MASARYK UNIVERSITY FACULTY OF SCIENCE
HABILITATION THESIS
Solution spaces of almost periodic
homogeneous linear diﬀerence
and diﬀerential systems
Brno 2015 Michal Vesel´y
Preface
In this work, a problem from the qualitative theory of almost periodic diﬀerence and
diﬀerential equations is solved. More precisely, using special constructions of almost periodic
(and limit periodic) sequences and functions, non-almost periodic solutions of almost
periodic homogeneous linear difference and diﬀerential systems are studied. The aim is
to ﬁnd systems, whose all solutions can be almost periodic, and to prove that, in any
neighbourhood of such a system, there exists a system which does not possess an almost
periodic solution other than the trivial one.
All results presented in this work are due to the author and are taken from papers
denoted as (2)–(5), (7), (9), (13), (18), and (21) on pages 168–169. Note that four of
the papers have a co-author, namely P. Hasil. In all cases, the contributions of the both
authors are equivalent. Certain parts of this work are also taken from the Ph.D. thesis of
the author (see (PhD) or directly papers (2)–(5)).
The history and basic motivation of the treated topic are included at the beginnings
of chapters and sections. For reader’s convenience, the end of each proof, example, and
remark is identiﬁed by symbol , ♦, and , respectively. The used notations are collected
in sections titled Preliminaries. Deﬁnitions, theorems, lemmas, corollaries, examples, and
remarks are numbered consecutively within each chapter.
Brno, September 21, 2015
Michal Vesel´y
Contents
Abstracts 4
Abstract of Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Abstract of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Abstract of Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Abstract of Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Abstract of Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Abstract of Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Abstract of Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1 Almost periodic and limit periodic sequences in pseudometric spaces 6
1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Generalizations of pure periodicity . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Almost periodic sequences . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.2 Asymptotically almost periodic sequences . . . . . . . . . . . . . . 11
1.2.3 Limit periodic sequences . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Constructions of almost periodic and limit periodic sequences . . . . . . . 13
1.4 Application related to almost periodic diﬀerence systems . . . . . . . . . . 20
2 Solutions of almost periodic diﬀerence systems 33
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2 General homogeneous linear diﬀerence systems . . . . . . . . . . . . . . . . 35
2.2.1 Transformable groups . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2.2 Strongly and weakly transformable groups . . . . . . . . . . . . . . 42
2.3 Systems without almost periodic solutions . . . . . . . . . . . . . . . . . . 46
3 Values of almost periodic and limit periodic sequences 70
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.2 Sequences with given values . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.3 Applications related to almost periodic diﬀerence systems . . . . . . . . . . 78
4 Solutions of limit periodic diﬀerence systems 82
4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2 General homogeneous linear diﬀerence systems . . . . . . . . . . . . . . . . 84
4.3 Systems without asymptotically almost periodic solutions . . . . . . . . . . 87
4.4 Systems with non-almost periodic solutions . . . . . . . . . . . . . . . . . . 99
2
5 Almost periodic and limit periodic functions in pseudometric spaces 117
5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.2 Generalizations of pure periodicity . . . . . . . . . . . . . . . . . . . . . . 118
5.2.1 Almost periodic functions . . . . . . . . . . . . . . . . . . . . . . . 118
5.2.2 Limit periodic functions . . . . . . . . . . . . . . . . . . . . . . . . 122
5.3 Constructions of almost periodic functions . . . . . . . . . . . . . . . . . . 123
6 Solutions of almost periodic diﬀerential systems 129
6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.2 Skew-Hermitian systems without almost periodic solutions . . . . . . . . . 130
6.3 Skew-symmetric systems without almost periodic solutions . . . . . . . . . 146
7 Values of almost periodic and limit periodic functions 158
7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.2 Functions with given values . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Author’s papers 168
References 170
3
Abstracts
Chapter 1: We introduce the notions of almost periodic and limit periodic sequences in
pseudometric spaces. Especially, we modify the Bochner deﬁnition of almost periodicity so
that it remains equivalent with the Bohr one. We present an easily modiﬁable method for
constructing almost periodic and limit periodic sequences with prescribed properties. We
apply the method to construct an almost periodic homogeneous linear diﬀerence system
which does not have any non-trivial almost periodic solution.
Chapter 2: Almost periodic homogeneous linear diﬀerence systems are analysed, where
the coeﬃcient matrices belong to a group. The goal is to ﬁnd such groups that the systems
having no non-trivial almost periodic solution form a dense subset of the set of all considered
systems. A closer examination of the used methods reveals that the problem can be treated
in such a generality that the entries of the coeﬃcient matrices are allowed to belong to any
complete metric ﬁeld. The concepts of transformable and weakly transformable groups
of matrices are introduced and these concepts enable us to derive eﬃcient conditions for
determining matrix groups with the required property.
Chapter 3: Limit periodic sequences with values in pseudometric spaces are considered.
We construct limit periodic sequences with given values. For any totally bounded
and countable set, we ﬁnd a limit periodic sequence which attains each value from this set
periodically. For any totally bounded countable set which is dense in itself, we construct
a limit periodic bijective map from the integers into this set. The corresponding results
about almost periodic sequences are explicitly formulated as well. As corollaries, we obtain
new results about non-almost periodic solutions of complex almost periodic transformable
diﬀerence systems.
Chapter 4: As in Chapter 2, we study homogeneous linear diﬀerence systems, where
the coeﬃcient matrices belong to a bounded group. Now we consider limit periodic systems
and we ﬁnd groups of matrices with the property that the systems, which do not possess
non-zero asymptotically almost periodic solutions, form a dense subset in the space of all
considered systems. Since the used method is substantially diﬀerent from the processes
applied in Chapter 2, we obtain new results also for almost periodic systems.
