are ready to formulate and to prove the main result of this section.
Theorem 2.6.3. Let [M(r1)]p−1
M(s1) = q−p
.
(i) If there exist R, S ∈ R such that
1
α
t+α
t
r2(u) du ≥ R,
1
α
t+α
t
s2(u) du ≥ S (2.6.31)
for all sufficiently large t and that
[M(r1)]p−1
S +
p − 1
qp
R
M(r1)
>
q1−p
2
, (2.6.32)
then Eq. (2.6.6) is oscillatory.
(ii) If there exist R, S ∈ R such that
1
α
t+α
t
r2(u) du ≤ R,
1
α
t+α
t
s2(u) du ≤ S (2.6.33)
for all sufficiently large t and that
[M(r1)]p−1
S +
p − 1
qp
R
M(r1)
<
q1−p
2
, (2.6.34)
then Eq. (2.6.6) is non-oscillatory.
2.6. RIEMANN–WEBER EQUATIONS 99
Proof. Let us consider the function ψ given by (2.6.20), where ϕ is an arbitrary solution
of Eq. (2.6.19) on R+
e . It holds
ψ (t) =
1
α
[ϕ(t + α) − ϕ(t)] =
1
α
t+α
t
ϕ (u) du
=
1
α
t+α
t
1
u log u
r1(u) +
r2(u)
[log(log u)]2 | cosp ϕ(u)|p
− Φ (cosp ϕ(u)) sinp ϕ(u) + s1(u) +
s2(u)
[log(log u)]2
|sinp ϕ(u)|p
p − 1
du
=
1
α


t+α
t
1
u log u
r1(u) +
r2(u)
[log(log u)]2 | cosp ϕ(u)|p
d
−
t+α
t
1
u log u
Φ (cosp ϕ(u)) sinp ϕ(u) du
+
t+α
t
1
u log u
s1(u) +
s2(u)
[log(log u)]2
|sinp ϕ(u)|p
p − 1
du


(2.6.35)
for any t ∈ R+
e . Let ε ∈ R+
be arbitrarily given.
We have
1
α
t+α
t
1
u log u
r1(u) +
r2(u)
[log(log u)]2 | cosp ϕ(u)|p
du
−
1
α t log t
t+α
t
r1(u) +
r2(u)
[log(log t)]2 du | cosp ψ(t)|p
2.6. RIEMANN–WEBER EQUATIONS 100
≤
1
α
t+α
t
1
u log u
r1(u) +
r2(u)
[log(log u)]2 | cosp ϕ(u)|p
du
−
1
α
t+α
t
1
t log t
r1(u) +
r2(u)
[log(log u)]2 | cosp ϕ(u)|p
du
+
1
α
t+α
t
1
t log t
r1(u) +
r2(u)
[log(log u)]2 | cosp ϕ(u)|p
du
−
1
α
t+α
t
1
t log t
r1(u) +
r2(u)
[log(log u)]2 | cosp ψ(t)|p
du
+
1
α
t+α
t
1
t log t
r1(u) +
r2(u)
[log(log u)]2 | cosp ψ(t)|p
du
−
1
α
t+α
t
1
t log t
r1(u) +
r2(u)
[log(log t)]2 | cosp ψ(t)|p
du
for all t ∈ R+
e . Since
lim
t→∞
t(t + α) log t
1
t log t
−
1
(t + α) log(t + α)
= α, (2.6.36)
we obtain (see (2.6.8), (2.6.10), and (2.6.16))
1
α
t+α
t
1
u log u
−
1
t log t
r1(u) +
r2(u)
[log(log u)]2 | cosp ϕ(u)|p
du
≤
1
α
t+α
t
1
t log t
−
1
(t + α) log(t + α)
r1(u) +
|r2(u)|
[log(log t)]2 | cosp ϕ(u)|p
du
≤ 2
t+α
t
1
t2 log t
r+
1 +
|r2(u)|
[log(log t)]2 du <
1
t
3
2
t+α
t
1
√
t log t
r+
1 + |r2(u)| du <
1
t
3
2
for all large t. Especially, we can assume that
1
α
t+α
t
1
u log u
−
1
t log t
r1(u) +
r2(u)
[log(log u)]2 | cosp ϕ(u)|p
du
<
ε
t log t [log(log t)]2 .
(2.6.37)
2.6. RIEMANN–WEBER EQUATIONS 101
We recall that the half-linear trigonometric functions sinp and cosp are periodic and
continuously differentiable. In particular, these facts imply the existence of a positive
number L such that
|| sinp x| − | sinp y|| ≤ L |x − y| , || cosp x| − | cosp y|| ≤ L |x − y| , (2.6.38)
and
|| sinp x|p
− | sinp y|p
| ≤ L |x − y| , || cosp x|p
− | cosp y|p
| ≤ L |x − y| , (2.6.39)
for any x, y ∈ R. Applying the second inequality in (2.6.39), we have (see (2.6.21) in
Lemma 2.6.1 and again (2.6.8) and (2.6.10))
1
α
t+α
t
1
t log t
r1(u) +
r2(u)
[log(log u)]2 [| cosp ϕ(u)|p
− | cosp ψ(t)|p
] du
≤
L
α
t+α
t
1
t log t
r+
1 +
|r2(u)|
[log(log t)]2 |ϕ(u) − ψ(t)| du
≤
L
α
t+α
t
1
t log t
r+
1 +
|r2(u)|
[log(log t)]2
1
√
t log t
du
<
ε
t log t [log(log t)]2
(2.6.40)
for sufficiently large t.
Using
lim
t→∞
t log t
1
[log(log t)]2 −
1
[log(log[t + α])]2 = 0
and (2.6.8), we obtain the estimation
1
α
t+α
t
r2(u)
[log(log t)]2 du −
1
α
t+α
t
r2(u)
[log(log u)]2 du
≤
1
α
t+α
t
|r2(u)|
1
[log(log t)]2 −
1
[log(log[t + α])]2 du
≤
1
t log t
t+α
t
|r2(u)| du ≤
1
√
t log t
2.6. RIEMANN–WEBER EQUATIONS 102
for every large t, which gives (consider also (2.6.16))
1
α
t+α
t
1
t log t
r1(u) +
r2(u)
[log(log u)]2 | cosp ψ(t)|p
du
−
1
α
t+α
t
1
t log t
r1(u) +
r2(u)
[log(log t)]2 | cosp ψ(t)|p
du
≤
1
αt log t
t+α
t
r2(u)
[log(log u)]2 −
r2(u)
[log(log t)]2 | cosp ψ(t)|p
du
≤
1
t log t
3
2
<
ε
t log t [log(log t)]2
(2.6.41)
for all large t.
Thus (see (2.6.37), (2.6.40), and (2.6.41)), we have
1
α
t+α
t
1
u log u
r1(u) +
r2(u)
[log(log u)]2 | cosp ϕ(u)|p
du
−
1
α t log t
t+α
t
r1(u) +
r2(u)
[log(log t)]2 du | cosp ψ(t)|p
<
3ε
t log t [log(log t)]2
(2.6.42)
for all large t.
Analogously (cf. (2.6.8) and (2.6.9)), one can show that
1
α(p − 1)
t+α
t
1
u log u
s1(u) +
s2(u)
[log(log u)]2 | sinp ϕ(u)|p
du
−
1
α(p − 1) t log t
t+α
t
s1(u) +
s2(u)
[log(log t)]2 du | sinp ψ(t)|p
<
3ε
t log t [log(log t)]2
(2.6.43)
for all large t.
2.6. RIEMANN–WEBER EQUATIONS 103
For large t, we have (see (2.6.16) and (2.6.36))
1
α
t+α
t
Φ(cosp ϕ(u)) sinp ϕ(u)
t log t
du −
1
α
t+α
t
Φ(cosp ϕ(u)) sinp ϕ(u)
u log u
du
≤
1
α
t+α
t
1
t log t
−
1
u log u
|Φ(cosp ϕ(u)) sinp ϕ(u)| du
≤
1
α
t+α
t
1
t log t
−
1
(t + α) log(t + α)
du
=
1
t log t
−
1
(t + α) log(t + α)
≤
2α
t2 log t
(2.6.44)
and (see (2.6.16), (2.6.21) in Lemma 2.6.1, and (2.6.38))
Φ (cosp ψ(t)) sinp ψ(t) −
1
α
t+α
t
Φ(cosp ϕ(u)) sinp ϕ(u) du
≤ Φ (cosp ψ(t)) sinp ψ(t) −
1
α
t+α
t
Φ(cosp ψ(t)) sinp ϕ(u) du
+
1
α
t+α
t
Φ(cosp ψ(t)) sinp ϕ(u) du −
1
α
t+α
t
Φ(cosp ϕ(u)) sinp ϕ(u) du
≤
1
α
t+α
t
|sinp ψ(t) − sinp ϕ(u)| du +
1
α
t+α
t
|Φ(cosp ψ(t)) − Φ(cosp ϕ(u))| du
≤
L
α
t+α
t
|ψ(t) − ϕ(u)| du +
K
α
t+α
t
|ψ(t) − ϕ(u)|min{p−1,1}
du
≤
1
√
t
min{p−1,1}
(2.6.45)
for some K ∈ R+
.
2.6. RIEMANN–WEBER EQUATIONS 104
Hence (see (2.6.44) and (2.6.45)), it holds
1
t log t
Φ(cosp ψ(t)) sinp ψ(t) −
1
α
t+α
t
1
u log u
Φ(cosp ϕ(u)) sinp ϕ(u) du
≤
1
t log t
Φ(cosp ψ(t)) sinp ψ(t) −
1
α
t+α
t
Φ(cosp ϕ(u)) sinp ϕ(u) du
+
1
αt log t
t+α
t
Φ(cosp ϕ(u)) sinp ϕ(u) du
−
1
α
t+α
t
1
u log u
Φ(cosp ϕ(u)) sinp ϕ(u) du
≤
1
t log t
1
√
t
min{p−1,1}
+
2α
t2 log t
<
ε
t log t [log(log t)]2
(2.6.46)
for all large t.
Finally (see (2.6.35), (2.6.42), (2.6.43), and (2.6.46)), we have
ψ (t) −
1
α
t+α
t
1
t log t
r1(u) +
r2(u)
[log(log t)]2 | cosp ψ(t)|p
−Φ (cosp ψ(t)) sinp ψ(t) + s1(u) +
s2(u)
[log(log t)]2
|sinp ψ(t)|p
p − 1
du
<
7ε
t log t [log(log t)]2
(2.6.47)
for any sufficiently large t.
Part (i). Let ϑ ∈ R+
be such that (see (2.6.32))
[M(r1)]p−1
(S − ϑ) +
p − 1
qp
R − ϑ
M(r1)
>
q1−p
2
. (2.6.48)
We consider ε ∈ R+
such that
7ε < ϑ, 7ε (p − 1) < ϑ. (2.6.49)
2.6. RIEMANN–WEBER EQUATIONS 105
For large t, we have (see (1.1.5), (2.6.31), (2.6.47), and (2.6.49))
ψ (t) >
1
αt log t
t+α
t
r1(u) +
r2(u)
[log(log t)]2 | cosp ψ(t)|p
− Φ (cosp ψ(t)) sinp ψ(t) + s1(u) +
s2(u)
[log(log t)]2 ×
×
|sinp ψ(t)|p
p − 1
du −
7ε
t log t [log(log t)]2
=
1
αt log t