4
Chapter 5: We introduce almost periodic and limit periodic functions with values
in a pseudometric space X. We mention the Bohr and the Bochner deﬁnition of almost
periodicity and the fundamental properties of almost periodic functions. In particular, we
prove the equivalence of the Bohr and the Bochner concept and we brieﬂy describe the
connection between almost periodic functions and sequences. Similarly as in Chapter 1,
we present a modiﬁable method for constructing almost periodic functions in X.
Chapter 6: We analyse solutions of almost periodic skew-Hermitian and skew-symmetric
homogeneous linear diﬀerential systems. It is known that the systems, whose all
solutions are almost periodic, form an everywhere dense subset in space of all almost periodic
skew-Hermitian or skew-symmetric systems (in the uniform topology). Applying
a construction from Chapter 5, we prove that, in any neighbourhood of an almost periodic
skew-Hermitian system, there exists a diﬀerent almost periodic skew-Hermitian system
which does not possess a non-trivial almost periodic solution. In addition, using a modiﬁcation
of the iterative process presented in Chapter 5, we obtain the same result for almost
periodic skew-symmetric systems as well.
Chapter 7: In pseudometric spaces, limit periodic and almost periodic functions with
given values are constructed. More precisely, for an arbitrary uniformly continuous function
which attains ﬁnitely many values on Z and whose range is totally bounded, we construct
an almost periodic function with the same range on Z and R which attains all value
periodically. In addition, if the uniformly continuous function with a totally bounded range
attains a value periodically, then we prove that the resulting function can be constructed
as limit periodic.
5
Chapter 1
Almost periodic and limit periodic
sequences in pseudometric spaces
In this chapter, we introduce the notions of limit periodicity and (asymptotic) almost
periodicity for sequences in a general pseudometric space. At ﬁrst, we mention the article
[65] by K. Fan which considers asymptotically almost periodic sequences of elements
of a metric space (based on the Fr´echet concept from [74, 75]) and the article [177] by
H. Tornehave about almost periodic functions of the real variable with values in a metric
space. In these papers, it is shown that many theorems which are valid for complex valued
sequences and functions are no longer true. For continuous functions, it is observed that
the important property is the local connection by arcs of the space of values and also its
completeness.
However, we do not use their results or other theorems and we introduce the considered
generalizations of periodic sequences in pseudometric spaces without any additional
restrictions; i.e., the deﬁnitions are similar to the classical ones, only the modulus being
replaced by the distance. We can also refer to [88, 131, 132, 147, 186, 189]. We add that the
concept of almost periodic functions of several variables with respect to Hausdorﬀ metrics
can be found in [165] which is an extension of [60] (see also [61], [149]).
In Banach spaces, a sequence {ϕk}k∈Z is almost periodic if and only if any sequence
of translates of {ϕk} has a subsequence which converges and its convergence is uniform
with respect to k in the sense of the norm. In 1933, the continuous case of the previous
result was proved by S. Bochner in [22], where the fundamental theorems of the theory of
almost periodic functions with values in Banach spaces are proved as well (see, e.g., [6], [7,
pp. 3–25] or [111], where the theorems are redemonstrated by the methods of the functional
analysis). We remark that the discrete version of this result can be proved similarly as
in [22] (or see directly papers [156, 179]).
In pseudometric spaces, it is easy to show that the above result is not generally true.
Nevertheless, by a simple modiﬁcation of the Bochner proof of this result, one can verify
that a necessary and suﬃcient condition for a sequence {ϕk}k∈Z to be almost periodic is
that any sequence of translates of {ϕk} has a subsequence satisfying the Cauchy condition
uniformly with respect to k.
6
1.1 Preliminaries 7
In this chapter, we also analyse systems of the form
xk+1 = Ak · xk, k ∈ Z (or k ∈ N0),
where {Ak} is almost periodic. The aim is to prove that there exists a system of the
above form which does not have an almost periodic solution other than the trivial one (see
Theorem 1.38 below). A closer examination of the method used in our construction reveals
that the problem can be treated in possibly the most general setting:
almost periodic sequences attain values in a pseudometric space;
the entries of almost periodic matrices are elements of an inﬁnite ring with a unit.
Note that many theorems about the existence of almost periodic solutions of general almost
periodic diﬀerence systems are published in [21, 83, 87, 163, 182, 186, 187, 189] and several
these existence theorems are proved there in terms of discrete Lyapunov functions. Here,
we can also refer to the monograph [183] and [190, Theorems 3.6, 3.7, 3.8]. For linear
systems with k ∈ N0, see [5, 168].
This chapter is organized as follows. In the ﬁrst section, we mention the notation which
is used throughout the whole chapter. Section 1.2 presents the deﬁnitions of (asymptotically)
almost periodic and limit periodic sequences in a pseudometric space, the above
necessary and suﬃcient condition for the almost periodicity of a sequence {ϕk}k∈Z, and
some basic properties of almost periodic sequences in pseudometric spaces.
In Section 1.3, we show a way one can construct limit periodic and almost periodic
sequences with given properties. We remark that our process is comprehensible and easily
modiﬁable and that methods of generating almost periodic sequences are mentioned in [138,
Section 4] as well.
Finally, in Section 1.4, we use results from the second and the third section to obtain
a theorem which plays an important role in Chapter 2, where it is proved that the almost
periodic homogeneous linear diﬀerence systems which do not have any non-zero almost
periodic solution form a dense subset of the set of all considered systems. Using our
constructions, we obtain generalizations of results from the paper of the author denoted
as (1) and from [176], where unitary and orthogonal systems are studied.