t+α
t
r1(u) +
r2(u) − 7ε
[log(log t)]2 | cosp ψ(t)|p
− Φ (cosp ψ(t)) sinp ψ(t)
+ s1(u) +
s2(u) − 7ε(p − 1)
[log(log t)]2
|sinp ψ(t)|p
p − 1
du
=
1
t log t
M(r1) +
1
α
t+α
t
r2(u) du − 7ε
[log(log t)]2 | cosp ψ(t)|p
− Φ (cosp ψ(t)) sinp ψ(t)
+ M(s1) +
1
α
t+α
t
s2(u) du − 7ε(p − 1)
[log(log t)]2
|sinp ψ(t)|p
p − 1
>
1
t log t
M(r1) +
R − ϑ
[log(log t)]2 | cosp ψ(t)|p
− Φ (cosp ψ(t)) sinp ψ(t)
+ M(s1) +
S − ϑ
[log(log t)]2
|sinp ψ(t)|p
p − 1
.
(2.6.50)
It suffices to use Lemma 2.6.3, (i) (compare (2.6.25) with (2.6.48) and Eq. (2.6.26) with
the form of (2.6.50) for X = R − ϑ, Y = S − ϑ). Since the Pr¨ufer angle ϕ is unbounded
from above (consider (2.6.21) in Lemma 2.6.1), Eq. (2.6.6) is oscillatory. Therefore, the
first part of the theorem is proved.
Part (ii). We consider ϑ ∈ R+
such that (see (2.6.34))
[M(r1)]p−1
(S + ϑ) +
p − 1
qp
R + ϑ
M(r1)
<
q1−p
2
(2.6.51)
2.6. RIEMANN–WEBER EQUATIONS 106
and ε ∈ R+
satisfying (2.6.49). We can proceed analogously as in the first case.
For large t, we have (see (1.1.5), (2.6.33), (2.6.47), and (2.6.49))
ψ (t) <
1
αt log t
t+α
t
r1(u) +
r2(u)
[log(log t)]2 | cosp ψ(t)|p
− Φ (cosp ψ(t)) sinp ψ(t)
+ s1(u) +
s2(u)
[log(log t)]2
|sinp ψ(t)|p
p − 1
du
+
7ε
t log t [log(log t)]2
<
1
t log t
M(r1) +
R + ϑ
[log(log t)]2 | cosp ψ(t)|p
− Φ (cosp ψ(t)) sinp ψ(t)
+ M(s1) +
S + ϑ
[log(log t)]2
|sinp ψ(t)|p
p − 1
.
(2.6.52)
Using Lemma 2.6.1 and Lemma 2.6.3, (ii) (cf. (2.6.27), (2.6.28) and (2.6.51), (2.6.52) for
V = R + ϑ, W = S + ϑ), we know that the Pr¨ufer angle is bounded from above, which
implies the non-oscillation of Eq. (2.6.6). The proof is complete.
§ 2.6.3 Corollaries and examples
In this paragraph, we illustrate the novelty of Theorem 2.6.3 on corollaries and examples
which are not covered by any previously known criteria. As a corollary of Theorem
2.6.3, we obtain the following new result which solves the oscillation behavior of the
non-perturbed equation in the critical case (cf. Theorem 2.5.3).
Corollary 2.6.1. If r : R → R+
and s : R → R are continuous α-periodic functions such
that
[M (r)]p−1
M(s) = q−p
,
then the equation
r−p
q (t)tp−1
Φ(x ) +
s(t)
t logp
t
Φ(x) = 0 (2.6.53)
is non-oscillatory.
Proof. It suffices to consider r1(t) = r(t), r2(t) ≡ 0, s1(t) = s(t), and s2(t) ≡ 0 in
Eq (2.6.6) and to put R := 0 and S := 0 in Theorem 2.6.3, (ii).
Example 2.6.1. We can apply Corollary 2.6.1, e.g., to the equation
2
1 + 2 sin2
t
p
q
tp−1
Φ(x ) +
q−p
+ p sin t − q cos t
t logp
t
Φ(x) = 0 (2.6.54)
2.6. RIEMANN–WEBER EQUATIONS 107
which is in the form of Eq. (2.6.53), where
M (r) = M
1 + 2 sin2
t
2
= 1,
M(s) = M q−p
+ p sin t − q cos t = q−p
.
Since [M (r)]p−1
M(s) = q−p
, Eq. (2.6.54) is in the critical case which means that it is
non-oscillatory.
Now, we formulate a direct consequence of Theorems 2.5.3 and 2.6.3 and Corollary
2.6.1 for linear equations.
Corollary 2.6.2. Consider the equations
tx
r1(t)
+
s1(t)x
t log2
t
= 0, (2.6.55)
[log (log t)]2
tx
r1(t) [log (log t)]2
+ r2(t)
+
1
t log2
t
s1(t) +
s2(t)
[log (log t)]2 x = 0 (2.6.56)
with continuous α-periodic coefficients r1 : R → R+
, s1 : R → R and with continuous
coefficients r2, s2 : R+
e → R satisfying (2.6.7), (2.6.8), and (2.6.9).
(i) If 4M(r1)M(s1) > 1, then Eq. (2.6.55) is oscillatory.
(ii) If 4M(r1)M(s1) ≤ 1, then Eq. (2.6.55) is non-oscillatory.
(iii) If 4M(r1)M(s1) = 1 and if there exist R, S ∈ R satisfying
S
M(s1)
+
R
M(r1)
> 1 and
1
α
t+α
t
r2(u) du ≥ R,
1
α
t+α
t
s2(u) du ≥ S, t ∈ R+
e ,
then Eq. (2.6.56) is oscillatory.
(iv) If 4M(r1)M(s1) = 1 and if there exist R, S ∈ R satisfying
S
M(s1)
+
R
M(r1)
< 1 and
1
α
t+α
t
r2(u) du ≤ R,
1
α
t+α
t
s2(u) du ≤ S, t ∈ R+
e ,
then Eq. (2.6.56) is non-oscillatory.
Example 2.6.2. Let a ∈ R+
1 and b, c, d ∈ R+
. From Corollary 2.6.2, we know that the
equation
tx
a + sin(cx)
+
b + cos(cx)
t log2
t
x = 0
2.6. RIEMANN–WEBER EQUATIONS 108
is oscillatory if and only if 4ab > 1. Note that the case 4ab = 1 is covered by Corollary 2.6.1
and the case 4ab = 1 by Theorem 2.5.3. In addition, applying Corollary 2.6.2, we know
that the equation
tx
a + sin(cx)
+
1
t log2
t
1
4a
+ cos(cx) +
d + sin(cx) cos(cx)
[log (log t)]2 x = 0
is oscillatory for 4ad > 1 and non-oscillatory for 4ad < 1.
To formulate the next corollary, we need the definitions of almost periodicity 2.4.1 and
2.4.2 and the definition of asymptotic almost periodicity 2.4.3. For more detail, we refer
to books [12, 29].
From Definition 2.4.3, it is seen that (2.6.8) and (2.6.9) hold for all asymptotically
almost periodic functions r2, s2. At the same time, (2.6.7) is valid for all large t if r2 is
asymptotically almost periodic. Therefore, we can use Theorem 2.6.3 for any equation
of the form (2.6.6) with α-periodic coefficients r1, s1 and asymptotically almost periodic
coefficients r2, s2. To be as clear as possible, we use in Corollary 2.6.3 and Example 2.6.3
below the fact, that any asymptotically almost periodic function has its mean value in
the sense of the following definition. Note that Theorem 2.6.3 can be applied also for
equations with coefficients which have mean values (see Definition 2.1.1) and which are
not asymptotically almost periodic.
Corollary 2.6.3. Let R1 : R → R+
, S1 : R → R be continuous α-periodic functions such
that
[M (R1)]p−1
M(S1) = q−p
and let R2, S2 : R+
∪ {0} → R be asymptotically almost periodic functions.
(i) If
[M(R1)]p−1
M(S2) +
p − 1
qp
M(R2)
M(R1)
>
q1−p
2
,
then the equation
R1(t) +
R2(t)
[log(log t)]2
−p
q
tp−1
Φ(x )
+
1
t logp
t
S1(t) +
S2(t)
[log(log t)]2 Φ(x) = 0
(2.6.57)
is oscillatory.
(ii) If
[M(R1)]p−1
M(S2) +
p − 1
qp
M(R2)
M(R1)
<
q1−p
2
,
then Eq. (2.6.57) is non-oscillatory.
2.6. RIEMANN–WEBER EQUATIONS 109
Proof. The corollary follows from Theorem 2.6.3 as well. It suffices to replace α by nα
for a sufficiently large number n ∈ N and to use the definition of the mean value given in
(2.1.6) and the existence of δ ∈ R+
with the property that
[M(R1)]p−1
δ +
p − 1
qp
δ
M(R1)
< [M(R1)]p−1
M(S2) +
p − 1
qp
M(R2)
M(R1)
−
q1−p
2
if
[M(R1)]p−1
M(S2) +
p − 1
qp
M(R2)
M(R1)
=
q1−p
2
.
Example 2.6.3. Let a, b, c ∈ R and u, v ∈ R \ {0} determine the coefficients of the
equation
3 + Φ(sin t)
3
+
a + sin(bt) + sin(ct)
[log(log[t + 1])]2
−p
q
tp−1
Φ(x )
+
1
t logp
t
2 sin2
t
qp
+
sin(ut) cos(ut) + v + t−2
log(log t)
2
Φ(x) = 0
(2.6.58)
which has the form of Eq. (2.6.57) for
R1(t) =
3 + Φ(sin t)
3
, S1(t) =
2 sin2
t
qp
,
R2(t) = [a + sin(bt) + sin(ct)]
log(log t)
log(log[t + 1])
2
, S2(t) = sin(ut) cos(ut) + v +
1
t2
2
.
One can verify that R2, S2 are asymptotically almost periodic functions and that
M(R1) = 1, M(S1) = q−p
, M(R2) = a, M(S2) =
8v2
+ 1
8
.
Especially, [M(R1)]p−1
M(S1) = q−p
. Hence, we can apply Corollary 2.6.3 which gives the
oscillation of Eq. (2.6.58) for
8v2
+ 1
8
+
a(p − 1)
qp
>
q1−p
2
and its non-oscillation for
8v2
+ 1
8
+
a(p − 1)
qp
<
q1−p
2
.
In the final corollary and example, we consider Eq. (2.6.6) with constant coefficients
r1, s1 and periodic coefficients r2, s2, which do not need to have any common period. We
point out that we get a new result even in the case when the periods of r2, s2 are same.
2.6. RIEMANN–WEBER EQUATIONS 110
Corollary 2.6.4. Let a, b ∈ R+
satisfy ap−1
b = q−p
. Let R, S : R → R be periodic
continuous functions.
(i) If
ap−1
M(S) +
p − 1
aqp
M(R) >
q1−p
2
,
then the equation
a +
R(t)
[log (log t)]2
−p
q
tp−1
Φ(x )
+
1
t logp
t
b +
S(t)
[log (log t)]2 Φ(x) = 0
(2.6.59)
is oscillatory.
(ii) If
ap−1
M(S) +
p − 1
aqp
M(R) <
q1−p
2
,
then Eq. (2.6.59) is non-oscillatory.
Proof. The corollary is a special case of Corollary 2.6.3.
Example 2.6.4. We illustrate Corollary 2.6.4 by the equation
1 +
c + d sin t
[log(log t)]2
−p
q
tp−1
Φ(x )
+
1
t logp
t
q−p
+
C + D cos
√
2 t
[log(log t)]2 Φ(x) = 0,
(2.6.60)
where c, d, C, D ∈ R are arbitrary constants. For a := 1, b := q−p
, R(t) := c + d sin t,
and S(t) := C + D cos
√
2 t , we have ap−1
b = q−p
and M(R) = c, M(S) = C. Hence,
Eq. (2.6.60) is oscillatory for
C +
p − 1
qp
c >
q1−p
2
, i.e., Cqp
+ (p − 1) c >
q
2
,
and non-oscillatory for
C +
p − 1
qp
c <
q1−p
2
, i.e., Cqp
+ (p − 1) c <
q
2
.
Chapter
3 Difference equations
3.1 Half-linear difference equations with coefficients
having mean values
This chapter is devoted to oscillatory properties of the half-linear difference equations
∆[rkΦ(∆xk)] + ckΦ(xk+1) = 0, (3.1.1)
where rk is positive for all considered k. Throughout the whole chapter, we consider
integers k ≥ a for a sufficiently large number a ∈ N. For reader’s convenience, we use
the notation Na := {n ∈ N : n ≥ a} for a ∈ N. Our aim is to find explicit oscillation
constants for all equations (3.1.1) from a large class of equations with
{ck}k∈Na ≡
γsk
(k + 1)(p)
k∈Na
,
i.e., in the form
∆[rkΦ(∆xk)] +
γsk
(k + 1)(p)
Φ(xk+1) = 0, (3.1.2)
where γ ∈ R and k(p)
stands for the generalized power function (also called the falling
factorial power) given by
k(p)
=
Γ(k + 1)
Γ(k + 1 − p)
, Γ(x) =
∞
0
e−s
sx−1
ds, x > 0.
For details about k(p)
, see, e.g., [51, Chapter 2]. The basic terminology is analogous to
the one from the continuous case. Eq. (3.1.2) is said to be conditionally oscillatory if
there exists a (positive) constant Γ such that Eq. (3.1.2) is oscillatory for γ > Γ and
non-oscillatory for γ < Γ. The constant Γ (which is dependent on coefficients) is called
the critical oscillation constant of Eq. (3.1.2).
Concerning the conditional oscillation of the studied difference equations, the first
result comes from [62], where the equation
∆2
xk +
γ
(k + 1)k
xk+1 = 0
111
3.1. COEFFICIENTS WITH MEAN VALUES 112
was proved to be conditionally oscillatory with the oscillation constant Γ = 1/4. Equations
with non-constant coefficients were analyzed in [58]. In [36], the conditional oscillation
of the linear equation
∆(rk∆xk) +
γsk
(k + 1)k
xk+1 = 0 (3.1.3)
with almost periodic coefficients was obtained. In [37], this result was generalized for halflinear
equations of the form (3.1.2) with positive asymptotically almost periodic sequences
{rk}, {sk}. From other results from the oscillation theory of difference equations, we
mention at least papers [6, 13, 50, 63, 80].
Since the main result of [37] is one of the basic motivations for the research presented
here, we reformulate it as follows. We remark that the symbol M(·) stands for the mean
values of indicated sequences clarified in the below given Definition 3.1.2 and that the
definition of asymptotic almost periodicity is mentioned in Definition 3.1.4 below.
Theorem 3.1.1. Let γ ∈ R be given and let {rk}k∈Na and {sk}k∈Na be arbitrary positive
asymptotically almost periodic sequences such that
inf {rk; k ∈ Na} > 0, lim sup
k→∞
sk > 0.
Let
Γ :=
p − 1
p
p
M r
1
1−p
k
1−p
[M ({sk})]−1
.
If γ > Γ, then Eq. (3.1.2) is oscillatory. If γ < Γ, then Eq. (3.1.2) is non-oscillatory.
Our second basic motivation comes from the continuous case which is described in
chapters 1 and 2. The most relevant result for the research contained in this chapter is
Theorem 2.1.3 which can be reformulated as follows.
Theorem 3.1.2. Let r : Ra → R be a continuous function, for which mean value
M r
1
1−p exists and for which it holds
0 < inf
t∈Ra
r(t) ≤ sup
t∈Ra
r(t) < ∞,
and let s : Ra → R be a continuous function having mean value M(s). Let
Γ :=
p − 1
p
p
M r
1
1−p
1−p
=
p − 1
p
p

 lim
t→∞
1
t
a+t
a
r
1
1−p (τ) dτ


1−p
.
Consider the equation
[r(t)Φ(x )] +
s(t)
tp
Φ(x) = 0. (3.1.4)
Eq. (3.1.4) is oscillatory if M(s) > Γ, and non-oscillatory if M(s) < Γ.
3.1. COEFFICIENTS WITH MEAN VALUES 113
In this chapter, we intend to generalize Theorem 3.1.1 into the case when the coefficients
have mean values and the second coefficient can change sign. It means that our aim
is to prove the discrete counterpart of Theorem 3.1.2. For this purpose, we improve the
method from [37]. Since we study equations with coefficients from more general classes,
we have to prove some new auxiliary results and inequalities (especially, we need Lemmata
3.1.1 and 3.1.2 below). Note that we partially apply the processes used in [37] (see
the proof of Theorem 3.1.5 below, where it is explicitly mentioned).
The chapter is organized as follows. In the next paragraph, we state the necessary
background and we recall the discrete Riccati technique, which is essential for our investigation.
In § 3.1.2, the reader can find preparatory lemmata, results, and corollaries. In
§ 3.1.3, we collect illustrative examples.
§ 3.1.1 Preliminaries
In this paragraph, we mention the needed background concerning the oscillation theory
of half-linear difference equations. For more details, we refer to [1, Chapter 3] and [21,
Chapter 8] with references cited therein. In addition, we recall the concept of mean values
which is necessary to find general oscillation constants. We also state the concept of the
(adapted) discrete half-linear Riccati equation which is the main tool in our investigation.
At first, we recall the basic notions from the oscillation theory of the half-linear equa-
tion
∆[rkΦ(∆xk)] + ckΦ(xk+1) = 0, (3.1.5)
where rk > 0 for all considered k ∈ Na. An interval (l, l + 1], l ∈ Na, contains the
generalized zero of a solution {xk} of Eq. (3.1.5) if xl = 0 and xlxl+1 ≤ 0. We say that
Eq. (3.1.5) is disconjugate on a set {l, l + 1, . . . , l + n} if any solution of Eq. (3.1.5) has at
most one generalized zero on (l, l + n + 1] and a solution {˜xk} given by the initial value
˜xl = 0 has no generalized zero on (l, l + n + 1]. Otherwise, Eq. (3.1.5) is called conjugate
on {l, l + 1, . . . , l + n}. Now, we can formulate the following definition.
Definition 3.1.1. Eq. (3.1.5) is called non-oscillatory if there exists l ∈ N with the
property that Eq. (3.1.5) is disconjugate on {l, l + 1, . . . , l + n} for all n ∈ N. In the
opposite case, Eq. (3.1.5) is called oscillatory.
The Sturm type separation theorem (see, e.g., [1, Theorem 3.3.6]) enables us to give
Definition 3.1.1, because the oscillation of an arbitrary non-zero solution of Eq. (3.1.5)
implies the oscillation of all solutions of Eq. (3.1.5). We will also use a consequence of
the Sturm type comparison theorem. We mention only the form that is prepared for our
purpose.
Theorem 3.1.3. Let {yk}k∈Na , {Yk}k∈Na , {zk}k∈Na , {Zk}k∈Na be sequences satisfying the
inequalities yk ≥ Yk > 0, Zk ≥ zk for all sufficiently large k. Let us consider the equations
∆[ykΦ(∆xk)] + zkΦ(xk+1) = 0, (3.1.6)
∆[YkΦ(∆xk)] + ZkΦ(xk+1) = 0. (3.1.7)
3.1. COEFFICIENTS WITH MEAN VALUES 114
If Eq. (3.1.7) is non-oscillatory, then Eq. (3.1.6) is non-oscillatory as well.
Proof. The theorem follows, e.g., from [1, Theorem 3.3.5].
To obtain explicit oscillation constants, we need the definition of the mean value of
a sequence.
Definition 3.1.2. Let a sequence {fk}k∈Na ⊂ R be given and let the limit
M ({fk}) := lim
n→∞
1
n
n+i−1
k=i
fk (3.1.8)
be finite and exist uniformly with respect to i ∈ Na. The number M({fk}) introduced
in (3.1.8) is called the mean value of {fk}.
An important class of sequences having mean values is formed by asymptotically
almost periodic sequences (see also [37]). Hence, we formulate the next definitions.
Definition 3.1.3. A sequence {fk}k∈Z ⊂ R is called almost periodic if, for any ε > 0,
there exists P(ε) ∈ N such that any set of the form {i, i+1, . . . , i+P(ε)−1} ⊂ Z contains
an integer l for which |fk − fk+l| < ε, k ∈ Z.
Definition 3.1.4. We say that a sequence {fk}k∈Na ⊂ R is asymptotically almost periodic
if there exists a pair of sequences {f1
k }k∈Z, {f2
k }k∈Na ⊂ R such that {f1
k } is almost periodic,
{f2
k } satisfies limk→∞ f2
k = 0, and {fk}k∈Na ≡ {f1
k + f2
k }k∈Na .
Finally, we describe the half-linear Riccati equation and its adapted version. Using
the Riccati substitution
wk = rkΦ
∆xk
xk
to Eq. (3.1.5), we obtain the associated Riccati equation
∆wk + ck + wk 1 −
rk
Φ[Φ−1(rk) + Φ−1(wk)]
= 0, (3.1.9)
where Φ−1
denotes the inverse function of Φ, i.e., Φ−1
(x) = |x|q−1
sgn x. Under the
condition wk + rk > 0, we can express (see [1, Lemma 3.2.6, (I8)])
wk 1 −
rk
Φ[Φ−1(rk) + Φ−1(wk)]
=
(p − 1) |wk|q
|βk|p−2
Φ[Φ−1(rk) + Φ−1(wk)]
,
where βk is between Φ−1
(rk) and Φ−1
(rk) + Φ−1
(wk), i.e., for wk + rk > 0, we have the
Riccati equation (3.1.9) associated to Eq. (3.1.5) in the form
∆wk + ck +
(p − 1) |wk|q
|βk|p−2
Φ[Φ−1(rk) + Φ−1(wk)]
= 0. (3.1.10)
The following theorem is typically known as the discrete Riccati method. It shows
the way in which the non-oscillation of Eq. (3.1.5) is connected to the solvability of
Eq. (3.1.10).
3.1. COEFFICIENTS WITH MEAN VALUES 115
Theorem 3.1.4. Eq. (3.1.5) is non-oscillatory if and only if there exist an integer b and
a sequence of wk which solves Eq. (3.1.10) and satisfies wk + rk > 0 for k ∈ Nb.
Proof. The theorem is a consequence of the well-known discrete Reid roundabout theorem
(see, e.g., [1, Theorem 3.3.4] or directly [21, Theorem 8.2.5]).
Taking into account the second substitution
ζk = −k(p−1)
wk (3.1.11)
together with the Riccati equation (3.1.10), we obtain the adapted Riccati equation associated
to Eq. (3.1.5) as
∆ζk =
1
k − p + 2
(p − 1)ζk + (k + 1)(p)
ck
+
(k + 1)(p − 1) |βk|p−2
|ζk|q
[k(p−1)]
q−1
Φ Φ−1(rk) + Φ−1 − ζk
k(p−1)
,
(3.1.12)
where βk is between Φ−1
(rk) and Φ−1
(rk) + Φ−1
− ζk
k(p−1) .
In fact, we will consider Eq. (3.1.5) in the form
∆ [rkΦ(∆xk)] +
sk
(k + 1)(p)
Φ(xk+1) = 0, (3.1.13)
where sequence {rk}k∈Na has mean value M({r1−q
k }) = 1 and
0 < r−
:= inf
k∈Na
rk ≤ sup
k∈Na
rk =: r+
< ∞ (3.1.14)
and where sequence {sk}k∈Na has a positive mean value, i.e., M({sk}) > 0. Therefore, we
will deal with the Riccati equation associated to Eq. (3.1.13) in the form (see Eq. (3.1.10))
∆wk +
sk
(k + 1)(p)
+
(p − 1) |wk|q
|βk|p−2
Φ [Φ−1 (rk) + Φ−1(wk)]
= 0 (3.1.15)
and with the adapted Riccati equation (see Eq. (3.1.12))
∆ζk =
1
k − p + 2
(p − 1)ζk + sk +
(k + 1)(p − 1) |βk|p−2
|ζk|q
[k(p−1)]
q−1
Φ Φ−1 (rk) + Φ−1 − ζk
k(p−1)
. (3.1.16)
More precisely, we will study Eq. (3.1.13) using Eq. (3.1.16).
3.1. COEFFICIENTS WITH MEAN VALUES 116
§ 3.1.2 Results
To prove the main results, we need the following lemmata.
Lemma 3.1.1. Let a sequence {fk}k∈Na ⊂ R have mean value M({fk}). There exists
a positive number K({fk}) for which |fk| < K({fk}), k ∈ Na.
Proof. The existence of M({fk}) gives m ∈ N such that
M({fk}) −
1
m + l
i+m+l−1
k=i
fk < 1, i ∈ Na, l ∈ N ∪ {0}. (3.1.17)
From (3.1.17) it follows
1
m
i+m
k=i+1
fk −
1
m
i+m−1
k=i
fk < 2, i ∈ Na,
1
m + 1
i+m+1
k=i+1
fk −
1
m + 1
i+m
k=i
fk < 2, i ∈ Na.
Especially,
| fi − fm+i | < 2m, i ∈ Na,
| fi − fm+i+1 | < 2(m + 1), i ∈ Na.
Thus, we have
| fm+i − fm+i+1 | < 4m + 2, i ∈ Na.
Finally, it holds
| fi − fi+1 | ≤ L({fk}) := max {4m + 2, |fa − fa+1|, . . . , |fm+a−1 − fm+a|}, i ∈ Na.
On contrary, let us suppose that lim supk→∞ |fk| = ∞. If
fi ≥ M({fk}) + 1 + (m − 1)L({fk})
for some i ∈ Na, then
fi+j ≥ M({fk}) + 1, j ∈ {0, 1, . . . , m − 1}. (3.1.18)
Analogously, if
fi ≤ M({fk}) − 1 − (m − 1)L({fk})
for some i ∈ Na, then
fi+j ≤ M({fk}) − 1, j ∈ {0, 1, . . . , m − 1}. (3.1.19)
Of course, each one of inequalities (3.1.18), (3.1.19) gives a contradiction with (3.1.17)
for l = 0. It means that it suffices to put
K({fk}) := |M({fk})| + 1 + (m − 1)L({fk})
for m from (3.1.17).
3.1. COEFFICIENTS WITH MEAN VALUES 117
Henceforth, let m ∈ N be such that
3M({sk})
2
>
1
m + l
i+m+l−1
k=i
sk >
M({sk})
2
> 0, i ∈ Na, l ∈ N ∪ {0}. (3.1.20)
We also put (cf. (3.1.14))
s+
:= sup
k∈Na
|sk| < ∞, (3.1.21)
where we use Lemma 3.1.1.
Lemma 3.1.2. If Eq. (3.1.13) is non-oscillatory, then there exist L ∈ N and a negative
solution {ζk}k∈NL
of Eq. (3.1.16) such that
lim
k→∞
ζk
k(p−1)
= 0. (3.1.22)
Proof. Considering Theorem 3.1.4, the non-oscillation of Eq. (3.1.13) implies that there
exist L ∈ N and a solution {wk}k∈NL
of Eq. (3.1.15) such that wk + rk > 0 for k ≥ L.
Considering (3.1.11), it gives the solution {ζk}k∈NL
≡ {−wkk(p−1)
}k∈NL
of Eq. (3.1.16).
We show that this solution {ζk} is negative and satisfies (3.1.22).
In fact, we show that the sequences {wmk}, {wmk+1}, . . . , {wmk+m−1} are decreasing
for sufficiently large k and tend to zero. Let j ∈ {0, 1, . . . , m − 1} be arbitrarily given.
From Eq. (3.1.15), we have
w(k+1)m+j − wmk+j = −
(k+1)m+j−1
i=mk+j
si
(i + 1)(p)
−
(k+1)m+j−1
i=mk+j
(p − 1) |wi|q
|βi|p−2
Φ [Φ−1 (ri) + Φ−1(wi)]
(3.1.23)
for all considered k. Since {sk} is bounded (consider Lemma 3.1.1) and
lim
l→∞
(l + m)(p)
(l + 1)(p)
= 1,
using (3.1.20), we obtain
−
(k+1)m+j−1
i=mk+j
si
(i + 1)(p)
< 0 (3.1.24)
for all large k. Considering wk + rk > 0 for k ≥ L, it is seen that
Φ Φ−1
(rk) + Φ−1
(wk) > 0, k ≥ L. (3.1.25)
From (3.1.23), (3.1.24), (3.1.25), we get that w(k+1)m+j < wmk+j for all large k. Thus,
there exist limits (as real numbers or −∞)
L0 := lim
k→∞
wmk, L1 := lim
k→∞
wmk+1, . . . , Lm−1 := lim
k→∞
wmk+m−1.
3.1. COEFFICIENTS WITH MEAN VALUES 118
Our aim is to prove that Lj = 0 for each j ∈ {0, 1, . . . , m − 1}. On contrary, let us
consider that Lj = 0 for some j. Denote W := maxk∈NL
wk.
Let Lj > ε > 0. We know that
Φ−1
(W) + Φ−1
(r+
) ≥ βkm+j ≥ Φ−1
(rkm+j) ≥ Φ−1
(r−
) > 0 (3.1.26)
and that
Φ Φ−1
(rkm+j) + Φ−1
(wmk+j) < Φ Φ−1
r+
+ Φ−1
(W) (3.1.27)
for all k. Hence (see (3.1.26)), there exists Bj > 0 with the property that
|βmk+j|p−2
> Bj, k ≥ L. (3.1.28)
In addition, for large k, we have (see (3.1.23), (3.1.24), (3.1.25), (3.1.27), and (3.1.28))
w(m+1)k+j − wmk+j < −
(k+1)m+j−1
i=mk+j
(p − 1) |wi|q
|βi|p−2
Φ [Φ−1 (ri) + Φ−1(wi)]
< −
(p − 1) |wmk+j|q
|βmk+j|p−2
Φ [Φ−1 (rmk+j) + Φ−1(wmk+j)]
< −
(p − 1)εq
Bj
Φ [Φ−1 (r+) + Φ−1(W)]
.
But, we obtain the contradiction Lj = −∞, because the last term is a negative constant.
Let Lj < −ε < 0, i.e., wmk+j < −ε for large k. In this case, for large k, we have
w(m+1)k+j − wmk+j < −
(k+1)m+j−1
i=mk+j
(p − 1) |wi|q
|βi|p−2
Φ [Φ−1 (ri) + Φ−1(wi)]
< −
(p − 1) |wmk+j|q
|βmk+j|p−2
Φ [Φ−1 (rmk+j) + Φ−1(wmk+j)]
< −
(p − 1) |ε|q
[Φ−1
(rmk+j) + Φ−1
(wmk+j)]
p−2
Φ [Φ−1 (rmk+j) + Φ−1(wmk+j)]
= −
(p − 1) |ε|q
Φ−1 (rmk+j) + Φ−1(wmk+j)
< −
(p − 1)εq
Φ−1 (r+)
if p ≥ 2; and
w(m+1)k+j − wmk+j < −
(p − 1) |wmk+j|q
|βmk+j|p−2
Φ [Φ−1 (rmk+j) + Φ−1(wmk+j)]
< −
(p − 1) |ε|q
[Φ−1
(rmk+j)]
p−2
Φ [Φ−1 (rmk+j)]
< −
(p − 1)εq
Φ−1 (r+)
if p ∈ (1, 2). Again, for any p > 1, we get Lj = −∞ which cannot be true, because
wk + rk > 0 for all k and {rk} is bounded.
Altogether, we know that {wk} is positive and
lim
k→∞
wk = 0. (3.1.29)
Thus, {ζk} is negative and (3.1.22) follows from (3.1.29).
3.1. COEFFICIENTS WITH MEAN VALUES 119
We remark that, in the case when the sequence of sk is positive, the statement
of Lemma 3.1.2 follows from [64, Lemma 1, (v) and Theorem 1] combined with [1,
Lemma 3.5.9] or with [21, Lemma 8.2.2].
Lemma 3.1.3. If there exists a negative solution {ζk}k∈NL
of Eq. (3.1.16), then Eq. (3.1.13)
is non-oscillatory.
Proof. A negative solution {ζk}k∈NL
of Eq. (3.1.16) gives {wk}k∈NL
≡ {−ζk/k(p−1)
}k∈NL
which is a positive solution of Eq. (3.1.15). Thus, the lemma follows from Theorem 3.1.4.
Applying the above lemmata, we can obtain the announced result. For readers’ convenience,
we recall the assumptions on coefficients.
Theorem 3.1.5. Let sequence {rk}k∈Na have mean value M({r1−q
k }) = 1 and satisfy
(3.1.14) and let sequence {sk}k∈Na have mean value M({sk}) > 0. Then, Eq. (3.1.13) is
oscillatory for M({sk}) > q−p
and non-oscillatory for M({sk}) < q−p
.
Proof. At first, let us approach the oscillatory part of the theorem. By contradiction, we
suppose that M({sk}) > q−p
and that Eq. (3.1.13) is non-oscillatory. Using Lemma 3.1.2,
we obtain the existence of a negative solution {ζk}k∈NL
of Eq. (3.1.16), i.e., it holds
∆ζk =
1
k − p + 2
(p − 1)ζk + sk
+
(k + 1)(p − 1)βp−2
k |ζk|q
[k(p−1)]
q−1
Φ Φ−1(rk) + Φ−1 − ζk
k(p−1)
, k ≥ L,
(3.1.30)
where
0 < Φ−1
(rk) ≤ βk ≤ Φ−1
(rk) + Φ−1
−
ζk
k(p−1)
, k ≥ L. (3.1.31)
From Lemma 3.1.2 (see (3.1.14), (3.1.22), and (3.1.31)), we also obtain
1 ≤ lim inf
k→∞
βk
Φ−1(rk)
≤ lim sup
k→∞
βk
Φ−1(rk)
≤ lim
k→∞
Φ−1
(rk) + Φ−1
− ζk
k(p−1)
Φ−1(rk)
= 1. (3.1.32)
From (3.1.32) it follows
lim
k→∞
βp−2
k rq−1
k
Φ Φ−1(rk) + Φ−1 − ζk
k(p−1)
= lim
k→∞
rq−1
k
p−2
rq−1
k
rk
= lim
k→∞
r2−q
k rq−1
k
rk
= 1. (3.1.33)
It is well-known that
lim
k→∞
k + 1
[k(p−1)]
q−1 = lim
k→∞
k
[k(p−1)]
q−1 = lim
k→∞
k
[kp−1]q−1 = 1. (3.1.34)
3.1. COEFFICIENTS WITH MEAN VALUES 120
Thus (see (3.1.14), (3.1.33), and (3.1.34)), we can assume that L > p − 2 is so large that
1
√
2r+
q−1 ≤
1
√
2rk
q−1 ≤
βp−2
k
Φ Φ−1(rk) + Φ−1 − ζk
k(p−1)
≤
√
2
rk
q−1
≤
√
2
r−
q−1
, k ≥ L,
(3.1.35)
and that
1
√
2
q−1
≤
k + 1
[k(p−1)]
q−1 ≤ 2
q−1
2 , k ≥ L. (3.1.36)
Combining (3.1.35) and (3.1.36), we obtain
(p − 1)|ζk|q
[2r+]q−1 ≤
(k + 1)(p − 1)βp−2
k |ζk|q
[k(p−1)]
q−1
Φ Φ−1(rk) + Φ−1 − ζk
k(p−1)
≤
2q−1
(p − 1)|ζk|q
[r−]q−1 (3.1.37)
for all k ≥ L.
Considering (3.1.30) and (3.1.37), we have (see also (3.1.21))
∆ζk ≥
1
k − p + 2
(p − 1)ζk − s+
+
(p − 1)|ζk|q
[2r+]q−1 , k ≥ L, (3.1.38)
and
∆ζk ≤
1
k − p + 2
s+
+
2q−1
(p − 1)|ζk|q
[r−]q−1 , k ≥ L. (3.1.39)
If
ζi < X1 := −2r+
2 +
s+
p − 1
1
q−1
− 1 (3.1.40)
for some i ≥ L, then
ζi −
s+
p − 1
+
|ζi|q
[2r+]q−1 > |ζi| −1 +
|ζi|
2r+
q−1
−
s+
p − 1
> |ζi| (−1 + 2) > −X1 > 0.
(3.1.41)
Thus, in this case, it holds ζi+1 > ζi. Indeed, (3.1.38) and (3.1.41) give
∆ζi ≥
p − 1
i − p + 2
ζi −
s+
p − 1
+
|ζi|q
[2r+]q−1 >
p − 1
i − p + 2
|X1| > 0 (3.1.42)
if (3.1.40) is valid. Let us consider the smallest integer l0 ≥ L such that ζl0 ≥ X1. Note
that such a number l0 has to exist because its existence follows from (3.1.42) and from
∞
i=L
p − 1
i − p + 2
|X1| ≥ (p − 1)
∞
j=L−p+2
1
j
= ∞.
3.1. COEFFICIENTS WITH MEAN VALUES 121
Using (3.1.38), we have
ζl0+1 ≥ X2 := X1 +
1
l0 − p + 2
(p − 1)X1 − s+
.
Analogously, one can get that ζj+1 ≥ X2 if ζj ≥ X1 for some j ≥ l0. Hence, there exists
N > 0 satisfying
ζk ∈ (−N, 0), k ≥ L. (3.1.43)
In fact, it suffices to put
N := min{ζL, ζL+1, . . . , ζl0 , X2}.
Trivially, from (3.1.38) and (3.1.39) (or directly from (3.1.30)), it is seen that
|∆ζk| ≤
1
k − p + 2
(p − 1)N + s+
+
2q−1
(p − 1)Nq
[r−]q−1 , k ≥ L. (3.1.44)
Therefore, there exists P > 0 for which
|∆ζk| <
P
k
, k ≥ L. (3.1.45)
Especially, (3.1.45) gives Q > 0 such that
|ζk+i − ζk+j| <
Q
k
, i, j ∈ {0, . . . , m − 1}, k ≥ L. (3.1.46)
Indeed (consider (3.1.44)), inequalities (3.1.45) and (3.1.46) are valid for
P := sup
k∈NL
k
k − p + 2
(p − 1)N + s+
+
2q−1
(p − 1)Nq
[r−]q−1
= max 1,
L
L − p + 2
(p − 1)N + s+
+
2q−1
(p − 1)Nq
[r−]q−1
and Q := (m − 1) P.
In addition (see Definition 3.1.2), we can assume that m ∈ N from (3.1.20) is so large
that
1
m + j
i+m+j−1
k=i
sk > q−p 1
m + l
i+m+l−1
k=i
r1−q
k
1−p
, i ∈ Na, j, l ∈ N ∪ {0}. (3.1.47)
To obtain the oscillatory part of the theorem, one can proceed as in the proof of [37,
Theorem 4.1], where only (3.1.20), (3.1.33), (3.1.34), (3.1.43), (3.1.46), and (3.1.47) are
used to get a contradiction with the existence of the negative solution {ζk}k∈NL
(in fact,
these inequalities are used only in special forms therein).
3.1. COEFFICIENTS WITH MEAN VALUES 122
In the non-oscillatory part of the proof, we consider that m ∈ N satisfies
1
m + j
i+m+j−1
k=i
sk < q−p 1
m + l
i+m+l−1
k=i
r1−q
k
1−p
, i ∈ Na, j, l ∈ N ∪ {0}. (3.1.48)
Let {ζk}k∈NL
be the solution of the Cauchy problem given by Eq. (3.1.16) and
ζL = −
q
m
L+m−1
i=L
r1−q
i
1−p
,
where L ∈ N is sufficiently large. As in the first part of the proof, we can obtain (3.1.38)
and (3.1.39) if ζk is negative. Thus, we can assume that L is so large that
2ζL < ζL+i < 0, i ∈ {1, . . . , m}. (3.1.49)
In addition (compare (3.1.49) with (3.1.43)), as in the first part, one can estimate
|∆ζL+i| <
P
L
, i ∈ {0, 1, . . . , m − 1}, (3.1.50)
|ζL+i − ζL+j| <
Q
L
, i, j ∈ {0, 1, . . . , m}, (3.1.51)
for some P, Q > 0. Now, the process from the proof of [37, Theorem 4.1] gives that ζk
is negative for all k ≥ L, because only (3.1.20), (3.1.33), (3.1.34), (3.1.48), (3.1.50), and
(3.1.51) are used therein. Hence, to complete the proof, it suffices to apply Lemma 3.1.3.
We slightly improve Theorem 3.1.5 into the following form (more common in the literature).
In particular, we remove the requirement on sequence {sk} that it has a positive
mean value.
Theorem 3.1.6. Let us consider the equation
∆ [˜rkΦ(∆xk)] +
˜sk
(k + 1)(p)
Φ(xk+1) = 0, (3.1.52)
where the coefficients {˜rk}k∈Na , {˜sk}k∈Na be such that the mean values of sequences ˜r1−q
k ,
{˜sk} exist and {˜rk} is bounded and positive. Let us denote
Γ := q−p
M {˜r1−q
k }
1−p
. (3.1.53)
Eq. (3.1.52) is oscillatory if M({˜sk}) > Γ. Eq. (3.1.52) is non-oscillatory if M({˜sk}) < Γ.
3.1. COEFFICIENTS WITH MEAN VALUES 123
Proof. Considering Lemma 3.1.1 for ˜r1−q
k and the boundedness of {˜rk}, we know that
0 < inf
k∈Na
˜r1−q
k ≤ sup
k∈Na
˜r1−q
k < ∞, i.e., ∞ > sup
k∈Na
˜rk ≥ inf
k∈Na
˜rk > 0. (3.1.54)
We use Theorem 3.1.5. Therefore, we assume that M({˜sk}) > 0.
We divide Eq. (3.1.52) by the constant value M ˜r1−q
k
1−p
> 0 (see (3.1.54)). We
obtain the equation
∆
˜rkΦ(∆xk)
M ˜r1−q
k
1−p +
˜skΦ(xk+1)
(k + 1)(p) M ˜r1−q
k
1−p = 0 (3.1.55)
which has the form of Eq. (3.1.13) with the coefficients
rk =
˜rk
M ˜r1−q
k
1−p , sk =
˜sk
M ˜r1−q
k
1−p .
Especially, we have
M r1−q
k = M
˜r1−q
k
M ˜r1−q
k
(1−p)(1−q)
= M ˜r1−q
k
−1
M ˜r1−q
k = 1,
where the identity
(1 − p)(1 − q) = 1 (3.1.56)
is used.
According to Theorem 3.1.5, Eq. (3.1.55) is oscillatory if
M({sk}) = M
˜sk
M({˜r1−q
k })
1−p = M ˜r1−q
k
p−1
M ({˜sk}) > q−p
,
and non-oscillatory if
M({sk}) = M ˜r1−q
k
p−1
M({˜sk}) < q−p
.
This fact implies the statement of Theorem 3.1.6 for any positive mean value of {˜sk}.
Now, let M({˜sk}) ≤ 0. Then, there exists a positive constant C such that
0 < M({˜sk}) + C = M({˜sk + C}) < Γ.
We consider the non-oscillatory equation
∆ [˜rkΦ(∆xk)] +
˜sk + C
(k + 1)(p)
Φ(xk+1) = 0,
which is a majorant equation of Eq. (3.1.52). Thus, the proof can be completed by the
application of Theorem 3.1.3.
3.1. COEFFICIENTS WITH MEAN VALUES 124
Since the presented results are new also for linear difference equations (the case when
p = q = 2), we mention the following direct corollary of Theorem 3.1.6.
Corollary 3.1.1. Let us consider the equation
∆ [rk∆xk] +
skxk+1
(k + 1)k
= 0, (3.1.57)
where the sequences {rk}k∈Na and {sk}k∈Na have the properties that M({r−1
k }) and M({sk})
exist and {rk} is bounded and positive. Then, Eq. (3.1.57) is oscillatory for
M r−1
k M({sk}) >
1
4
and non-oscillatory for
M r−1
k M({sk}) <
1
4
. (3.1.58)
Based on results of [75] (see also [77, 76]), the conjecture is given in paper [37] that
the border case M r−1
k M({sk}) = 1/4 from Corollary 3.1.1 is not solvable for general
coefficients, i.e., in the border case, there exist oscillatory equations in the form of
Eq. (3.1.57) and, at the same time, there exist non-oscillatory equations in this form.
(The situation is similar to the behavior of the continuous equations.)
In addition, using the Sturm type comparison theorem, we get the next new result
concerning non-oscillatory half-linear difference equations when the coefficient in the difference
term does not need to be bounded.
Theorem 3.1.7. Let us consider Eq. (3.1.52) and Γ introduced in (3.1.53). Let the
coefficients {˜rk}k∈Na , {˜sk}k∈Na be such that the mean values of sequences ˜r1−q
k , {˜sk}
exist and {˜rk} is positive. Then, Eq. (3.1.52) is non-oscillatory if M({˜sk}) < Γ.
Proof. The inequality M({˜sk}) < Γ can be trivially rewritten into the form
M({˜sk}) < q−p
M {˜r1−q
k }
1−p
− δ
for some δ > 0. In particular, there exists ϑ > 0 for which
M({˜sk}) < q−p
M {˜r1−q
k } + ϑ
1−p
. (3.1.59)
From Definition 3.1.2 and (3.1.56), it is seen that the sequence { ˜Rk}k∈Na given by
˜Rk := ˜r1−q
k + ϑ
1−p
, k ∈ Na,
satisfies
M ˜R1−q
k = M ˜r1−q
k + ϑ = M ˜r1−q
k + ϑ > 0. (3.1.60)
3.1. COEFFICIENTS WITH MEAN VALUES 125
In addition, sequence { ˜Rk} is bounded. Thus, we can apply Theorem 3.1.6 which guarantees
that the equation
∆ ˜RkΦ(∆xk) +
˜sk
(k + 1)(p)
Φ(xk+1) = 0 (3.1.61)
is non-oscillatory (see (3.1.59) and (3.1.60)). Of course, Eq. (3.1.61) is a majorant of
Eq. (3.1.52) because ˜Rk ≤ ˜rk for all considered k (see again (3.1.56)). Finally, it suffices
to use Theorem 3.1.3.
Again, from the theorem above, we obtain a new result in the linear case. The linear
version of Theorem 3.1.7 reads as follows.
Corollary 3.1.2. Let us consider Eq. (3.1.57) with the coefficients {rk}k∈Na and {sk}k∈Na
such that M({r−1
k }) and M({sk}) exist and {rk} is positive. Then, Eq. (3.1.57) is nonoscillatory
if (3.1.58) is valid.
§ 3.1.3 Examples
In this paragraph, we give some simple examples of oscillatory and non-oscillatory equations
whose oscillatory properties do not follow from any previously known oscillation or
non-oscillation criteria. To illustrate Theorems 3.1.5, 3.1.6, 3.1.7 and Corollaries 3.1.1,
3.1.2, we mention Examples 3.1.1, 3.1.2, 3.1.4 and Examples 3.1.3, 3.1.5, respectively.
Example 3.1.1. Let a, b > 0 be arbitrary. The equation
∆ [Φ (∆xk)] +
a |sin k| + b cos k
(k + 1)(p)
Φ(xk+1) = 0 (3.1.62)
has evidently the form of Eq. (3.1.13). Since
M ({a |sin k| + b cos k}) = aM ({|sin k|}) + bM ({cos k}) =
2a
π
,
Eq. (3.1.62) is oscillatory for 2a > q−p
π and non-oscillatory for 2a < q−p
π.
Example 3.1.2. Let λ, µ ∈ R be arbitrarily given, where |µ| > |γ|. Let us consider the
equation
∆ sin
(3k + 1)π
9
−p
q
Φ (∆xk) +
λ + (−1)k
µ
k(p)
Φ(xk+1) = 0 (3.1.63)
which has the form of Eq. (3.1.52) for
˜rk = sin
(3k + 1)π
9
−p
q
, ˜sk = λ + (−1)k
µ
(k + 1)(p)
k(p)
.
3.1. COEFFICIENTS WITH MEAN VALUES 126
Since
M ˜r1−q
k = M sin
(3k + 1)π
9
=
1
6
6
k=1
sin
(3k + 1)π
9
=
1
3
sin
π
9
+ sin
2π
9
+ sin
4π
9
=
2
3
sin
π
9
+ sin
2π
9
and
M({˜sk}) = M λ + (−1)k
µ = λ,
considering Theorem 3.1.6, we know that Eq. (3.1.63) is oscillatory for
λ > Γ := q−p 2
3
sin
π
9
+ sin
2π
9
1−p
and non-oscillatory for λ < Γ.
Example 3.1.3. Let K1, L1, K2, L2 > 0. We define the sequence {rk}k∈N by the formula
rk :=
K1 + L1, k ∈ {2n
; n ∈ N};
K1, k ∈ N {2n
; n ∈ N}
and the sequence {sk}k∈N by
sk :=
K2 + (−1)n
L2, k ∈ {3n
; n ∈ N};
K2, k ∈ N {3n
; n ∈ N}.
If we consider these functions as the coefficients in Eq. (3.1.57), then this equation is
oscillatory for
M r−1
k M({sk}) =
K2
K1
>
1
4
and non-oscillatory for K1 > 4K2. Indeed, we can apply Corollary 3.1.1.
Example 3.1.4. Let γ > 0. We use Theorem 3.1.7 for the following equation
∆
1
1 + cos k · sin
√
2k
·
∆xk
|∆xk|
+
1
γ
√
k3
·
xk+1
|xk+1|
= 0, (3.1.64)
where p = 3/2 (i.e., q = 3) and
˜rk =
1
1 + cos k · sin
√
2k
, ˜sk =
(k + 1)(3/2)
γ
√
k3
3.1. COEFFICIENTS WITH MEAN VALUES 127
for all large k ∈ N. One can easily verify that
M ˜r−2
k = M 1 + cos k · sin
√
2k
2
= 1 + M cos2
k · sin2
√
2k =
5
4
and that
M ({˜sk}) =
1
γ
lim
k→∞
(k + 1)(3/2)
√
k3
=
1
γ
.
Thus, Eq. (3.1.64) is non-oscillatory if 2γ > 3
√
15.
Example 3.1.5. For any c < 1/4, the linear equation
∆
∆xk
1 + cos k2
+
c
(k + 1)k
xk+1 = 0 (3.1.65)
satisfies all assumptions of Corollary 3.1.2. It is seen that
M r−1
k M({sk}) = M 1 + cos k2
M({c}) = c <
1
4
,
which means that Eq. (3.1.65) is non-oscillatory.
Now, we briefly explain why the oscillatory problems in the above examples are not
covered by any previously known results (see also Theorem 3.1.1). In both of Examples
3.1.1 and 3.1.2, the second coefficient changes its sign. In Example 3.1.3, the coefficients
are not asymptotically almost periodic. In Example 3.1.4, the coefficient in the
difference term is not bounded. In the last example, the first coefficient is not asymptotically
almost periodic and, at the same time, it is not bounded.
As a final remark, we focus our attention on the denominators of the potentials considered
in Examples 3.1.2 and 3.1.4, where (k + 1)(p)
and (k + 1)(3/2)
is replaced by k(p)
and
√
k3, respectively. In fact, all presented results remain true if we replace the coefficients
{sk} or {˜sk} by {fk · sk} or {fk · ˜sk} for any sequence of real numbers fk satisfying
limk→∞ fk = 1. Indeed, the existence of M({hk}) implies that M({hk · gk}) = M({hk})
whenever limk→∞ gk = 1 (consider Definition 3.1.2 and Lemma 3.1.1). Note that we
consider the denominator (k + 1)(p)
due to the form of previously known results.
Chapter
4 Dynamic equations on time scales
4.1 Half-linear Euler-type dynamic equations
on time scales
In this chapter, we analyze oscillatory properties of the second-order half-linear Euler-type
dynamic equation
r(t)Φ(y∆
)
∆
+ c(t)Φ(yσ
) = 0, Φ(y) = |y|p−1
sgn y, p > 1, (4.1.1)
on time scale T with
c(t) =
γs(t)
t(p−1)σ(t)
, (4.1.2)
where t(p)
is generalized power function (for the definition see below), the functions r, s
are rd-continuous, positive, α-periodic with inf{r(t), t ∈ T} > 0 and γ ∈ R is an arbitrary
constant.
Among others, the Sturmian theory extends verbatim for dynamic half-linear equations,
therefore we can classify equations as oscillatory and non-oscillatory. For the full
theory background and comprehensive literature overview, we refer to [2, 3, 21] (see also
[66, 67, 69]).
As well as in the previous chapters, we are interested in the conditional oscillation of
equation (4.1.1) with (4.1.2). It means that our aim is to prove that there exists a socalled
critical constant, dependent only on coefficients r and s, which establishes a sharp
borderline between oscillation and non-oscillation of these equations. More precisely, let
us consider the equation
ˆr(t)Φ(y∆
)
∆
+ ˆγd(t)Φ(yσ
) = 0, ˆγ ∈ R. (4.1.3)
We say, that equation (4.1.3) is conditionally oscillatory, if there exists a positive constant
Γ such that equation (4.1.3) is oscillatory if ˆγ > Γ and non-oscillatory if ˆγ < Γ. Since the
Sturmian theory (especially the comparison theorem) is valid in the theory of half-linear
dynamic equations, conditionally oscillatory equations are good testing equations. E.g.,
let r, ˆr ≡ 1, and let d be arbitrary positive rd-continuous function. Then equation (4.1.1)
is oscillatory if lim inft→∞ c(t)/d(t) > Γ and non-oscillatory if lim supt→∞ c(t)/d(t) < Γ
(see Corollary 4.1.1).
We note, that the case γ = Γ is resolved for differential equations (i.e., for T = R)
as non-oscillatory. However, the oscillation behavior of the discrete equation (T = Z)
128
4.1. EQUATIONS WITH PERIODIC COEFFICIENTS 129
for γ = Γ is generally not known (see Chapter 3). Moreover, it can be shown that even
differential equations cannot be generally classified as (non-)oscillatory in the critical case
for larger classes of coefficients.
Our aim is to prove that equation (4.1.1) with (4.1.2) is conditionally oscillatory. We
will also find its critical constant Γ. Evidently, this result covers the mentioned (i.e.,
p = 2) case and results for equations (1.2.3) and (3.1.3) as well as the result of [78] for
the Euler-type dynamic equation with α-periodic positive coefficients
[r(t)y∆
]∆
+
γs(t)
tσ(t)
yσ
= 0 (4.1.4)
and its critical oscillation constant
Γ =
α2
4
a+α
a
∆t
r(t)
−1 a+α
a
s(t)∆t
−1
.
We note that, in the literature, one can find Euler type half-linear dynamic equation
in forms different from the one treated in this chapter. More precisely, the potential
(4.1.2) is sometimes considered with the standard power function in the denominator
(i.e., c(t) = γs(t)/tp
or c(t) = γs(t)/(σ(t))p
) or in differential form (see, e.g., [67]).
Nevertheless, we have selected the potential in the form of (4.1.2), because there is a direct
correspondence with the difference as well as with differential equations and for p = 2 it
corresponds to Euler-type dynamic equation (4.1.4).
The chapter is organized as follows. The notion of time scales is recalled in the next
paragraph together with the definition of the generalized power function. The (non-)oscillation
theory for half-linear dynamic equation with lemmata that we need in the rest of
the chapter can the reader find in § 4.1.1 as well. Then, in § 4.1.2, we formulate and
prove the main result concerning the conditional oscillation of the mentioned Euler-type
half-linear dynamic equation (4.1.1) with (4.1.2) and illustrate it with an example. The
chapter is finished by corollaries and concluding remarks given in § 4.1.3.
§ 4.1.1 Preliminaries
At the beginning, let us remind a notation on time scales. The theory of time scales
was introduced by Stefan Hilger in his Ph.D. thesis in 1988, see [47], in order to unify
the continuous and discrete calculus. Nowadays, it is well-known calculus and it is often
studied in applications. Remind that a time scale T is an arbitrary nonempty closed subset
of reals. Note that [a, b]T := [a, b]∩T (resp. [a, ∞)T := [a, ∞)∩T) stands for an arbitrary
finite (resp. infinite) time scale interval. Symbols σ, ρ, µ, fσ
, f∆
, and
b
a
f(t)∆t stand
for the forward jump operator, backward jump operator, graininess, f ◦σ, ∆-derivative of
f, and ∆-integral of f from a to b, respectively. Further, we use the symbols Crd(T) and
C1
rd(T) for the class of rd-continuous and rd-continuous ∆-differentiable functions defined
on the time scale T. Recall that the time scale T is α-periodic if there exists constant
α > 0 such that if t ∈ T then t ± α ∈ T. We note, that any α-periodic time scale T is
4.1. EQUATIONS WITH PERIODIC COEFFICIENTS 130
infinite and, naturally, unbounded from above. For further information and background
on time scale calculus, see [48], which is the initiating paper of the time scale theory, and
the books [9, 10], which contain a lot of information on time scale calculus.
For further reading, it is necessary to remind a definition of n-th composition of
operator ρ, see also [9]. We define
ρ−1
(t) := σ(t), ρ0
(t) := t, ρ1
(t) := ρ(t), ρ2
(t) := ρ(ρ(t)), . . . , ρn
(t) = ρ(ρn−1
(t)).
If −∞ < a = min T, then we define ρn
(a) = a for each n ∈ N.
Definition 4.1.1. For arbitrary t ∈ T and p ∈ N, we define the generalized power function
on time scales as
t(p)
:= tρ(t) · · · ρp−1
(t).
For p = 0, we define t(0)
:= 1.
The following definition naturally extends the previous one for arbitrary real p ≥ 0.
Definition 4.1.2. Let p ∈ R and p denote the greatest integer less then or equal to
p (the floor function). For arbitrary t ∈ T and p ≥ 0, we define the generalized power
function on time scales as
t(p)
:= t( p )
ρ p−1
(t)
1−p+ p
· ρ p
(t)
p− p p− p
.
Example 4.1.1. Let us illustrate the generalized power function with two simple examples
involving the backward and the forward jump operator, respectively.
(i) t(7/3)
= t(2)
(ρ(t))2/3
· (ρ2
(t))1/3 1/3
= t · (ρ(t))11/9
· (ρ2
(t))1/9
,
(ii) t(3/4)
= (σ(t))1/4
· t3/4 3/4
= (σ(t))3/16
· t9/16
.
Note that for T = R we get the classic power function and for T = Z, p ∈ N, we get
generalized discrete power function, see Section 3.1 or directly [51, Chapter 2]. In the
following, we show some properties of the generalized power function, which will be useful
later.
Lemma 4.1.1. Let T be an α-periodic time scale and p ≥ 0. Then the function f(p) = t(p)
is continuous and increasing in p for large t ∈ T and
lim
t→∞
t(p)
tp
= 1. (4.1.5)
Proof. For the sake of clarity, we will use p ∈ [1, 2] in the first part of the proof and
p ∈ [1, 2) in the second part. Nevertheless, for any other intervals [k, k + 1] and [k, k + 1),
k ∈ N ∪ {0}, it can be verified analogously.
4.1. EQUATIONS WITH PERIODIC COEFFICIENTS 131
Let p ∈ [1, 2]. We show a continuity from the right-side at a point p = 1 and a continuity
from the left-side at a point p = 2 (for any other p ∈ (1, 2) the continuity is
obvious)
lim
p→1+
t(p)
= t lim
p→1+
t2−p
· (ρ(t))p−1 p−1
= t = t(1)
and
lim
p→2−
t(p)
= t lim
p→2−
t2−p
· (ρ(t))p−1 p−1
= tρ(t) = t(2)
.
Next, we show that f is increasing for p ∈ [1, 2). Let p1, p2 ∈ [1, 2), p1 < p2. On the
contrary, let t(p1)
> t(p2)
, i.e.,
t2−p1
· (ρ(t))p1−1 p1−1
> t2−p2
· (ρ(t))p2−1 p2−1
.
It is easy to see that the last inequality can be written in the form
tp1−p2
· (t/ρ(t))(p1−p2)(2−p1−p2)
> 1. (4.1.6)
Hence, for the arbitrary fixed p1 and p2, we can see that tp1−p2
→ 0 as t → ∞ and
(t/ρ(t))(p1−p2)(2−p1−p2)
→ 1 as t → ∞,
thus the inequality (4.1.6) is not valid for large t ∈ T and we get a contradiction.
Finally, for arbitrary fixed p ∈ [1, 2), we show that (4.1.5) holds. Let p ∈ [1, 2), then
t(p)
tp
=
t {t2−p
· (ρ(t))p−1
}
p−1
tp
=
t {t2−p
· tp−1
[1 − (µ(t)/t)]p−1
}
p−1
tp
= [1 − (µ(t)/t)](p−1)2
.
Hence, in view of µ(t)/t → 0 as t → ∞ (due to µ(t) ≤ α for every t), we get (4.1.5).
Now, we recall basic elements of the oscillation theory of dynamic equations on time
scales. Throughout this chapter, we assume that the time scale T is α-periodic, which
implies sup T = ∞. Consider the second order half-linear dynamic equation
[r(t)Φ(y∆
)]∆
+ c(t)Φ(yσ
) = 0, Φ(y) = |y|p−1
sgn y, p > 1, (4.1.7)
on a time scale T, where c, r ∈ Crd(T) and inf{r(t), t ∈ T} > 0. We note that Φ−1
(y) =
|y|q−1
sgn y, where q > 1 is the conjugate number of p, i.e., p + q = pq. It is easy to see
that any solution y of (4.1.7) satisfies r Φ(y∆
) ∈ C1
rd(T).
Further, we note that it is not sufficient to assume only r(t) > 0 (instead of inf{r(t), t ∈
T} > 0), because it may happen that limt→t0− r(t) = 0 and r(t0) > 0, which would not be
convenient in our case. Indeed, we need 1/r ∈ Crd(T) due to the integration of 1/rq−1
(t),
which is now fulfilled, see also [68], where this and similar problems are discussed.
Definition 4.1.3. We say that a nontrivial solution y of (4.1.7) has a generalized zero at
t if
r(t)y(t)y(σ(t)) ≤ 0.
If y(t) = 0, we say that solution y has a common zero at t (the common zero is a special
case of the generalized zero).
4.1. EQUATIONS WITH PERIODIC COEFFICIENTS 132
Definition 4.1.4. We say that a solution y of equation (4.1.7) is non-oscillatory on T if
there exists τ ∈ T such that there does not exist any generalized zero at t for t ∈ [τ, ∞)T.
Otherwise, we say that it is oscillatory.
Remark 4.1.1. Oscillation may be equivalently defined as follows. A nontrivial solution y
of (4.1.7) is called oscillatory on T, if y has a generalized zero on [τ, ∞)T for every τ ∈ T.
From the Sturm type separation theorem (see, e.g., [65]) it follows that if one solution
of (4.1.7) is oscillatory (resp. non-oscillatory), then every solution of (4.1.7) is oscillatory
(resp. non-oscillatory). Hence we can speak about oscillation or non-oscillation of
equation (4.1.7).
Next, let us recall the generalization of the Sturm type comparison theorem for time
scale calculus, which will be useful later.
Theorem 4.1.1 (Sturm type comparison theorem, [65, p. 388]). Consider the equation
[R(t)Φ(y∆
)]∆
+ C(t)Φ(yσ
) = 0 (4.1.8)
and equation (4.1.7), where R, C ∈ Crd(T) with inf{|R(t)|, t ∈ T} > 0.
(i) Let R(t) ≥ r(t) and C(t) ≤ c(t) for every t ∈ T. If (4.1.7) is non-oscillatory then
(4.1.8) is also non-oscillatory.
(ii) Let R(t) ≤ r(t) and C(t) ≥ c(t) for every t ∈ T. If (4.1.7) is oscillatory then (4.1.8)
is also oscillatory.
Our approach to the oscillatory and non-oscillatory problems of (4.1.7) is based mainly
on the application of the generalized Riccati dynamic equation
w∆
(t) + c(t) + S[w, r, µ](t) = 0, (4.1.9)
where
S[w, r, µ] = lim
λ→µ
w
λ
1 −
r
Φ(Φ−1(r) + λΦ−1(w))
.
It is not difficult to observe that
S[w, r, µ](t) =