1.1 Preliminaries
As usual, R+
denotes the set of all positive reals, R+
0 the set of all non-negative real
numbers, and N0 the set of all natural numbers including the zero. Let X = ∅ be an
arbitrary set and let d : X × X → R+
0 be a pseudometric on X. For given ε > 0 and
x ∈ X, in the same way as in metric spaces, we deﬁne the ε-neighbourhood of x in X as
the set {y ∈ X; d(x, y) < ε}. The ε-neighbourhood is denoted by Oε(x).
We consider sequences in X. The scalar (and vector) valued sequences are denoted
by the lower-case letters, the matrix valued sequences by the capital letters (X is a set
of matrices in this case), and each one of the scalar and matrix valued sequences by
symbols {ϕk}, {ψk}, {χk}.
1.2 Generalizations of pure periodicity 8
1.2 Generalizations of pure periodicity
Now we introduce the concept of (asymptotically) almost periodic and limit periodic
sequences in the pseudometric space X. We remark that the approach is very general and
that the presented theory of almost periodic sequences does not distinguish between x ∈ X
and y ∈ X if d(x, y) = 0.
1.2.1 Almost periodic sequences
We begin with a “natural” generalization of almost periodicity.
Deﬁnition 1.1. A sequence {ϕk} is called almost periodic if for any ε > 0, there exists
a positive integer p (ε) such that any set consisting of p (ε) consecutive integers (non-negative
integers if k ∈ N0) contains at least one integer l with the property that
d(ϕk+l, ϕk) < ε, k ∈ Z (or k ∈ N0).
The number l is called an ε-translation number of {ϕk}. For any ε > 0, the set of all
ε-translation numbers of a sequence {ϕk} is denoted by T({ϕk}, ε).
Remark 1.2. If X is a Banach space, then a necessary and suﬃcient condition for a
sequence {ϕk}k∈Z to be almost periodic is it to be normal; i.e., {ϕk} is almost periodic if
and only if any sequence of translates of {ϕk} has a subsequence, uniformly convergent for
k ∈ Z in the sense of the norm. This result and Theorem 1.3 below are not valid if {ϕk} is
deﬁned for k ∈ N0 and if we consider only translates to the right (consider X = R, ϕ0 = 1,
and ϕk = 0, k ∈ N). But, if we consider translates to the left, then the both results are
valid for k ∈ N0 as well.
It is seen that the result mentioned in Remark 1.2 is no longer valid if the space of
values fails to be complete. Especially, in a pseudometric space (X, d), it is not generally
true that a sequence {ϕk}k∈Z is almost periodic if and only if it is normal. Nevertheless,
applying the methods from any one of the proofs of the results [7, Statement (ζ)], [46,
Theorem 1.10], and [72, Theorem 1.14], one can easily prove that every normal sequence
{ϕk}k∈Z is almost periodic. Further, we can prove the next theorem (a generalization of
the theorem called the Bochner concept) which we will need later. We add that its proof
is a modiﬁcation of the proof of [46, Theorem 1.26].
Theorem 1.3. Let {ϕk}k∈Z be given. For an arbitrary sequence {hn}n∈N ⊆ Z, there exists
a subsequence {˜hn}n∈N ⊆ {hn}n∈N with the Cauchy property with respect to {ϕk}, i.e., for
any ε > 0, there exists M = M(ε) ∈ N for which the inequality
d ϕk+˜hi
, ϕk+˜hj
< ε
holds for all i, j, k ∈ Z, i, j > M, if and only if {ϕk}k∈Z is almost periodic.
1.2 Generalizations of pure periodicity 9
Proof. If any sequence of translates of {ϕk} has a subsequence which has the Cauchy
property, then {ϕk} is almost periodic. It can be proved similarly as [46, Theorem 1.10],
where it is not used that X is complete. To prove the opposite implication, we assume that
{ϕk} is almost periodic and we use the well-known method of the diagonal extraction.
Let {hn}n∈N ⊆ Z and ϑ > 0 be arbitrary. By Deﬁnition 1.1, there exists a positive
integer p such that, in any set {hn − p, hn − p + 1, . . . , hn}, there exists a ϑ-translation
number ln. We know that 0 ≤ hn − ln ≤ p for all n ∈ N. We put kn := hn − ln, n ∈ N.
Clearly, kn = c = const. (a constant value from {0, 1, . . . , p}) for inﬁnitely many values
of n. Since
d (ϕk+hn , ϕk+kn ) = d ϕ(k+hn−ln)+ln , ϕk+hn−ln < ϑ, k ∈ Z,
there exists a subsequence {h1
n} of {hn} and an integer c1 such that
d ϕk+h1
n
, ϕk+c1 < ϑ, k ∈ Z, n ∈ N. (1.1)
Consider now a sequence of positive numbers ϑ1 > ϑ2 > · · · > ϑn > · · · converging
to 0. From the sequence {ϕk+hn }, we extract a subsequence {ϕk+h1
n
} which satisﬁes (1.1)
for ϑ = ϑ1. From this sequence, we extract a subsequence {ϕk+h2
n
} for which an inequality
analogous to (1.1) is valid. Of course, c is not same, but it depends on the subsequence.
We proceed further in the same way. Next, we form the sequence {ϕk+hn
n
}n∈N. Assume
that ε > 0 is given and that we have 2ϑm < ε for m ∈ N. As a result, for i, j > m, i, j ∈ N,
we obtain
d ϕk+hi
i
, ϕk+hj
j
≤ d ϕk+hi
i
, ϕk+cm + d ϕk+cm , ϕk+hj
j
< ε, k ∈ Z,
where cm is the number corresponding to the sequence {ϕk+hm
n
}n∈N and ϑm.