p−1
Φ−1(r)
|w|q
(t) at right-dense t,
w
µ
1 − r
Φ(Φ−1(r)+µΦ−1(w))
(t) at right-scattered t.
Note that using the Lagrange mean value theorem on time scales (see, e.g., [10]), one can
show that the operator S can be written in the form
S[w, r, µ](t) =
(p − 1)|w(t)|q
|η(t)|p−2
Φ[Φ−1(r(t)) + µ(t)Φ−1(w(t))]
, (4.1.10)
where η(t) is between Φ−1
(r(t)) and Φ−1
(r(t))+µ(t)Φ−1
(w(t)). The form (4.1.10) will be
convenient for our purpose.
4.1. EQUATIONS WITH PERIODIC COEFFICIENTS 133
The relation between (4.1.7) and (4.1.9) is the following. If y(t) is a solution of (4.1.7)
with y(t)yσ
(t) = 0 for t ∈ [t1, t2]T and we denote
w(t) =
r(t)Φ(y∆
(t))
Φ(y(t))
,
then, for t ∈ [t1, t2]T, w = w(t) satisfies equation (4.1.9). Now, we are ready to formulate
the “time scale version” of the Reid roundabout theorem, which can be understood as
a central statement of the oscillation theory for equation (4.1.7).
Theorem 4.1.2 (Roundabout theorem, [65, p. 383]). Let a ∈ T. The following statements
are equivalent.
(i) Every nontrivial solution of (4.1.7) has at most one generalized zero on [a, ∞)T.
(ii) Equation (4.1.7) has a solution having no generalized zeros on [a, ∞)T.
(iii) Equation (4.1.9) has a solution w with
Φ−1
(r) + µ Φ−1
(w) (t) > 0 for t ∈ [a, ∞)T. (4.1.11)
The following theorem is a consequence of the Roundabout theorem 4.1.2 and the
Sturm type comparison theorem 4.1.1. The method of oscillation theory for (4.1.7),
which uses the ideas of this theorem, is usually referred to as the Riccati technique.
Theorem 4.1.3 (Riccati technique, [65, p. 390]). The following statements are equiva-
lent.
(i) Equation (4.1.7) is non-oscillatory.
(ii) There is a ∈ T and a function w : [a, ∞)T → R such that (4.1.11) holds and w(t)
satisfies (4.1.9) for t ∈ [a, ∞)T.
(iii) There is a ∈ T and a function w : [a, ∞)T → R such that (4.1.11) holds and w(t)
satisfies
w∆
(t) + c(t) + S[w, r, µ](t) ≤ 0 for t ∈ [a, ∞)T.
For further considerations, the following lemma plays an important role (see also [65],
where the similar result can be found).
Lemma 4.1.2. Let the equation
[r(t)Φ(y∆
)]∆
+ c(t)Φ(yσ
) = 0, (4.1.12)
where coefficients c, r ∈ Crd(T) are positive and
0 < inf{r(t), t ∈ T} ≤ sup{r(t), t ∈ T} < ∞, (4.1.13)
be non-oscillatory. Then for every solution w(t) of the associated generalized Riccati
equation (4.1.9), there exists T ∈ T such that w(t) > 0 for t ∈ [T, ∞)T. Moreover, w(t)
is decreasing for large t with
lim
t→∞
w(t) = 0.
4.1. EQUATIONS WITH PERIODIC COEFFICIENTS 134
Proof. At first, let us suppose that y is a positive solution of non-oscillatory equation
(4.1.12), i.e., y(t) > 0 for t ∈ [S, ∞)T, where S ∈ T is sufficiently large. By contradiction,
we prove that there exists T ∈ [S, ∞)T such that y∆
(t) > 0 for t ∈ [T, ∞)T.
(i) Let y∆
(t) < 0 for t ∈ [S, ∞)T. Because c(t)Φ(yσ
(t)) > 0 for t ∈ [S, ∞)T, we have
r(t)Φ(y∆
(t))
∆
< 0 for t ∈ [S, ∞)T.
Integrating the last inequality from S to t, we have
r(t)Φ(y∆
(t)) − r(S)Φ(y∆
(S)) =
t
S
r(s)Φ(y∆
(s))
∆
∆s ≤ 0.
Hence
y∆
(t) ≤
rq−1
(S)y∆
(S)
rq−1(t)
(4.1.14)
for t ∈ [S, ∞)T. Integrating (4.1.14) for t ≥ S, we get
lim
t→∞
y(t) − y(S) =
∞
S
y∆
(s)∆s ≤ rq−1
(S)y∆
(S)
∞
S
∆s
rq−1(s)
= −∞.
Note that the last integral is equal to infinity in view of (4.1.13). Hence y(t) → −∞ as
t → ∞, which is a contradiction. Therefore y∆
(t) < 0 cannot hold for large t.
(ii) Let y∆
(t) > 0 for large t, i.e., there exists T0 ∈ [S, ∞)T such that y∆
(T0) ≤ 0.
Thanks to c(t) > 0 for t ∈ T, we have
lim inf
t→∞
t
S
c(s)∆s > 0.
Since (4.1.12) is non-oscillatory, then due to Theorem 4.1.3, the function
w(t) =
r(t)Φ(y∆
(t))
Φ(y(t))
(4.1.15)
satisfies (4.1.9) with {Φ−1
(r) + µ Φ−1
(w)} (t) > 0 for t ∈ [S, ∞)T. Integrating (4.1.9) from
T0 to t, t ≥ T0, we get
w(t) = w(T0) −
t
T0
c(s)∆s −
t
T0
S[w, r, µ](s)∆s. (4.1.16)
Since w(T0) ≤ 0, the first integral in (4.1.16) is positive for large t, and the second
integral in (4.1.16) is nonnegative for large t, we obtain lim supt→∞ w(t) < 0. For the
nonnegativity of function S see [65, Lemma 13]. Hence, there exists T1 ∈ [S, ∞)T such
4.1. EQUATIONS WITH PERIODIC COEFFICIENTS 135
that w(t) < 0 for t ∈ [T1, ∞)T, thus y∆
(t) < 0 for t ∈ [T1, ∞)T, which is a contradiction
to the case (i). We proved that for positive y there exists T ∈ T such that y∆
(t) > 0 for
t ∈ [T, ∞)T.
Let y(t) be any negative solution of (4.1.12) for large t. Then −y(t) > 0 is a positive
solution of (4.1.12) with just proven property (the solution space of half linear equations
is homogeneous). Hence y∆
(t) < 0 for t ∈ [T, ∞)T.
In any case, we get (see (4.1.15)) that w(t) > 0 and satisfies (4.1.9) together with
(4.1.11) for t ∈ [T, ∞)T. Moreover, since
w∆
= −c(t) − S[w, r, µ](t) < 0,
w(t) is decreasing for t ∈ [T, ∞)T.
Finally, we show that w(t) → 0 as t → ∞. Suppose that a solution y is positive and
increasing for large t (the case y is negative and decreasing can be proven analogously).
Then it either converges to a positive constant L or diverges to ∞. First, we suppose
that y(t) → ∞ as t → ∞. Then, since r(t)Φ(y∆
(t)) is decreasing (see (4.1.12)), we have
w(t) =
r(t)Φ(y∆
(t))
Φ(y(t))
<
r(T)Φ(y∆
(T))
Φ(y(t))
→ 0 as t → ∞.
Hence w(t) → 0 as t → ∞. Second, if y(t) → L as t → ∞, then y∆
(t) → 0 as t → ∞.
Thus r(t)Φ(y∆
(t)) → 0 as t → ∞ and consequently, w(t) tends to zero as t → ∞ (see
(4.1.15)).
In the proof of the main result, we use the so-called adapted generalized Riccati
equation. Putting
z(t) = −tp−1
w(t)
and using the form of (4.1.9) with (4.1.10), a direct calculation leads to the adapted
generalized Riccati equation
z∆
(t) = c(t)(σ(t))p−1
+
(p − 1)(σ(t))p−1
|η(t)|p−2
|z(t)|q
tp Φ[Φ−1(r(t)) + µ(t)Φ−1(−z(t)/tp−1)]
+
(p − 1)(ζ(t))p−2
z(t)
tp−1
,
(4.1.17)
where η(t) is between Φ−1
(r(t)) and Φ−1
(r(t)) + µ(t)Φ−1
(−z(t)/tp−1
) and ζ(t) is defined
as
ζ(t) :=
(tp−1
)
∆
p − 1
1
p−2
. (4.1.18)
Note that using the Lagrange mean value theorem on time scales, we can (after rewriting
(4.1.18) on (tp−1
)
∆
= (p − 1)(ζ(t))p−2
) see that ζ(t) exists and satisfies t ≤ ζ(t) ≤ σ(t).
Now, we state two auxiliary lemmata concerning equation (4.1.17), which can be
regarded as consequences of Lemma 4.1.2.
4.1. EQUATIONS WITH PERIODIC COEFFICIENTS 136
Lemma 4.1.3. Let (4.1.12) be non-oscillatory. Then for every solution z(t) of the associated
adapted generalized Riccati equation (4.1.17), there exists sufficiently large t0 ∈ T
such that z(t) < 0 for all t ∈ [t0, ∞)T.
Proof. The statement of the lemma follows from Lemma 4.1.2.
Lemma 4.1.4. If there exists a solution z(t) of the equation (4.1.17) satisfying z(t) < 0
for all t ∈ [t0, ∞)T, then its original equation (4.1.12) is non-oscillatory. Moreover,
z(t)/tp−1
→ 0 as t → ∞.
Proof. From z(t) < 0 it follows that {Φ−1
(r) + µ Φ−1
(w)} (t) > 0 for all t ∈ [t0, ∞)T.
Hence, thanks to Theorem 4.1.2, we get that every solution of (4.1.12) is non-oscillatory
and (4.1.12) is non-oscillatory as well. Further, z(t)/tp−1
= −w(t) → 0 as t → ∞ follows
from Lemma 4.1.2.
§ 4.1.2 Conditional oscillation
In this paragraph, we formulate and prove the main result of the chapter. At first, for
reader’s convenience, let us recall, that we deal with the Euler-type half-linear dynamic
equation
r(t)Φ(y∆
)
∆
+
γs(t)
t(p−1)σ(t)
Φ(yσ
) = 0, Φ(y) = |y|p−1
sgn y, p > 1, (4.1.19)
on an α-periodic (α > 0) time scale interval [a, ∞)T, a ∈ T with a > 0, where t(p)
is
generalized power function, the functions r, s are rd-continuous, positive, α-periodic with
inf{r(t), t ∈ [a, ∞)T} > 0, and γ ∈ R is an arbitrary constant. Now, we can formulate
the main theorem as follows.
Theorem 4.1.4. Let γ ∈ R be a given constant and let r, s ∈ Crd([a, ∞)T) be positive
α-periodic functions satisfying inf{r(t), t ∈ [a, ∞)T} > 0. Further let
Γ :=
α
q
p


a+α
a
r1−q
(t)∆t


1−p 

a+α
a
s(t)∆t


−1
. (4.1.20)
Then the Euler-type half-linear dynamic equation (4.1.19) is oscillatory for γ > Γ and
non-oscillatory for γ < Γ.
Proof. Since the functions r and s are α-periodic, we have that µ(t) ≤ α for every
t ∈ [a, ∞)T and that a written in limits of integrals in (4.1.20) can be replace by arbitrary
τ ∈ [a, ∞)T with same resulting value Γ.
Throughout the proof, we will use the following estimates in which we assume that
γ > 0 and z(t) < 0 for large t. Denote
r+
:= sup{r(t), t ∈ [a, ∞)T}, r−
:= inf{r(t), t ∈ [a, ∞)T}
4.1. EQUATIONS WITH PERIODIC COEFFICIENTS 137
and
s+
:= sup{s(t), t ∈ [a, ∞)T}, s−
:= inf{s(t), t ∈ [a, ∞)T}.
Note that due to rd-continuity and α-periodicity of the functions r and s,
0 < r−
≤ r+
< ∞ and 0 ≤ s−
≤ s+
< ∞
hold. In view of (4.1.17), the adapted Riccati equation associated to (4.1.19) has the form
z∆
(t) =
γs(t)(σ(t))p−2
t(p−1)
+
(p − 1)(σ(t))p−1
|η(t)|p−2
|z(t)|q
tp Φ[Φ−1(r(t)) + µ(t)Φ−1(−z(t)/tp−1)]
+
(p − 1)(ζ(t))p−2
z(t)
tp−1
,
(4.1.21)
where η(t) is between Φ−1
(r(t)) and Φ−1
(r(t))+µ(t)Φ−1
(−z(t)/tp−1
), and t ≤ ζ(t) ≤ σ(t).
Let us define the function
h(t) := µ(t)r1−q
(t)Φ−1
(−z(t)/tp−1
).
It is easy to see (in view of Lemma 4.1.4) that
0 ≤ h(t) → 0 as t → ∞. (4.1.22)
Therefore, equation (4.1.21) can be written in the form
z∆
(t) =
γs(t)(σ(t))p−2
t(p−1)
+ (p − 1)|z(t)|
((σ(t))p−1
/t)|η(t)|p−2
|z(t)|q−1
− (ζ(t))p−2
F(t)
tp−1F(t)
,
(4.1.23)
where
F(t) := Φ[Φ−1
(r(t)) + µ(t)Φ−1
(−z(t)/tp−1
)] = r(t)[1 + h(t)]p−1
> 0. (4.1.24)
Hence, we get for large t and for p ≥ 2
z∆
(t) ≥
γs−
σ(t)
+ (p − 1)|z(t)| ·
(σ(t))p−2
|η(t)|p−2
|z(t)|q−1
− (σ(t))p−2
r(t)[1 + h(t)]p−1
tp−1r(t)[1 + h(t)]p−1
>
γs−
σ(t)
+ (p − 1)|z(t)|(σ(t))p−2
·
r(q−1)(p−2)
(t)|z(t)|q−1
− 2p−1
r(t)
tp−1r(t)[1 + h(t)]p−1
=
γs−
σ(t)
+ (p − 1)|z(t)|(σ(t))p−2
r2−q
(t) ·
|z(t)|q−1
− 2p−1
rq−1
(t)
tp−1r(t)[1 + h(t)]p−1
.
4.1. EQUATIONS WITH PERIODIC COEFFICIENTS 138
Analogously, for large t and for p < 2, we have
z∆
(t) ≥
γs−
σ(t)
+ (p − 1)|z(t)| ·
(σ(t))p−2
|η(t)|p−2
|z(t)|q−1
− (σ(t))p−2
r(t)[1 + h(t)]p−1
tp−1r(t)[1 + h(t)]p−1
>
γs−
σ(t)
+ (p − 1)|z(t)| ·
(σ(t))p−2
(2rq−1
(t))p−2
|z(t)|q−1
− tp−2
2p−1
r(t)
tp−1r(t)[1 + h(t)]p−1
=
γs−
σ(t)
+ (p − 1)|z(t)|(σ(t))p−2
2p−2
r2−q
(t) ·
|z(t)|q−1
− (σ(t)/t)2−p
2rq−1
(t)
tp−1r(t)[1 + h(t)]p−1
≥
γs−
σ(t)
+ (p − 1)|z(t)|(σ(t))p−2
2p−2
r2−q
(t) ·
|z(t)|q−1
− (1 + α)2−p
2rq−1
(t)
tp−1r(t)[1 + h(t)]p−1
and thus
z∆
(t) >
γs−
σ(t)
if z(t) < min −2(p−1)2
r+
, −2
p
q (1 + α)
2−p
q−1 r+
. (4.1.25)
Simultaneously, we estimate |z∆
(t)| for z(t) ∈ (−C, 0) and large t. We denote
D := max sup
σ(t)
t
, t ∈ [a, ∞)T , sup
(σ(t))p−2
tp−2
, t ∈ [a, ∞)T ,
sup
t(σ(t))p−2
t(p−1)
, t ∈ [a, ∞)T > 0.
Then, we get thanks to (4.1.23) for p ≥ 2 (i.e., q ≤ 2)
|z∆
(t)| <
γs+
D
t
+ (p − 1)C
(σ(t))p−2
D[2rq−1
(t)]p−2
· Cq−1
+ (σ(t))p−2
2p−1
r(t)
tp−1r(t)
≤
γs+
D
t
+
2p−2
C(p − 1)(σ(t))p−2
[Cq−1
Dr2−q
(t) + 2r(t)]
tp−1r−
≤
γs+
D
t
+
2p−2
C(p − 1)D[Cq−1
D(r+
)2−q
+ 2r+
]
tr−
=
γs+
r−
D + 2p−2
C(p − 1)D[Cq−1
D(r+
)2−q
+ 2r+
]
tr−
,
(4.1.26)
and for p < 2 (i.e., q > 2)
|z∆
(t)| <
γs+
D
t
+ (p − 1)C
(σ(t))p−2
D[rq−1
(t)]p−2
· Cq−1
+ tp−2
2p−1
r(t)
tp−1r(t)
≤
γs+
D
t
+
(p − 1)Cq
D2
(r−
)2−q
tr−
+
(p − 1)2p−1
r+
C
tr−
=
γs+
r−
D + (p − 1)Cq
D2
(r−
)2−q
+ (p − 1)2p−1
r+
C
tr−
.
(4.1.27)
4.1. EQUATIONS WITH PERIODIC COEFFICIENTS 139
Therefore, we have
|z∆
(t)| <
H(C)
t
, (4.1.28)
where
H(C) := max
γs+
r−
D + 2p−2
C(p − 1)D[Cq−1
D(r+
)2−q
+ 2r+
]
r−
,
γs+
r−
D + (p − 1)Cq
D2
(r−
)2−q
+ (p − 1)2p−1
r+
C
r−
(4.1.29)
is a positive constant which exists due to (4.1.26) and (4.1.27).
Next, from (4.1.25) and (4.1.28) it follows that if z(t) < 0 for every t ∈ [t0, ∞)T,
t0 ≥ a, then there exists a constant K > 0 such that
z(t) ∈ (−K, 0) for every t ∈ [t0, ∞)T. (4.1.30)
Indeed, according to (4.1.25), z(t) is increasing if z(t) is sufficiently small. Otherwise,
thanks to (4.1.28), z(t) cannot drop arbitrarily low.
Next, using the fact that the graininess µ(t) ≤ α for all t ∈ [a, ∞)T together with the
definition of ζ given in (4.1.18) and taking into account that η(t) is between Φ−1
(r(t))
and Φ−1
(r(t)) + µ(t)Φ−1
(−z(t)/tp−1
), we obtain (see also Lemma 4.1.1), that there exists
a constant ε ∈ [0, 1/2) such that
1 − ε ≤
(σ(t))p−2
t(p−1)/t
≤ 1 + ε, 1 − ε ≤
(σ(t))p−1
|η(t)|p−2
tp−1r2−q(t)
≤ 1 + ε,
1 − ε ≤
(ζ(t))p−2
tp−2
≤ 1 + ε
(4.1.31)
are fulfilled for arbitrary p > 1 and large t. More precisely, ε can be chosen arbitrarily
near to zero in (4.1.31) if t is sufficiently large.
Using the above estimates, we can turn our attention to the proof of the theorem. We
start with the oscillatory part. In this part of the proof, let γ > Γ. By contradiction, we
suppose that (4.1.19) is non-oscillatory. According to Lemma 4.1.3, for every solution z(t)
of the associated adapted Riccati equation (4.1.21) there exists sufficiently large t0 ∈ T
such that z(t) < 0 for t ∈ [t0, ∞)T. Moreover, from previous estimates, there exists K > 0
such that (4.1.30) holds. Using (4.1.28) and (4.1.29), we get
|z∆
(t)| <
H(K)
t
, t ∈ [t0, ∞)T. (4.1.32)
Now, we introduce the average value ξ(t) of the function z(t) on an arbitrary interval
[t, t + α]T, where t is sufficiently large. Using ξ(t), we will obtain a contradiction with
z(t) ∈ (−K, 0). Obviously,
ξ(t) ∈ (−K, 0) and ξ(t) :=
1
α
t+α
t
z(τ)∆τ, t ∈ [t0, ∞)T. (4.1.33)
4.1. EQUATIONS WITH PERIODIC COEFFICIENTS 140
Using (4.1.23), (4.1.24), (4.1.31), and (4.1.33) we get
ξ∆
(t) =
1
α
t+α
t
z∆
(τ)∆τ
=
1
α
t+α
t
1
τ
γs(τ)(σ(τ))p−2
τ(p−1)/τ
+
(p − 1)(σ(τ))p−1
|η(τ)|p−2
|z(τ)|q
τp−1r(τ)[1 + h(τ)]p−1
∆τ
+
1
α
t+α
t
1
τ
·
(p − 1)(ζ(τ))p−2
z(τ)
τp−2
∆τ
≥
1
α
·
1 − ε
t + α
t+α
t
γs(τ) +
(p − 1)r1−q
(τ)|z(τ)|q
[1 + h(τ)]p−1
∆τ
+
1
α
·
1 + ε
t
t+α
t
(p − 1)z(τ)∆τ
=
1 − ε
t + α