Corollary 1.4. Let p ∈ N be arbitrarily given and let {ϕk}k∈Z be almost periodic. For
any ε > 0, the set {lp; l ∈ N} ∩ T({ϕk}, ε) is inﬁnite.
Proof. It suﬃces to apply Theorem 1.3 for hn := pn, n ∈ N. Indeed, it holds
sup
k∈Z
d ϕk+hi
, ϕk+hj
= sup
k∈Z
d ϕk+hi−hj
, ϕk , i, j ∈ N.
In Chapter 2, we consider almost periodic sequences in complete metric spaces. Thus,
we also need the following consequence of the so-called Bochner theorem.
Corollary 1.5. An arbitrary sequence {ϕk}k∈Z in a complete metric space is almost periodic
if and only if, from any sequence of the form {{ϕk+hi
}k∈Z}i∈N, where {hi}i∈N ⊆ Z,
one can extract a subsequence converging uniformly for k ∈ Z.
Many results about almost periodic sequences with values in C are extendable to sequences
with values in a complete metric space (or in a pseudometric space). We mention
the following results which we will need later and which can be easily proved using methods
from the classical theory of almost periodic functions (see [7, 46] for the classical cases and,
e.g., [13] for generalizations). We also refer to [134, 189].
1.2 Generalizations of pure periodicity 10
Theorem 1.6. Let X1, X2 be arbitrary pseudometric spaces and Φ : X1 → X2 be a uniformly
continuous map. If {ϕk} ⊆ X1 is almost periodic, then the sequence {Φ(ϕk)} is
almost periodic as well.
Proof. Taking ε > 0 arbitrarily, let δ(ε) > 0 be the number corresponding to ε from the
deﬁnition of the uniform continuity of Φ. Now, Theorem 1.6 follows from the fact that the
set of all ε-translation numbers of {Φ(ϕk)} contains the set of all δ(ε)-translation numbers
of {ϕk}, i.e., from the inclusion
T ({ϕk}, δ(ε)) ⊆ T ({Φ(ϕk)}, ε) .
Theorem 1.7. For every sequence of almost periodic sequences
{ϕ1
k}, . . . , {ϕi
k}, . . . ,
the sequence of lim
i→∞
ϕi
k is almost periodic if the convergence is uniform with respect to k.
Proof. The proof can be easily obtained by a modiﬁcation of the proof of [46, Theorem
6.4].
Theorem 1.8. Let (X, d) be a complete metric space. For an almost periodic sequence
{ϕk}k∈Z and an arbitrary sequence of integers h1, . . . , hi, . . . , there exists a subsequence
{˜hi}i∈N of {hi}i∈N such that
lim
j→∞
lim
i→∞
ϕk+˜hi−˜hj
= ϕk.
Proof. Since {ϕk} is normal, we know that there exists {¯hi}i∈N ⊆ {hi}i∈N for which
the sequence {{ϕk+¯hi
}k∈Z}i∈N converges uniformly to an almost periodic sequence (see
Theorem 1.7), denoted as {ψk}. Applying Corollary 1.5 again, we obtain a subsequence
{˜hi}i∈N ⊆ {¯hi}i∈N with the property that the sequence {{ψk−˜hi
}k∈Z}i∈N is uniformly convergent.
We denote the limit as {χk}.
Now we choose ε > 0 arbitrarily. We have
ψk−˜hi
, χk <
ε
2
, ψk, ϕk+˜hj
<
ε
2
, k ∈ Z,
if i, j > n (i, j ∈ N) for some suﬃciently large n = n(ε) ∈ N. Thus, for all k ∈ Z, it is true
(ϕk, χk) ≤ ϕk, ψk−˜hi
+ ψk−˜hi
, χk < ε.
Because of the arbitrariness of ε > 0, we get the identity {ϕk} ≡ {χk}.
Remark 1.9. It is possible to prove that a sequence {ϕk}k∈Z is almost periodic if and only
if every pair of sequences {hi}i∈N, {li}i∈N ⊆ Z have common subsequences {˜hi}i∈N, {˜li}i∈N
with the property that
lim
j→∞
lim
i→∞
ϕk+˜hi+˜lj
= lim
i→∞
ϕk+˜hi+˜li
pointwise for k ∈ Z. (1.2)
1.2 Generalizations of pure periodicity 11
In fact, condition (1.2) is necessary and suﬃcient in each one of the two modes of the
convergence; i.e., in the strongest version, this condition is necessary in the uniform sense
and suﬃcient in the pointwise sense. For almost periodic functions deﬁned on R with values
in C, the above result is due to S. Bochner and it can be found in [23]. The proof from the
paper can be generalized for complete metric spaces (see [142, 143]). If this necessary and
suﬃcient condition is applied only to the case {li} ≡ {−hi} (as in Theorem 1.8), then one
gets a diﬀerent class of sequences called almost automorphic sequences (for more details,
see [56, 151]; concerning the linear systems treated in Chapter 2, see [35, 126]).
Taking n ∈ N and using Theorem 1.3 (and Remark 1.2) n-times, one can easily prove
the corollaries below.
Corollary 1.10. Let sequences {ϕ1
k}, . . . , {ϕn
k } be given. Then, the sequence {ψk} which
is deﬁned by
ψk := ϕi+1
j for all considered k,
where k = jn + i, j ∈ Z, i ∈ {0, . . . , n − 1}, is almost periodic if and only if all sequences
{ϕ1
k}, . . . , {ϕn
k } are almost periodic.