γ
α
t+α
t
s(τ)∆τ +
1
α
t+α
t
(p − 1)r1−q
(τ)|z(τ)|q
[1 + h(τ)]p−1
∆τ


+
(1 + ε)(p − 1)ξ(t)
t
=
1 − ε
t + α



γ
α
t+α
t
s(τ)∆τ −
Ap
(t)
p
+
1
α
t+α
t
(p − 1)r1−q
(τ)|z(τ)|q
[1 + h(τ)]p−1
∆τ −
Bq
(t)
q
+
1 + ε
1 − ε
·
t + α
t
(p − 1)ξ(t) +
Ap
(t)
p
+
Bq
(t)
q
,
(4.1.34)
where
A(t) = (p − 1)

 p
α
t+α
t
r1−q
(τ)∆τ


−1/q
,
B(t) = |ξ(t)|

 p
α
t+α
t
r1−q
(τ)∆τ


1/q
, t ≥ t0.
(4.1.35)
We will estimate ξ∆
(t) using (4.1.34) in three steps.
4.1. EQUATIONS WITH PERIODIC COEFFICIENTS 141
Step I. We show that there exists M > 0 such that
γ
α
t+α
t
s(τ)∆τ −
Ap
(t)
p
= M (4.1.36)
holds for every t ∈ [t0, ∞)T. Using p/q = p − 1, we have for every t ∈ [t0, ∞)T
γ
α
t+α
t
s(τ)∆τ −
Ap
(t)
p
=
γ
α
t+α
t
s(τ)∆τ −
(p − 1)p
p

 p
α
t+α
t
r1−q
(τ)∆τ


−p/q
=

 1
α
t+α
t
s(τ)∆τ




γ −
(p − 1)p
α1+p/q
p1+p/q


t+α
t
r1−q
(τ)∆τ


−p/q 

t+α
t
s(τ)∆τ


−1



=

 1
α
t+α
t
s(τ)∆τ



γ −
p − 1
p
p
αp


t+α
t
r1−q
(τ)∆τ


1−p 

t+α
t
s(τ)∆τ


−1

=

 1
α
t+α
t
s(τ)∆τ



γ −
α
q
p


a+α
a
r1−q
(t)∆t


1−p 

a+α
a
s(t)∆t


−1
 = S(γ − Γ),
where
S :=
1
α
t+α
t
s(τ)∆τ > 0. (4.1.37)
Hence there exists M = S(γ − Γ) > 0 such that (4.1.36) holds for t ∈ [t0, ∞)T.
Step II. We prove the existence of t1 ∈ T, t1 ≥ t0, satisfying
1
α
t+α
t
(p − 1)r1−q
(τ)|z(τ)|q
[1 + h(τ)]p−1
∆τ −
Bq
(t)
q
≥ −
M
4
, t ∈ [t1, ∞)T, (4.1.38)
where M is taken from Step I. To do it, we need three further auxiliary estimates. First,
in view of (4.1.22), we can write
1
[1 + h(t)]p−1
=
1
1 + ˜h(t)
= 1 −
˜h(t)
1 + ˜h(t)
= 1 − ˆh(t), (4.1.39)
where ˜h(t) and ˆh(t) are convenient functions. It is obvious that 0 ≤ ˜h(t) → 0 as t → ∞
and
0 ≤ ˆh(t) → 0 as t → ∞. (4.1.40)
4.1. EQUATIONS WITH PERIODIC COEFFICIENTS 142
Second, since the function y = |x|q
is continuously differentiable on (−K, 0), there exists
λ > 0 for which
|z|q
− |ξ|q
≥ −λ|z − ξ|, where z, ξ ∈ (−K, 0). (4.1.41)
Third, from (4.1.32) we have
|z(tm) − z(tn)| =
tm
tn
z∆
(τ)∆τ ≤
tm
tn
|z∆
(τ)|∆τ ≤
t+α
t
|z∆
(τ)|∆τ
<
t+α
t
H(K)
τ
∆τ ≤
1
t
t+α
t
H(K)∆τ =
H(K)α
t
(4.1.42)
for every tm, tn ∈ [t, t + α]T, where t ≥ t0 and (see (4.1.29)) H(K) > 0. Because (see
(4.1.33))
ξ(t) ∈ [zmin(t), zmax(t)],
where
zmin(t) := min{z(τ), τ ∈ [t, t + α]T}, zmax(t) := max{z(τ), τ ∈ [t, t + α]T},
there exist tm, tn ∈ [t, t + α]T (see (4.1.42)) such that for every τ ∈ [t, t + α]T
|z(τ) − ξ(t)| ≤ |z(tm) − z(tn)| <
H(K)α
t
. (4.1.43)
Now, we are ready to finish Step II. Using (4.1.35), (4.1.39), (4.1.41), (4.1.43), and again
4.1. EQUATIONS WITH PERIODIC COEFFICIENTS 143
p/q = p − 1, we can estimate
1
α
t+α
t
(p − 1)r1−q
(τ)|z(τ)|q
[1 + h(τ)]p−1
∆τ −
Bq
(t)
q
=
1
α
t+α
t
(p − 1)r1−q
(τ)|z(τ)|q
(1 − ˆh(τ))∆τ −
|ξ(t)|q
p
αq
t+α
t
r1−q
(τ)∆τ
=
p − 1
α
t+α
t
r1−q
(τ)|z(τ)|q
(1 − ˆh(τ)) − r1−q
(τ)|ξ(t)|q
∆τ
=
p − 1
α
t+α
t
|z(τ)|q
− |ξ(t)|q
− ˆh(τ)|z(τ)|q
rq−1(τ)
∆τ
≥ −
λ(p − 1)
α
t+α
t
|z(τ) − ξ(t)|
rq−1(τ)
∆τ −
(p − 1)
α
t+α
t
ˆh(τ)|z(τ)|q
rq−1(τ)
∆τ
> −
λ(p − 1)H(K)
t
t+α
t
1
rq−1(τ)
∆τ −
(p − 1)
α
t+α
t
ˆh(τ)|z(τ)|q
rq−1(τ)
∆τ
≥ −
λ(p − 1)H(K)α
t(r−)q−1
−
(p − 1)
α(r−)q−1
t+α
t
ˆh(τ)|z(τ)|q
∆τ, t ∈ [t0, ∞)T.
(4.1.44)
Finally, (4.1.44) (see also (4.1.40), which ensures that the value of the last integral in
(4.1.44) tends to zero for large t) implies that there exists t1 ≥ t0 such that (4.1.38) is
fulfilled.
Step III. From Young’s inequality (Ap
/p + Bq
/q ≥ AB), from the fact that
(p − 1)|ξ(t)| = A(t)B(t)
(see (4.1.35)), and from (4.1.33), we obtain that
t + α
t
(p − 1)ξ(t) +
Ap
(t)
p
+
Bq
(t)
q
=
Ap
(t)
p
+
Bq
(t)
q
− (p − 1)|ξ(t)| −
α(p − 1)
t
|ξ(t)|
=
Ap
(t)
p
+
Bq
(t)
q
− A(t)B(t) +
α(p − 1)ξ(t)
t
> −
α(p − 1)K
t
is fulfilled for t ∈ [t0, ∞)T. Hence there exists t2 ∈ T, t2 ≥ t1, such that
t + α
t
(p − 1)ξ(t) +
Ap
(t)
p
+
Bq
(t)
q
≥ −
M
8
, t ∈ [t2, ∞)T, (4.1.45)
4.1. EQUATIONS WITH PERIODIC COEFFICIENTS 144
where M is taken from Step I. Finally, we know that the constant ε in (4.1.34) can be
taken arbitrarily near to zero for sufficiently large t. Hence, and in view of (4.1.45), there
exists t3 ∈ T, t3 ≥ t2, such that
1 + ε
1 − ε
·
t + α
t
(p − 1)ξ(t) +
Ap
(t)
p
+
Bq
(t)
q
≥ −
M
4
, t ∈ [t3, ∞)T. (4.1.46)
Altogether, from the previous three steps, we show that ξ(t) → ∞ if t → ∞. Indeed,
in view of (4.1.34) and estimates (4.1.36), (4.1.38), and (4.1.46), we can easily see that
ξ∆
(t) ≥
1 − ε
t + α
M −
M
4
−
M
4
=
M(1 − ε)
2(t + α)
>
M
4(t + α)
, t ∈ [t3, ∞)T. (4.1.47)
Integrating (4.1.47) from t3 to ∞, we get (thanks to µ(t) ≤ α)
lim
t→∞
ξ(t) − ξ(t3) ≥
M
4
∞
t3
∆t
t + α
≥
M
4
∞
n=1
α
nα + t3 + α
= ∞,
thus ξ(t) → ∞ if t → ∞. Therefore, ξ(t) > 0 for every sufficiently large t ∈ T, which
means that z(t) > 0 for every sufficiently large t ∈ T. This contradiction gives that
equation (4.1.19) is oscillatory for γ > Γ.
To prove the non-oscillatory part of the theorem, we start with γ ≤ 0. In this case,
Eq. (4.1.19) is non-oscillatory in view of Theorem 4.1.1, part (i). It suffices to consider
the non-oscillatory equation r(t)Φ(y∆
)
∆
= 0. Then
c(t) = 0 ≥
γs(t)
t(p−1)σ(t)
= C(t), t ∈ [a, ∞)T.
Therefore, using this comparison, (4.1.19) is non-oscillatory as well.
To prove the last part of the theorem, we show that Eq. (4.1.19) is non-oscillatory for
0 < γ < Γ. To do it, we show that there exists t∗
∈ T such that a solution z(t) of (4.1.21)
with
z(t∗
) = −

 q
α
t∗+α
t∗
∆τ
rq−1(τ)


1−p
:= −Z (4.1.48)
is negative for every t ∈ [t∗
, ∞)T. Since
−r+
< −
r+
qp−1
≤ −

 q
α
t∗+α
t∗
∆τ
rq−1(τ)


1−p
,
and using (4.1.25) and (4.1.28), there exists T1 ∈ T sufficiently large such that
z(t) ∈ (−2r+
, 0) for t ∈ [t∗
, t∗
+ α]T, t∗
≥ T1. (4.1.49)
4.1. EQUATIONS WITH PERIODIC COEFFICIENTS 145
More precisely, according to (4.1.25), z(t) is increasing if z(t) ∈ (−2r+
, −r+
). Otherwise,
from (4.1.28), we have that z(t) ∈ (−2r+
, 0) is varying arbitrarily small for large t. Hence,
for t ∈ [t∗
, t∗
+ α]T, (4.1.49) holds.
Next, using (4.1.49) (see also (4.1.28) and (4.1.42)), there exists constant c > 0 such
that
|z(tm) − z(tn)| <
c
t∗
for tm, tn ∈ [t∗
, t∗
+ α]T, t∗
≥ T1. (4.1.50)
Analogously as in the first part of the proof, we use the average value ξ(t∗
), i.e.,
ξ(t∗
) ∈ (−2r+
, 0) and ξ(t∗
) :=
1
α
t∗+α
t∗
z(τ)∆τ, t∗
≥ T1. (4.1.51)
From (4.1.50) it follows (compare (4.1.43))
|ξ(t∗
) − z(τ)| <
c
t∗
, τ ∈ [t∗
, t∗
+ α]T, t∗
≥ T1. (4.1.52)
Now (similarly as before, see (4.1.34)), we estimate ξ∆
(t∗
). Using (4.1.21), (4.1.31),
and (4.1.51), we get
ξ∆
(t∗
) =
1
α
t∗+α
t∗
z∆
(τ)∆τ
≤
1
α
·
1
t∗
t∗+α
t∗
γs(τ)(σ(τ))p−2
τ(p−1)/τ
+
(p − 1)(σ(τ))p−1
|η(τ)|p−2
|z(τ)|q
τp−1r(τ)[1 + h(τ)]p−1
∆τ
+
1
α
·
1
t∗ + α
t∗+α
t∗
(p − 1)(ζ(τ))p−2
z(τ)
τp−2
∆τ
=
1
t∗



γ
α
t∗+α
t∗
s(τ)(σ(τ))p−2
τ(p−1)/τ
∆τ −
Ap
(t∗
)
p
−
1
t∗ + α
t∗+α
t∗
(p − 1)(ζ(τ))p−2
z(τ)
τp−2
∆τ
+
1
α
t∗+α
t∗
(p − 1)(σ(τ))p−1
|η(τ)|p−2
|z(τ)|q
τp−1r(τ)[1 + h(τ)]p−1
∆τ −
Bq
(t∗
)
q
+
1
α
t∗+α
t∗
(p − 1)(ζ(τ))p−2
z(τ)
τp−2
∆τ +
Ap
(t∗
)
p
+
Bq
(t∗
)
q



,
(4.1.53)
4.1. EQUATIONS WITH PERIODIC COEFFICIENTS 146
where A(t) and B(t) are given in (4.1.31). Again, we will estimate ξ∆
(t∗
) using (4.1.53)
in three steps.
Step I. Let S > 0 be defined by (4.1.37) for t∗
. Then, using (4.1.31), (4.1.35), and
(4.1.51), we get
γ
α
t∗+α
t∗
s(τ)(σ(τ))p−2
τ(p−1)/τ
∆τ −
Ap
(t∗
)
p
−
1
t∗ + α
t∗+α
t∗
(p − 1)(ζ(τ))p−2
z(τ)
τp−2
∆τ
≤ γS(1 + ε) − ΓS +
2αr+
(p − 1)(1 + ε)
t∗ + α
= S (1 + ε)γ − Γ +
2αr+
(p − 1)(1 + ε)
t∗ + α
.
Therefore, there exist T2 ∈ T, T2 ≥ T1, and N > 0 such that for t∗
≥ T2 (t∗
∈ T) we have
γ
α
t∗+α
t∗
s(τ)(σ(τ))p−2
τ(p−1)/τ
∆τ −
Ap
(t∗
)
p
−
1
t∗ + α
t∗+α
t∗
(p − 1)(ζ(τ))p−2
z(τ)
τp−2
∆τ ≤ −N. (4.1.54)
Note that we use the fact that ε tends to zero for large t.
Step II. Using (4.1.31), (4.1.39), (4.1.49), and (4.1.52), we have
1
α
t∗+α
t∗
(p − 1)(σ(τ))p−1
|η(τ)|p−2
|z(τ)|q
τp−1r(τ)[1 + h(τ)]p−1
∆τ −
Bq
(t∗
)
q
=
1
α
t∗+α
t∗
(p − 1)(σ(τ))p−1
|η(τ)|p−2
|z(τ)|q
τp−1r(τ)[1 + h(τ)]p−1
∆τ − |ξ(t∗
)|q p
qα
t∗+α
t∗
r1−q
(τ)∆τ
≤
(1 + ε)(p − 1)
α
t∗+α
t∗
r1−q
(τ)|z(τ)|q
[1 + h(τ)]p−1
∆τ − (p − 1)
|ξ(t∗
)|q
α
t∗+α
t∗
r1−q
(τ)∆τ
=
(1 + ε)(p − 1)
α
t∗+α
t∗
r1−q
(τ)|z(τ)|q
(1 − ˆh(τ))∆τ
− (p − 1)
|ξ(t∗
)|q
α
t∗+α
t∗
r1−q
(τ)∆τ
=
p − 1
α
t∗+α
t∗
r1−q
(τ) |z(τ)|q
(1 − ˆh(τ)) − |ξ(t∗
)|q
∆τ
+
ε(p − 1)
α
t∗+α
t∗
r1−q
(τ)|z(τ)|q
(1 − ˆh(τ))∆τ ≤
N
4
(4.1.55)
for t∗
∈ [T3, ∞)T , where T3 ≥ T2 is sufficiently large. Indeed, T3 exists due to the facts,
that r, z, ξ are bounded, ˆh, ε tend to zero, and due to the continuity of the function |x|q
(compare (4.1.41)). Of course, the constant N is taken from Step I.
4.1. EQUATIONS WITH PERIODIC COEFFICIENTS 147
Step III. Using (4.1.31), (4.1.35), and (4.1.51) in this part of the proof, we have
1
α
t∗+α
t∗
(p − 1)(ζ(τ))p−2
z(τ)
τp−2
∆τ +
Ap
(t∗
)
p
+
Bq
(t∗
)
q
≤ (1 + ε)(p − 1)ξ(t∗
) +
(p − 1)p
p

 p
α
t∗+α
t∗
r1−q
(τ)∆τ


−p/q
+ (p − 1) |ξ(t∗
)|q 1
α
t∗+α
t∗
r1−q
(τ)∆τ,
which is, according to (4.1.52), asymptotically the same as
(1+ε)(p − 1)z(t∗
) +
(p − 1)p
p

 p
α
t∗+α
t∗
r1−q
(τ)∆τ


−p/q
+ (p − 1) |z(t∗
)|q 1
α
t∗+α
t∗
r1−q
(τ)∆τ
= −(1 + ε)(p − 1)q1−p

 1
α
t∗+α
t∗
r1−q
(τ)∆τ


1−p
+
p − 1
p
p

 1
α
t∗+α
t∗
r1−q
(τ)∆τ


1−p
+ (p − 1)q−p

 1
α
t∗+α
t∗
r1−q
(τ)∆τ


1−p
=

 1
α
t∗+α
t∗
r1−q
(τ)∆τ


1−p
−(1 + ε)(p − 1)q1−p
+
p − 1
p
p
+ (p − 1)q−p
.
By a direct calculation one can verify, that p + q = pq implies
(p − 1)q−p
− (p − 1)q1−p
+
p − 1
p
p
= 0.
Therefore, there exists T4 ∈ T, T4 ≥ T3, such that
1
α
t∗+α
t∗
(p − 1)(ζ(τ))p−2
z(τ)
τp−2
∆τ +
Ap
(t∗
)
p
+
Bq
(t∗
)
q
≤
N
4
, t∗
∈ [T4, ∞)T, (4.1.56)
4.1. EQUATIONS WITH PERIODIC COEFFICIENTS 148
where N is, again, taken from Step I.
Finally, using (4.1.54), (4.1.55), and (4.1.56) in (4.1.53), we have
ξ∆
(t∗
) ≤
1
t∗
−N +
N
4
+
N
4
= −
N
2t∗
, t∗
∈ [T4, ∞)T, (4.1.57)
and taking into account (4.1.57), we obtain
ξ∆
(t∗
) =
1
α
t∗+α
t∗
z∆
(τ)∆τ =
z(t∗
+ α) − z(t∗
)
α
< 0, t∗
∈ [T4, ∞)T,
i.e.,
z(t∗
+ α) < z(t∗
), t∗
∈ [T4, ∞)T. (4.1.58)
In particular, if (4.1.48) holds for some t∗
∈ [T4, ∞)T, then (4.1.49) and (4.1.58) assure
the negativity of z(t) for the whole period, more precisely,
z(t) < 0 for t ∈ [t∗
, t∗
+ α]T with z(t∗
+ α) < z(t∗
).
To finish the proof, it suffices to show the existence of ϑ > 0 (depending only on r and
α) such that if z(t) ∈ (−ϑ − Z, −Z) for some t ∈ (t∗
, ∞)T, t∗
> T4, then z(t + α) < z(t).
Immediately, we have that if z(t) ∈ (−ϑ − Z, −Z) then z(t + α) < −Z. Next, using
(4.1.50), if z(t) ≤ −ϑ − Z then z(t + α) ≤ −ϑ − Z as well. Further, the initial value −Z
was not used in (4.1.53), (4.1.54), and (4.1.55). Moreover, (4.1.56) is valid for (4.1.48)
with a sufficiently small negative perturbation depending only on the coefficient r and
the period α. Therefore, the number ϑ exists, which guarantees the existence of negative
solution z(t) of (4.1.21) for large t.
Altogether, we have shown, that the initial value problem (4.1.21), (4.1.48) has a solution
z(t) satisfying z(t) < 0 for every t ∈ [t∗
, ∞)T (where t∗
is sufficiently large), which,
combined with Lemma 4.1.4, means that equation (4.1.19) is non-oscillatory.
The following example demonstrates the previous theorem.
Example 4.1.2. Consider an arbitrary finite time scale interval [3, 3 + α]T with α > 0,
where 3 ∈ T and 3 + α ∈ T. Let us define infinite time scale interval [3, ∞)T such that
if t ∈ [3, 3 + α]T, then {t + αn}∞
n=1 ⊆ [3, ∞)T
and moreover, [3, ∞)T does not contain any other points. Consider the dynamic equation
3 − 2 cos
2πt
α
Φ(y∆
)
∆
+
γ 1 + 2
3
sin 2πt
α
t(p−1)σ(t)
Φ(yσ
) = 0 (4.1.59)
on [3, ∞)T. Then (4.1.59) is oscillatory if γ > ˜Γ and non-oscillatory if γ < ˜Γ, where
˜Γ =
α
q
p


3+α
3
3 − 2 cos
2πt
α
1−q
∆t


1−p 

3+α
3
1 +
2
3
sin
2πt
α
∆t


−1
.
4.1. EQUATIONS WITH PERIODIC COEFFICIENTS 149
For the concrete time scale interval [3, ∞)T and numbers α and p, we can compute
the exact value of constant ˜Γ. We illustrate this fact, e.g., for
T =
∞
k=0
[3 + 3k, 4 + 3k] ∪ {5 + 3k}∞
k=0,
α = 3, and p = 3/2 (which implies q = 3). For this choice we get
˜Γ =


6
3
3 − 2 cos
2πt
3
−2
∆t


−1/2 

6
3
1 +
2
3
sin
2πt
3
∆t


−1
=


4
3
3 − 2 cos
2πt
3
−2
dt


−1/2 

4
3
1 +
2
3
sin
2πt
3
dt


−1
+
5
k=4
3 − 2 cos
2kπ
3
−2 −1/2 5
k=4
1 +
2
3
sin
2kπ
3
−1
=
√
2 +
20
√
6π3
3(2π + 3) 5
√
3 + 24
√
5 arctan
√
15
.
= 2.513 492 637.
Note that we used a software to obtain this value (namely, we used Maple 16).
§ 4.1.3 Applications and concluding remarks
The result of Theorem 4.1.4 can be used as an oscillation test also to equations that are
not Euler-type. For example, we can combine Theorem 4.1.4 and Sturm type comparison
theorem 4.1.1 to obtain the following Kneser-type oscillation criteria.
Corollary 4.1.1. Let us consider the equation
Φ(y∆
)
∆
+ d(t)Φ(yσ
) = 0, (4.1.60)
where d ∈ Crd([a, ∞)T), a ∈ T, a > 0.
(i) If there exists a positive α-periodic function s ∈ Crd([a, ∞)T) such that
lim sup
t→∞
t(p−1)
σ(t)d(t)
s(t)
<
α
qp


a+α
a
s(t)∆t


−1
,
then Eq. (4.1.60) is non-oscillatory.
4.1. EQUATIONS WITH PERIODIC COEFFICIENTS 150
(ii) If there exists a positive α-periodic function s ∈ Crd([a, ∞)T) such that
lim inf
t→∞
t(p−1)
σ(t)d(t)
s(t)
>
α
qp


a+α
a
s(t)∆t


−1
,
then Eq. (4.1.60) is oscillatory.
Proof. Let the assumptions of the first part hold. We consider the Euler-type equation
Φ(y∆
)
∆
+
γs(t)
t(p−1)σ(t)
Φ(yσ
) = 0 (4.1.61)
together with its oscillation constant
Γ =
α
qp


a+α
a
s(t)∆t


−1
.
Then, for some positive number ε ∈ R, we have
d(t) <
(Γ − ε)s(t)
t(p−1)σ(t)
.
From Theorem 4.1.4 we have that Eq. (4.1.61) is non-oscillatory for γ = Γ−ε. Using Sturm
type comparison theorem 4.1.1, part (i), we obtain that Eq. (4.1.60) is non-oscillatory.
The second part follows from an analogical idea and Sturm type comparison theorem
4.1.1, part (ii).
Next, let us mention a corollary that (partially) covers the cases of negative coefficients.
Corollary 4.1.2. Let us consider Eq. (4.1.19) with rd-continuous, α-periodic functions
r, s satisfying
inf{|r(t)|, t ∈ [a, ∞)T} > 0, s(t) ≡ 0, t ∈ [a, ∞)T.
Further denote
¯Γ :=
α
q
p