Corollary 1.11. Let (X1, d1), . . . , (Xn, dn) be pseudometric spaces and {ϕ1
k}, . . . , {ϕn
k } be
arbitrary sequences with values in X1, . . . , Xn, respectively. The sequence {ψk}, with values
in X1 × · · · × Xn given by
ψk := ϕ1
k, . . . , ϕn
k for all considered k,
is almost periodic if and only if each one of sequences {ϕ1
k}, . . . , {ϕn
k } is almost periodic.
Corollary 1.12. Let ε > 0 be arbitrary and let the sequences {ϕ1
k}k∈Z, . . . , {ϕn
k }k∈Z be
almost periodic. Then, the set
T({ϕ1
k}, ε) ∩ · · · ∩ T({ϕn
k }, ε)
is relatively dense in Z.
We remark that it is possible to use Corollaries 1.10, 1.11, and 1.12 to obtain more
general versions of Theorems 1.23, 1.27, and 1.29 below.
1.2.2 Asymptotically almost periodic sequences
Now we mention the deﬁnition of asymptotic almost periodicity in pseudometric spaces.
Deﬁnition 1.13. A sequence {ϕk}k∈Z ⊆ X (or {ϕk}k∈N0 ⊆ X) is asymptotically almost
periodic if for every ε > 0, there exist positive integers r(ε) and R(ε) such that any set
consisting of r(ε) consecutive integers contains at least one number l for which
d (ϕk+l, ϕk) < ε, k, k + l ≥ R(ε), k ∈ N.
Directly from Deﬁnition 1.13, we obtain the following theorem.
1.2 Generalizations of pure periodicity 12
Theorem 1.14. The range of any asymptotically almost periodic sequence is totally boun-
ded.
The Bochner theorem for asymptotically almost periodic sequences reads as follows.
Theorem 1.15. Let {ϕk}k∈Z ⊆ X or {ϕk}k∈N0 ⊆ X be given. The sequence {ϕk} is asymptotically
almost periodic if and only if any sequence {ln}n∈N ⊆ N satisfying limn→∞ ln = ∞
has a subsequence {¯ln}n∈N ⊆ {ln} such that, for any ε > 0, there exists L(ε) ∈ N with the
property that it holds
d ϕk+¯li
, ϕk+¯lj
< ε, i, j > L(ε), k ∈ N.
Proof. See [65, Part 2].
Remark 1.16. In Banach spaces, a sequence is asymptotically almost periodic if and only
if it is the sum of an almost periodic sequence and a sequence vanishing at inﬁnity (see,
e.g., [184]).
1.2.3 Limit periodic sequences
Finally, we deﬁne limit periodicity in X.
Deﬁnition 1.17. A sequence {ϕk}k∈Z ⊆ X (or {ϕk}k∈N0 ⊆ X) is called limit periodic if
there exists a sequence of periodic sequences {ϕn
k }k∈Z ⊆ X (or {ϕn
k }k∈N0 ⊆ X) for n ∈ N
such that limn→∞ ϕn
k = ϕk uniformly with respect to k ∈ Z (or k ∈ N0).
Remark 1.18. Of course, the periods of sequences {ϕn
k } in Deﬁnition 1.17 do not need to
be the same for considered n.
Remark 1.19. In fact, limit periodic sequences coincide with the so-called semi-periodic
sequences. We refer to [17] (and to [8] in the continuous case).
Remark 1.20. In the literature, it is possible to ﬁnd another deﬁnition of limit periodicity
which leads to a larger class of sequences. See, e.g., [59, 130]. We consider Deﬁnition 1.17
because it is the standard one in the theory of almost periodic functions (and we obtain
the strongest results in this case). See, e.g., [18, 46].
Theorem 1.21. There exists an almost periodic sequence {fk}k∈Z ⊂ C (with respect to the
usual metric) which is not limit periodic.
Proof. It suﬃces to consider, e.g., the sequence {eik
}k∈Z (or also [46, Theorem 1.27]).
Theorem 1.22. Any limit periodic sequence is almost periodic and any almost periodic
sequence is asymptotically almost periodic.
Proof. It suﬃces to consider Theorems 1.3, 1.7, and 1.15 together with Deﬁnitions 1.1,
1.13, and 1.17.
1.3 Constructions of almost periodic and limit periodic sequences 13
1.3 Constructions of almost periodic and limit periodic
sequences
In this section, we prove several theorems which facilitate to ﬁnd almost periodic and
limit periodic sequences having certain speciﬁc properties. In Theorem 1.23, we consider
sequences for k ∈ N0; in Theorem 1.25 and Corollary 1.26, sequences for k ∈ Z obtained
from almost periodic sequences for k ∈ N0; and, in Theorems 1.27 and 1.29, sequences
for k ∈ Z.
Theorem 1.23. Let ϕ0, . . . , ϕm ∈ X and j ∈ N be arbitrarily given. Let {rn}n∈N be an
arbitrary sequence of non-negative real numbers such that
∞
n=1
rn < ∞. (1.3)
Then, any sequence {ϕk}k∈N0 ⊆ X, where
ϕk ∈ Or1 ϕk−(m+1) , k ∈ {m + 1, . . . , 2m + 1},
ϕk ∈ Or1 ϕk−2(m+1) , k ∈ {2(m + 1), . . . , 3(m + 1) − 1},
...
ϕk ∈ Or1 ϕk−j(m+1) , k ∈ {j(m + 1), . . . , (j + 1)(m + 1) − 1},
ϕk ∈ Or2 ϕk−(j+1)(m+1) , k ∈ {(j + 1)(m + 1), . . . , 2(j + 1)(m + 1) − 1},
ϕk ∈ Or2 ϕk−2(j+1)(m+1) , k ∈ {2(j + 1)(m + 1), . . . , 3(j + 1)(m + 1) − 1},
...
ϕk ∈ Or2 ϕk−j(j+1)(m+1) , k ∈ {j(j + 1)(m + 1), . . . , (j + 1)2
(m + 1) − 1},
...
ϕk ∈ Orn ϕk−(j+1)n−1(m+1) , k ∈ (j + 1)n−1
(m + 1),
. . . , 2(j + 1)n−1
(m + 1) − 1 ,
ϕk ∈ Orn ϕk−2(j+1)n−1(m+1) , k ∈ 2(j + 1)n−1
(m + 1),
. . . , 3(j + 1)n−1
(m + 1) − 1 ,
...
ϕk ∈ Orn ϕk−j(j+1)n−1(m+1) , k ∈ j(j + 1)n−1
(m + 1),
. . . , (j + 1)n
(m + 1) − 1 ,
...
is almost periodic.
1.3 Constructions of almost periodic and limit periodic sequences 14
Proof. Consider an arbitrary ε > 0. We need to prove that the set of all ε-translation
numbers of {ϕk} is relatively dense in N0. Using (1.3), one can ﬁnd n(ε) ∈ N for which
∞
n=n(ε)
rn <
ε
2
. (1.4)
We see that
ϕk+(j+1)n(ε)−1(m+1) ∈ Orn(ε)
(ϕk),
ϕk+2(j+1)n(ε)−1(m+1) ∈ Orn(ε)
(ϕk),
...
ϕk+j(j+1)n(ε)−1(m+1) ∈ Orn(ε)
(ϕk)
(1.5)
if
0 ≤ k < (j + 1)n(ε)−1
(m + 1).
Next, from (1.5) it follows (consider i ∈ {(j + 1)n
, . . . , (j + 1)n+1
− 1}, n ∈ N)
ϕk+(j+1)(j+1)n(ε)−1(m+1) ∈ Orn(ε)+rn(ε)+1
(ϕk),
ϕk+((j+1)2−1)(j+1)n(ε)−1(m+1) ∈ Orn(ε)+rn(ε)+1
(ϕk),
...
ϕk+i(j+1)n(ε)−1(m+1) ∈ Orn(ε)+rn(ε)+1+···+rn(ε)+n
(ϕk),
...
for k ∈ {0, . . . , (j + 1)n(ε)−1
(m + 1) − 1}. Therefore (consider (1.4)), we have
ϕk+l(j+1)n(ε)−1(m+1) ∈ Oε
2
(ϕk), 0 ≤ k < (j + 1)n(ε)−1
(m + 1), l ∈ N0. (1.6)
We put
q(ε) := (j + 1)n(ε)−1
(m + 1). (1.7)
Any p ∈ N0 can be expressed uniquely in the form
p = k(p) + l(p)q(ε) for some k(p) ∈ {0, . . . , q(ε) − 1} and l(p) ∈ N0.
Applying (1.6), we obtain
d ϕp, ϕp+lq(ε) = d ϕk(p)+l(p)q(ε), ϕk(p)+l(p)q(ε)+lq(ε)
≤ d ϕk(p)+l(p)q(ε), ϕk(p) + d ϕk(p), ϕk(p)+(l+l(p))q(ε) <
ε
2
+
ε
2
= ε,
(1.8)
where p, l ∈ N0 are arbitrary; i.e., lq(ε) is an ε-translation number of {ϕk} for all l ∈ N0.
The fact that the set {lq(ε); l ∈ N0} is relatively dense in N0 proves the theorem.
1.3 Constructions of almost periodic and limit periodic sequences 15
Remark 1.24. From the proof of Theorem 1.23 (see (1.7) and (1.8)), for any ε > 0 and
any sequence {ϕk} considered there, we get the existence of n(ε) ∈ N such that the set of
all ε-translation numbers of {ϕk} contains {l(j + 1)n(ε)−1
(m + 1); l ∈ N}; i.e., we have
T ({ϕk}, n(ε)) := l(j + 1)n(ε)−1
(m + 1); l ∈ N ⊆ T ({ϕk}, ε) (1.9)
for every ε > 0.
From Remark 1.24 (see (1.9)), we immediately obtain that the resulting sequence {ϕk}
in Theorem 1.23 is actually limit periodic.
Theorem 1.25. Let {ϕk}k∈N0 be an almost periodic sequence and let {rn}n∈N ⊂ R+
0 and
{ln}n∈N ⊆ N be such that
rnln → 0 as n → ∞. (1.10)
If for any n ∈ N, there exists a set T(rn) of some rn-translation numbers of {ϕk}
which is relatively dense in N0 and, for every non-zero l = l(rn) ∈ T(rn), there exists
i = i(l) ∈ {1, . . . , ln + 1} with the property that
ϕ(i−1)l+k ∈ Ornln (ϕil−k) , k ∈ {0, . . . , l}, (1.11)
then the sequence {ψk}k∈Z, given by the formula
ψk := ϕk for k ∈ N0; ψk := ϕ−k for k ∈ Z N0, (1.12)
is almost periodic.
If for any n ∈ N, there exists a set T(rn) of some rn-translation numbers of {ϕk}
which is relatively dense in N0 and, for every non-zero m = m(rn) ∈ T(rn), there exists
i = i (m) ∈ {1, . . . , ln + 1} with the property that
ϕ(i−1)m+k ∈ Ornln (ϕim−k−1) , k ∈ {0, . . . , m − 1}, (1.13)
then the sequence {χk}k∈Z, given by the formula
χk := ϕk for k ∈ N0; χk := ϕ−(k+1) for k ∈ Z N0, (1.14)
is almost periodic.