a+α
a
|r(t)|1−q
∆t


1−p 

a+α
a
|s(t)|∆t


−1
.
Then the following statements hold.
(i) If r(t) is positive for t ∈ [a, ∞)T and γ < ¯Γ, then Eq. (4.1.19) is non-oscillatory.
(ii) If s(t) is positive for t ∈ [a, ∞)T and γ > ¯Γ, then Eq. (4.1.19) is oscillatory.
4.1. EQUATIONS WITH PERIODIC COEFFICIENTS 151
Proof. The corollary comes directly from Theorem 4.1.4, Sturm type comparison theorem
4.1.1, and the fact that the absolute value preserves periodicity.
Finally, as a possible direction of future research, we conjecture that Eq. (4.1.19) with
more general coefficients remains conditionally oscillatory. This conjecture is based on
continuous and discrete cases. More precisely, in [37], there is found the oscillation constant
for Euler-type half-linear difference equations with asymptotically almost periodic
coefficients. Concerning the continuous case, in [35] is shown that Euler-type half-linear
differential equations with coefficients having mean values (which covers periodic and
almost periodic cases) are conditionally oscillatory. However, extension of these types for
dynamic equations on time scales appear to be much more technically difficult.
For another natural possible direction, we should mention papers [17, 19, 39], where
perturbed half-linear differential equations are studied. Typically, the perturbations are
placed in the potential of the given equation, which leads to the equations of the form
[r(t)Φ(y )] +
c(t)
t2
+
d(t)
t2 log2
t
Φ(y) = 0, T = R,
which is referred to as the Riemann–Weber half-linear equation (see also Section 2.6).
Eventually, the perturbation in the potential can be replaced by a more complex one involving
the iterated logarithms (i.e., log(log(. . . (log t)))). In the above mentioned papers,
it is proved that such equations are conditionally oscillatory and from the behavior of
the “more perturbed” equation, there is shown, that the “less perturbed” equation with
the critical constant is non-oscillatory, e.g., the results concerning the Riemann–Weber
equation give that the border case of the Euler equation is non-oscillatory.
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Appendix
B Curriculum vitae
Personal Details
Name Mgr. Petr Hasil, Ph.D.
Date of Birth 23. 3. 1982
Nationality Czech
E-mail hasil@mail.muni.cz
Web http://www.math.muni.cz/~hasil
Education
2010 Ph.D. Mathematical Analysis (Masaryk University)
Supervisor: prof. RNDr. Ondˇrej Doˇsl´y, DrSc.
2006 Master’s degree Mathematical Analysis (Masaryk University)
2004 Bachelor’s degree (General) Mathematics (Masaryk University)
Professional experience
since 2013 Assistant Professor Masaryk University
Department of Mathematics and Statistics
since 2010 Research Assistant Mendel University in Brno
Department of Mathematics
2008 – 2010 Assistant Mendel University in Brno
Department of Mathematics
Professional stays
March – July 2009 Ulm University (Ulm, Germany)
Teaching Assistant (Lineare Algebra II, summer semester 2009)
Membership in scientific societies
since 2010 International Society of Difference Equations (ISDE)
since 2013 Reviewer for MathSciNet (American Mathematical Society)
Main pedagogical activities
MU Lecturer, Examiner, Instructor,
Bachelor and master students supervisor
Member of bachelor and master exam committees
MENDELU Course supervisor, Lecturer, Examiner, Instructor
158
159
Teaching activities
Ulm University: Linear algebra
Faculty of Mathematics and Economics
Masaryk University: Mathematical Analysis
Faculty of Science
Masaryk University: Differential and Integral Calculus
Faculty of Informatics
Mendel University in Brno: Engineering mathematics
Faculty of Forestry and Wood Technology Advanced Mathematics
Applied Mathematics
Mendel University in Brno: Matematics
Faculty of Agronomy Rudiments of Higher Mathematics
Awards Related to Science and Research
2010 Prize of the Dean of the Faculty of Science (Masaryk University)
for excellent study results and success in the research
2005 Prize of the Department of Mathematics (Faculty of Science, Masaryk University)
for excellent study results and creative approach to solving problems
in mathematics
Conferences and invited talks
2008–2015 14 international conferences
(13 talks, 1 poster — CA, CZ, DE, ES, FR, LV, OM, PT, SK)
2008–2015 6 lectures at “Seminar on Differential Equations”
Dep. Math. Stat., Fac. Sci., MU
Publications (including accepted publications)
Total number of research papers 24
Papers in journals with IF 21
Citations without self-citations 38
Citations without self-citations on WoS 29
Citations without self-citations of all authors 26
h-index (SCOPUS) 6
h-index (WoS) 5
Pedagogical publications 4
Appendix
C Publications
Research Papers
1. Petr Hasil, Michal Vesel´y:
Values of limit periodic sequences and functions.
Accepted in Mathematica Slovaca, ISSN 0139-9918,
impact factor (2014): 0.409.
2. Petr Hasil, Michal Vesel´y:
Oscillation constant for modified Euler type half-linear equations.
Electronic Journal of Differential Equations (2015), ISSN 1072-6691, vol. 2015,
no. 220, pp. 1–14
impact factor (2014): 0.524.
3. Petr Hasil, Michal Vesel´y:
Oscillation constants for half-linear difference equations with coefficients having
mean values.
Advances in Difference Equations (2015), ISSN 1687-1847, vol. 2015, no. 210, pp. 1–18,
impact factor (2014): 0.640, doi: 10.1186/s13662-015-0544-1.
4. Petr Hasil, Michal Vesel´y:
Non-oscillation of perturbed half-linear differential equations with sums of periodic
coefficients.
Advances in Difference Equations (2015), ISSN 1687-1847, vol. 2015, no. 190, pp. 1–17,
impact factor (2014): 0.640, doi: 10.1186/s13662-015-0533-4.
5. Petr Hasil, Michal Vesel´y:
Limit periodic homogeneous linear difference systems.
Applied Mathematics and Computation (2015), ISSN 0096-3003, vol. 265, pp. 958–
972,
impact factor (2014): 1.551, doi: 10.1016/j.amc.2015.06.008.
Cited in
• Martin Chv´atal:
Asymptotically almost periodic solutions of limit and almost periodic linear
difference systems.
160
161
Electron. J. Qual. Theory Differ. Equ. (2015), ISSN 1417-3875, vol. 2015,
no. 75, pp. 1–17.
• Martin Chv´atal:
Non-almost periodic solutions of limit periodic and almost periodic homogeneous
linear difference systems.
Electron. J. Qual. Theory Differ. Equ. (2014), ISSN 1417-3875, vol. 2014,
no. 76, pp. 1–20.
6. Petr Hasil, Jiˇr´ı V´ıtovec:
Conditional oscillation of half-linear Euler-type dynamic equations on time scales.
Electronic Journal of Qualitative Theory of Differential Equations (2015), ISSN
1417-3875, vol. 2015, no. 6, pp. 1–21,
impact factor (2014): 0.817.
7. Petr Hasil, Michal Vesel´y:
Non-oscillation of half-linear differential equations with periodic coefficients.
Electronic Journal of Qualitative Theory of Differential Equations (2015), ISSN
1417-3875, vol. 2015, no. 1, pp. 1–24,
impact factor (2014): 0.817.
8. Petr Hasil, Michal Vesel´y:
Conditional oscillation of Riemann–Weber half-linear differential equations with
asymptotically almost periodic coefficients.
Studia Scientiarum Mathematicarum Hungarica (2014), ISSN 0081-6906, vol. 51,
no. 3, pp. 303–321,
impact factor (2014): 0.205, doi: 10.1556/SScMath.51.2014.3.1283.
Cited in
• Ondˇrej Doˇsl´y, Michal Vesel´y:
Oscillation and non-oscillation of Euler type half-linear differential equations.
J. Math. Anal. Appl. (2015), ISSN 0022-247X, vol. 429, no. 1, pp. 602–621,
doi: 10.1016/j.jmaa.2015.04.030.
9. Petr Hasil, Robert Maˇr´ık, Michal Vesel´y:
Conditional oscillation of half-linear differential equations with coefficients having
mean values.
Abstract and Applied Analysis (2014), ISSN 1085-3375, vol. 2014, no. 258159, pp. 1–14,
impact factor (2013): 1.274, doi: 10.1155/2014/258159.
Cited in
• Ondˇrej Doˇsl´y, Michal Vesel´y:
Oscillation and non-oscillation of Euler type half-linear differential equations.
J. Math. Anal. Appl. (2015), ISSN 0022-247X, vol. 429, no. 1, pp. 602–621,
doi: 10.1016/j.jmaa.2015.04.030.
162
• Masakazu Onitsuka, Tomomi Soeda:
Uniform asymptotic stability implies exponential stability for nonautonomous
half-linear differential systems.
Adv. Differ. Equ. (2015), ISSN 1687-1847, vol. 2015, no. 158, pp. 1–24.
doi: 10.1186/s13662-015-0494-7.
10. Petr Hasil, Michal Vesel´y:
Limit periodic linear difference systems with coefficient matrices from commutative
groups.
Electronic Journal of Qualitative Theory of Differential Equations (2014), ISSN
1417-3875, vol. 2014, no. 23, pp. 1–25,
impact factor (2014): 0.817.
Cited in
• Martin Chv´atal:
Asymptotically almost periodic solutions of limit and almost periodic linear
difference systems.
Electron. J. Qual. Theory Differ. Equ. (2015), ISSN 1417-3875, vol. 2015,
no. 75, pp. 1–17.
• Martin Chv´atal:
Non-almost periodic solutions of limit periodic and almost periodic homogeneous
linear difference systems.
Electron. J. Qual. Theory Differ. Equ. (2014), ISSN 1417-3875, vol. 2014,
no. 76, pp. 1–20.
11. Petr Hasil, Michal Vesel´y:
Oscillation and non-oscillation of asymptotically almost periodic half-linear difference
equations.
Abstract and Applied Analysis (2013), ISSN 1085-3375, vol. 2013, no. 432936, pp. 1–12,
impact factor (2013): 1.274, doi: 10.1155/2013/432936.
Cited in
• Ondˇrej Doˇsl´y, Michal Vesel´y:
Oscillation and non-oscillation of Euler type half-linear differential equations.
J. Math. Anal. Appl. (2015), ISSN 0022-247X, vol. 429, no. 1, pp. 602–621,
doi: 10.1016/j.jmaa.2015.04.030.
• Masakazu Onitsuka, Tomomi Soeda:
Uniform asymptotic stability implies exponential stability for nonautonomous
half-linear differential systems.
Adv. Differ. Equ. (2015), ISSN 1687-1847, vol. 2015, no. 158, pp. 1–24.
doi: 10.1186/s13662-015-0494-7.
• Jiˇr´ı V´ıtovec:
Critical oscillation constant for Euler-type dynamic equations on time scales.
163
Appl. Math. Comput. (2014), ISSN 0096-3003, vol. 243, no. 7, pp. 838–848,
doi: 10.1016/j.amc.2014.06.066.
12. Petr Hasil, Michal Vesel´y:
Oscillation of half-linear differential equations with asymptotically almost periodic
coefficients.
Advances in Difference Equations, ISSN 1687-1847, vol. 2013, no. 122, pp. 1–15,
impact factor (2013): 0.634, doi: 10.1186/1687-1847-2013-122.
Cited in
• Ondˇrej Doˇsl´y, Michal Vesel´y:
Oscillation and non-oscillation of Euler type half-linear differential equations.
J. Math. Anal. Appl. (2015), ISSN 0022-247X, vol. 429, no. 1, pp. 602–621,
doi: 10.1016/j.jmaa.2015.04.030.
• Jiˇr´ı V´ıtovec:
Critical oscillation constant for Euler-type dynamic equations on time scales.
Appl. Math. Comput. (2014), ISSN 0096-3003, vol. 243, no. 7, pp. 838–848,
doi: 10.1016/j.amc.2014.06.066.
• Ondˇrej Doˇsl´y, Hana Funkov´a:
Euler type half-linear differential equation with periodic coefficients.
Abstract Appl. Anal. (2013), ISSN 1085-3375, vol. 2013, no. 714263, pp. 1–6,
doi: 10.1155/2013/714263.
• Simona Fiˇsnarov´a, Robert Maˇr´ık:
Oscillation of half-linear differential equations with delay.
Abstract Appl. Anal. (2013), ISSN 1085-3375, vol. 2013, no. 583147, pp. 1–6,
doi: 10.1155/2013/583147.
13. Petr Hasil, Michal Vesel´y:
Critical oscillation constant for difference equations with almost periodic coefficients.
Abstract and Applied Analysis (2012), ISSN 1085-3375, vol. 2012, no. 471435, pp. 1–19,
impact factor (2012): 1.102, doi: 10.1155/2012/471435.
Cited in
• Ondˇrej Doˇsl´y, Michal Vesel´y:
Oscillation and non-oscillation of Euler type half-linear differential equations.
J. Math. Anal. Appl. (2015), ISSN 0022-247X, vol. 429, no. 1, pp. 602–621,
doi: 10.1016/j.jmaa.2015.04.030.
• Jiˇr´ı V´ıtovec:
Critical oscillation constant for Euler-type dynamic equations on time scales.
Appl. Math. Comput. (2014), ISSN 0096-3003, vol. 243, no. 7, pp. 838–848,
doi: 10.1016/j.amc.2014.06.066.
164
14. Petr Hasil, Michal Vesel´y:
Criticality of one term 2n-order self-adjoined differential equations.
Electronic Journal of Qualitative Theory of Differential Equations (2012), Proc.
9’th Coll. QTDE, ISSN 1417-3875, vol. 2012, no. 18, pp. 1–12,
impact factor (2012): 0.740.
15. Petr Hasil, Michal Vesel´y:
Almost periodic transformable difference systems.
Applied Mathematics and Computation (2012), ISSN 0096-3003, 2012, vol. 218,
no. 9, pp. 5562–5579,
impact factor: (2012): 1.349, doi: 10.1016/j.amc.2011.11.050.
Cited in
• Martin Chv´atal:
Asymptotically almost periodic solutions of limit and almost periodic linear
difference systems.
Electron. J. Qual. Theory Differ. Equ. (2015), ISSN 1417-3875, vol. 2015,
no. 75, pp. 1–17.
• Martin Chv´atal:
Non-almost periodic solutions of limit periodic and almost periodic homogeneous
linear difference systems.
Electron. J. Qual. Theory Differ. Equ. (2014), ISSN 1417-3875, vol. 2014,
no. 76, pp. 1–20.
• Michal Vesel´y:
Almost periodic skew-symmetric differential systems.
Electron. J. Qual. Theory Differ. (2012), ISSN 1417-3875, vol. 2012, no. 72,
pp. 1–16,
doi: 10.1504/IJDSDE.2011.038498.
16. Petr Hasil, Petr Zem´anek:
Friedrichs extension of operators defined by even order Sturm–Liouville equations
on time scales.
Applied Mathematics and Computation (2012), ISSN 0096-3003, vol. 218, no. 22,
pp. 10829–10842,
impact factor (2012): 1.349, doi: 10.1016/j.amc.2012.04.027.
Cited in
• Petr Zem´anek:
A note on the equivalence between even order Sturm–Liouville equations and
symplectic systems on time scales.
Appl. Math. Lett. (2013), ISSN 0893-9659, vol. 26, no. 1, pp. 134–139,
doi: 10.1016/j.aml.2012.04.009.
165
17. Petr Hasil:
Conjugacy of self-adjoint difference equations of even order.
Abstract and Applied Analysis (2011), ISSN 1085-3375, vol. 2011, no. 814962, pp. 1–16,
impact factor (2011): 1.318, doi: 10.1155/2011/814962.
Cited in
• Peter ˇSepitka, Roman ˇSimon Hilscher:
Recessive solutions for nonoscillatory discrete symplectic systems.
Linear Alg. Appl. (2015), ISSN 0024-3795, vol. 469, pp. 243–275,
doi: 10.1016/j.laa.2014.11.029.
18. Petr Hasil:
Criterion of p-criticality for one term 2n-order difference operators.
Archivum Mathematicum (2011), ISSN 0044-8753, vol. 47, no. 2, pp. 99–109,
Scopus, SciNet: MR2813536.
19. Ondˇrej Doˇsl´y, Petr Hasil:
Critical higher order Sturm–Liouville difference operators.
Journal of Difference Equations and Applications (2011), ISSN 1023-6198, vol. 17,
no. 9, pp. 1351–1363,
impact factor (2011): 0.800, doi: 10.1080/10236190903527251.
20. Ondˇrej Doˇsl´y, Petr Hasil:
Critical oscillation constant for half-linear differential equations with periodic coef-
ficients.
Annali di Matematica Pura ed Applicata (2011), ISSN 0373-3114, vol. 190, no. 3,
pp. 395–408,
impact factor (2011): 0.838, doi: 10.1007/s10231-010-0155-0.
Cited in
• Ondˇrej Doˇsl´y, Michal Vesel´y:
Oscillation and non-oscillation of Euler type half-linear differential equations.
J. Math. Anal. Appl. (2015), ISSN 0022-247X, vol. 429, no. 1, pp. 602–621,
doi: 10.1016/j.jmaa.2015.04.030.
• Ondˇrej Doˇsl´y, Naoto Yamaoka:
Oscillation constants for second-order ordinary differential equations related to
elliptic equations with p-Laplacian.
Nonlinear Anal.-Theory Methods Appl. (2015), ISSN 0362-546X, vol. 113,
pp. 115–136,
doi: 10.1016/j.na.2014.09.025.
• Jiˇr´ı V´ıtovec:
Critical oscillation constant for Euler-type dynamic equations on time scales.
Appl. Math. Comput. (2014), ISSN 0096-3003, vol. 243, no. 7, pp. 838–848,
doi: 10.1016/j.amc.2014.06.066.
166
• Jitsuro Sugie, Masakazu Onitsuka:
Growth conditions for uniform asymptotic stability of damped oscillators.
Nonlinear Anal.-Theory Methods Appl. (2014), ISSN 0362-546X, vol. 98,
pp. 83–103,
doi: 10.1016/j.na.2013.12.005.
• K¯odai Fujimoto, Naoto Yamaoka:
Global existence and nonexistence of solutions for second-order nonlinear differential
equations.
J. Math. Anal. Appl. (2014), ISSN 0022-247X, vol. 411, no. 2, pp. 707–718,
doi: 10.1016/j.jmaa.2013.10.022.
• Ondˇrej Doˇsl´y, Hana Funkov´a:
Euler type half-linear differential equation with periodic coefficients.
Abstract Appl. Anal. (2013), ISSN 1085-3375, vol. 2013, no. 714263, pp. 1–6,
doi: 10.1155/2013/714263.
• Naoto Yamaoka:
Oscillation criteria for second-order nonlinear difference equations of Euler
type.
Adv. Differ. Equ. (2012), ISSN 1687-1847, vol. 2012, no. 218, pp. 1–14,
doi: 10.1186/1687-1847-2012-218.
• Ondˇrej Doˇsl´y, Hana Haladov´a:
Half-linear Euler differential equations in the critical case.
Tatra Mt. Math. Publ. (2011), ISSN 1210-3195 vol. 48, pp. 41–49,
doi: 10.2478/v10127-011-0004-6.
• Ondˇrej Doˇsl´y, Simona Fiˇsnarov´a:
Half-linear oscillation criteria: Perturbation in term involving derivative.
Nonlinear Anal.-Theory Methods Appl. (2010), ISSN 0362-546X, vol. 73,
no. 12, pp. 3756–3766,
doi: 10.1016/j.na.2010.07.049.
21. Petr Hasil, Petr Zem´anek:
Critical second order operators on time scales.
Discrete and Continuous Dynamical Systems (2011), ISSN 1078-0947, vol. Suppl.
2011, no. 1, pp. 653–659,
impact factor (2011): 0.913.
22. Ondˇrej Doˇsl´y, Petr Hasil:
Friedrichs extension of operators defined by symmetric banded matrices.
Linear Algebra and its Applications (2009), ISSN 0024-3795, vol. 430, no. 8–9,
pp. 1966–1975,
impact factor (2009): 1.073, doi: 10.1016/j.laa.2008.11.005.
167
Cited in
• Peter ˇSepitka, Roman ˇSimon Hilscher:
Recessive solutions for nonoscillatory discrete symplectic systems.
Linear Alg. Appl. (2015), ISSN 0024-3795, vol. 469, pp. 243–275,
doi: 10.1016/j.laa.2014.11.029.
• Petr Zem´anek:
Krein-von Neumann and Friedrichs extensions for second order operators on
time scales.
Int. J. Dyn. Syst. Differ. Equ. (2011), ISSN 1752-3583, vol. 3, no. 1-2,
pp. 132–144,
doi: 10.1504/IJDSDE.2011.038498.
• Roman ˇSimon Hilscher, Petr Zem´anek:
Friedrichs extension of operators defined by linear Hamiltonian systems on unbounded
interval.
Math. Bohem. (2010), ISSN 0862-7959, vol. 135, no. 2, pp. 209–222.
23. Petr Hasil:
On positivity of the three term 2n-order difference operators.
Studies of the University of ˇZilina – Mathematical Series (2009), ISSN 1336-149X
vol. 23, no. 1, pp. 51–58,
SciNet: MR2741998.
24. Petr Hasil:
Conditional oscillation of half-linear differential equations with periodic coefficients.
Archivum Mathematicum (2008), ISSN 0044-8753, vol. 44, no. 2, pp. 119–131,
Scopus, SciNet: MR2432849.
Cited in
• Ondˇrej Doˇsl´y, Michal Vesel´y:
Oscillation and non-oscillation of Euler type half-linear differential equations.
J. Math. Anal. Appl. (2015), ISSN 0022-247X, vol. 429, no. 1, pp. 602–621,
doi: 10.1016/j.jmaa.2015.04.030.
• Jiˇr´ı V´ıtovec:
Critical oscillation constant for Euler-type dynamic equations on time scales.
Appl. Math. Comput. (2014), ISSN 0096-3003, vol. 243, no. 7, pp. 838–848,
doi: 10.1016/j.amc.2014.06.066.
• Ondˇrej Doˇsl´y, Hana Funkov´a:
Euler type half-linear differential equation with periodic coefficients.
Abstract Appl. Anal. (2013), ISSN 1085-3375, vol. 2013, no. 714263, pp. 1–6,
doi: 10.1155/2013/714263.
• Simona Fiˇsnarov´a, Robert Maˇr´ık:
Generalized picone and riccati inequalities for half-linear differential operators
168
with arbitrary elliptic matrices.
Electron. J. Differ. Equ. (2010), ISSN 1072-6691, vol. 2010, no. 111, pp. 1–13.
• Jitsuro Sugie, Saori Hata, Masakazu Onitsuka:
Global attractivity for half-linear differential systems with periodic coefficients.
J. Math. Anal. Appl. (2010), ISSN 0022-247X, vol. 371, no. 1, pp. 95–112,
doi: 10.1016/j.jmaa.2010.04.035.
Pedagogical publications
1. Petr Hasil, Petr Zem´anek:
Sb´ırka ˇreˇsen´ych pˇr´ıklad˚u z matematick´e anal´yzy I.
[Solved Problems in Mathematical Analysis I, in czech]
Elport´al (2012), ISSN 1802-128X, 527 p., ISBN 978-80-210-5882-8.
2. Petr Hasil:
Pozn´amky z matematiky.
[Notes in mathematics, in czech, cit. 2016-2-16]
http://um.mendelu.cz/akademie/predmety/matematika-na-af.html, 173 p.
3. Petr Hasil, Petr Zem´anek:
Sb´ırka pˇr´ıklad˚u z matematick´e anal´yzy II.
[Problems in Mathematical Analysis II, in czech, cit. 2016-2-16]
http://goo.gl/iXs0PD, 89 p.
4. Petr Hasil, Petr Zem´anek:
Demonstrativn´ı cviˇcen´ı.
[Demonstrations, in czech, cit. 2016-2-16]
http://www.math.muni.cz/~xvesely/data/Hasil-Vesely.pdf, 151 p.
169
Conferences
1. Equadiff 15, 2015 (Lyon, France),
talk: Critical oscillation constants for half-linear differential and difference equa-
tions.
2. CDSDEA 2014: 10th AIMS Conference on Dynamical Systems, Differential Equations
and Applications (Madrid, Spain),
talk: Critical oscillation constant for half-linear differential equations with coefficients
having mean values.
3. CDDEA 2014: Conference on Differential and Difference Equations and Applications
(Jasn´a, Slovakia),
talk: Oscillation constant for half-linear differential equations with coefficients
having mean values.
4. ICDEA 2013: 19th International Conference on Difference Equations and Applications
(Muscat, Sultanate of Oman),
talk: Conditional oscillation of half-linear equations with asymptotically almost periodic
coefficients.
5. Equadiff 13, 2013 (Praha, Czech Republic),
talk: Oscillation constant for half-linear equations with asymptotically almost periodic
coefficients.
6. CDDEA 2012: Conference on Differential and Difference Equations and Applications
(Terchov´a, Slovakia),
talk: Conditional oscillation of Euler-type difference equation with almost periodic
coefficients.
7. ICDEA 2011: 17th International Conference on Difference Equations and Applications
(Trois-Rivi`eres, Canada),
talk: Conjugacy criterion for self-adjoint difference equations.
8. ICDEA 2010: 16th International Conference on Difference Equations and Applications
(Riga, Latvia),
talk: Disconjugacy of critical difference operators.
9. ICDSDEA 2010: 8th AIMS International Conference on Dynamical Systems, Differential
Equations and Applications (Dresden, Germany),
talk: On criticality of higher order difference operators.
10. CDDEA 2010: Conference on Differential and Difference Equations and Applications
(Rajeck´e Teplice, Slovakia),
talk: Critical difference operators.
11. CDEIT 2010: Colloquium on Differential Equations and Integration Theory (Kˇrtiny,
Czech Republic),
170
poster: Critical oscillation constant for half-linear differential equations with periodic
coefficients.
12. ICDEA 2009: 15th International Conference on Difference Equations and Applications
(Estoril, Portugal),
talk: Critical higher order Sturm–Liouville difference operators.
13. Equadiff 12, 2009 (Brno, Czech Republic),
talk: Critical higher order Sturm–Liouville difference operators.
14. CDDEA 2008: Conference on Differential and Difference Equations and Applications
(Streˇcno, Slovakia),
talk: Conditional oscillation of half-linear differential equations with periodic coef-
ficients.
Lectures at “Seminar on Differential Equations”, Department of Mathematics
and Statistics, Faculty of Science, Masaryk University
1. 5.10.2015,
Oscilaˇcn´ı konstanta pro diferenci´aln´ı a diferenˇcn´ı rovnice
[Oscillation constant for differential and difference equations]
2. 29.4.2013,
Oscilatoriˇcnost pololine´arn´ıch diferenˇcn´ıch rovnic s asymptoticky skoroperiodick´ymi
koeficienty
[Oscillation of half-linear difference equations with asymptotically almost periodic
coefficients]
3. 15.10.2012,
Podm´ınˇen´a oscilatoriˇcnost diferenˇcn´ı rovnice Eulerova typu se skoroperiodick´ymi
koeficienty
[Conditional oscillation of Euler-type difference equation with almost periodic coef-
ficients]
4. 22.11.2010,
Kritick´a oscilaˇcn´ı konstanta pro pololine´arn´ı diferenci´aln´ı rovnice s periodick´ymi
koeficienty
[Critical oscillation constant for half-linear differential equations with periodic coef-
ficients]
5. 9.11.2009,
Konjugovanost Sturm–Liouvilleovych diferenˇcn´ıch rovnic vyˇsˇs´ıch ˇr´ad˚u
[Conjugacy of Sturm-Liouville difference equations of higher order]
6. 3.11.2008,
Podm´ınˇen´e oscilatorick´e pololine´arn´ı rovnice s periodick´ymi koeficienty
[Conditional oscillation of half-linear differential equations with periodic coefficients]