Proof. We prove only the ﬁrst part of Theorem 1.25. The proof of the second case (the
almost periodicity of {χk}) is analogous. Let ε > 0 be arbitrarily small. Consider n ∈ N
satisfying (see (1.10))
rnln <
ε
3
. (1.15)
We want to prove that the set T({ψk}, ε) of all ε-translation numbers of {ψk} contains the
numbers {±l; l ∈ T(rn)}; i.e., we want to obtain the inequality
d (ψk, ψk±l) < ε, l ∈ T(rn), k ∈ Z, (1.16)
which proves the theorem because {±l; l ∈ T(rn)} is relatively dense in Z.
1.3 Constructions of almost periodic and limit periodic sequences 16
First of all we choose arbitrary l ∈ T(rn). From the theorem, we have i = i(l). Without
loss of generality, we can consider only +l. (For −l, we can proceed similarly.) Because of
ln ∈ N and l ∈ T(rn), from (1.12) and (1.15) it follows
d (ψk, ψk+l) <
ε
3
, k /∈ {−l, . . . , −1}, k ∈ Z. (1.17)
Let k ∈ {−l, . . . , −1} be also arbitrarily chosen. Evidently, we have
k + (1 − i)l ∈ {−il, . . . , −(i − 1)l − 1}
and
d (ψk, ψk+l) ≤ d ψk, ψk+(1−i)l + d ψk+(1−i)l, ψk+l
= d ϕ−k, ϕ(i−1)l−k + d ϕ(i−1)l−k, ϕl+k .
(1.18)
The number (i − 1)l is an (ε/3)-translation number of {ϕk}. It follows from (1.15) and
from i ≤ ln + 1. Therefore, we have
d ϕ−k, ϕ(i−1)l−k <
ε
3
. (1.19)
Using (1.11) and (1.15), we get
d ϕ(i−1)l−k, ϕil+k < rnln <
ε
3
.
Thus, it holds
d ϕ(i−1)l−k, ϕl+k <
2ε
3
. (1.20)
Indeed, (i − 1)l is an (ε/3)-translation number of {ϕk} (consider again (1.15) and the
inequality i − 1 ≤ ln).
Altogether, from (1.18), (1.19), and (1.20), we obtain
d (ψk, ψk+l) <
ε
3
+
2ε
3
= ε. (1.21)
Since the choice of k, l is arbitrary (see (1.17)), (1.21) gives (1.16).
Corollary 1.26. Let m ∈ N0, j ∈ N, the sequence {ϕk}k∈N0 be from Theorem 1.23, and
M > 0 be arbitrary.
If for all n > M, n ∈ N, there exists at least one i ∈ {1, . . . , j} satisfying
ϕi(j+1)n(m+1)+k = ϕ(i+1)(j+1)n(m+1)−k, k ∈ {0, . . . , (j + 1)n
(m + 1)}, (1.22)
then the sequence {ψk}k∈Z given by (1.12) is almost periodic.
If for all n > M, n ∈ N, there exists at least one i ∈ {1, . . . , j} satisfying
ϕi(j+1)n(m+1)+k = ϕ(i+1)(j+1)n(m+1)−k−1, k ∈ {0, . . . , (j + 1)n
(m + 1) − 1}, (1.23)
then the sequence {χk}k∈Z given by (1.14) is almost periodic.
1.3 Constructions of almost periodic and limit periodic sequences 17
Proof. We put
rn :=
1
n
, ln := 1, T(rn) := T {ϕk}, n
rn
2
for all n ∈ N,
where T ({ϕk}, n(ε)) is deﬁned in (1.9). Since we can assume that n (1/2) > M − 1, it
suﬃces to consider Theorem 1.25 and Remark 1.24 (using (1.22) and (1.23), we get (1.11)
and (1.13), respectively).
Theorem 1.27. Let ϕ0, . . . , ϕn ∈ X and j ∈ N be given and let {ri}i∈N ⊂ R+
0 satisfy
∞
i=1
ri < ∞. (1.24)
Then, every sequence {ϕk}k∈Z for which
ϕk ∈ Or1 ϕk−(n+1) , k ∈ {n + 1, . . . , 2(n + 1) − 1},
...
ϕk ∈ Or1 ϕk−j(n+1) , k ∈ {j(n + 1), . . . , (j + 1)(n + 1) − 1},
ϕk ∈ Or2 ϕk+(j+1)(n+1) , k ∈ {−(j + 1)(n + 1), . . . , −1},
...
ϕk ∈ Or2 ϕk+j(j+1)(n+1) ,
k ∈ {−j(j + 1)(n + 1), . . . , −(j − 1)(j + 1)(n + 1) − 1},
ϕk ∈ Or3 ϕk−(j+1)2(n+1) ,
k ∈ {(j + 1)(n + 1), . . . , (j + 1)(n + 1) + (j + 1)2
(n + 1) − 1},
...
ϕk ∈ Or3 ϕk−j(j+1)2(n+1) ,
k ∈ {(j + 1)(n + 1) + (j − 1)(j + 1)2
(n + 1), . . . ,
(j + 1)(n + 1) + j(j + 1)2
(n + 1) − 1},
ϕk ∈ Or4 ϕk+(j+1)3(n+1) ,
k ∈ {−(j + 1)3
(n + 1) − j(j + 1)(n + 1), . . . , −j(j + 1)(n + 1) − 1},
...
ϕk ∈ Or4 ϕk+j(j+1)3(n+1) ,
k ∈ {−j(j + 1)3
(n + 1) − j(j + 1)(n + 1), . . . ,
− (j − 1)(j + 1)3
(n + 1) − j(j + 1)(n + 1) − 1},
...
1.3 Constructions of almost periodic and limit periodic sequences 18
ϕk ∈ Or2i
ϕk+(j+1)2i−1(n+1) ,
k ∈ {−((j + 1)2i−1
+ · · · + j(j + 1)3
+ j(j + 1))(n + 1), . . . ,
− (j(j + 1)2i−3
+ · · · + j(j + 1)3
+ j(j + 1))(n + 1) − 1},
...
ϕk ∈ Or2i
ϕk+j(j+1)2i−1(n+1) ,
k ∈ {−(j(j + 1)2i−1
+ · · · + j(j + 1)3
+ j(j + 1))(n + 1), . . . ,
− ((j − 1)(j + 1)2i−1
+ · · · + j(j + 1)3
+ j(j + 1))(n + 1) − 1},
ϕk ∈ Or2i+1
ϕk−(j+1)2i(n+1) ,
k ∈ {(j + 1)(n + 1) + j(j + 1)2
(n + 1) + · · · + j(j + 1)2i−2
(n + 1), . . . ,
(j + 1)(n + 1) + j(j + 1)2
(n + 1) + · · ·
+ j(j + 1)2i−2
(n + 1) + (j + 1)2i
(n + 1) − 1},
...
ϕk ∈ Or2i+1
ϕk−j(j+1)2i(n+1) ,
k ∈ {(j + 1)(n + 1) + j(j + 1)2
(n + 1) + · · ·
+ j(j + 1)2i−2
(n + 1) + (j − 1)(j + 1)2i
(n + 1), . . . ,
(j + 1)(n + 1) + j(j + 1)2
(n + 1) + · · · + j(j + 1)2i
(n + 1) − 1},
...
is limit periodic.
Proof. Let ε > 0 be arbitrarily given and let the number i(ε) ∈ N satisfy the condition
(see (1.24))
∞
i=i(ε)
ri <
ε
2
.
One can easily show that
l(j + 1)i(ε)−1
(n + 1); l ∈ Z ⊆ T ({ϕk}, ε) .
Of course, this inclusion proves the theorem.
For n = 0, j = 1, we get the most important case of Theorem 1.27 which reads as
follows.
Corollary 1.28. Let ψ0 ∈ X and {εi}i∈N ⊂ R+
0 satisfying
∞
i=1
εi < ∞ (1.25)
be arbitrary. Then, every sequence {ψk}k∈Z for which
ψk ∈ Oε1 (ψk−20 ) , k ∈ {1} = {2 − 1},
1.3 Constructions of almost periodic and limit periodic sequences 19
ψk ∈ Oε2 (ψk+21 ) , k ∈ {−2, −1},
ψk ∈ Oε3 (ψk−22 ) , k ∈ {2, . . . , 2 + 22
− 1},
ψk ∈ Oε4 (ψk+23 ) , k ∈ {−23
− 2, . . . , −2 − 1},
ψk ∈ Oε5 (ψk−24 ) , k ∈ {2 + 22
, . . . , 2 + 22
+ 24
− 1},
...
ψk ∈ Oε2i
(ψk+22i−1 ) , k ∈ {−22i−1
− · · · − 23
− 2, · · · , −22i−3
− · · · − 23
− 2 − 1},
ψk ∈ Oε2i+1
(ψk−22i ) , k ∈ {2 + 22
+ · · · + 22i−2
, . . . , 2 + 22
+ · · · + 22i−2
+ 22i
− 1},
...
is limit periodic.
Theorem 1.29. Let ϕ0, . . . , ϕm ∈ X be given, {ri}i∈N ⊂ R+
0 , {ji}i∈N ⊆ N, and n ∈ N0 be
arbitrary such that m + n is even and
∞
i=1
ri ji < ∞. (1.26)
For any ϕm+1, . . . , ϕm+n, if one puts
ψk := ϕk+m+n
2
, k ∈ −
m + n
2
, . . . ,
m + n
2
,
M :=
m + n
2
, N := m + n
and arbitrarily chooses
ψk ∈ Or1 (ψk+N+1) , k ∈ {−N − M − 1, . . . , −M − 1},
...
ψk ∈ Or1 (ψk+N+1) , k ∈ {−j1N − M − 1, . . . , −(j1 − 1)N − M − 1},
ψk ∈ Or1 (ψk−N−1) , k ∈ {M + 1, . . . , N + M + 1},
...
ψk ∈ Or1 (ψk−N−1) , k ∈ {(j1 − 1)N + M + 1, . . . , j1N + M + 1},
...
ψk ∈ Ori
(ψk+pi
) , k ∈ {−pi − pi−1 − · · · − p1, . . . , −pi−1 − · · · − p1},
...
ψk ∈ Ori
(ψk+pi
) , k ∈ {−jipi − pi−1 − · · · − p1, . . . , −(ji − 1)pi − pi−1 − · · · − p1},
ψk ∈ Ori
(ψk−pi
) , k ∈ {pi−1 + · · · + p1, . . . , pi + pi−1 + · · · + p1},
1.4 Application related to almost periodic diﬀerence systems 20
...
ψk ∈ Ori
(ψk−pi
) , k ∈ {(ji − 1)pi + pi−1 + · · · + p1, . . . , jipi + pi−1 + · · · + p1},
...
where
p1 := (j1N + M + 1) + 1, p2 := 2(j1N + M + 1) + 1,
p3 := (2j2 + 1)p2, . . . pi := (2ji−1 + 1)pi−1, . . . ,
then the resulting sequence {ψk}k∈Z is limit periodic.
Proof. Consider arbitrary ε > 0 and a positive integer n(ε) ≥ 2 for which (see (1.26))
∞
i=n(ε)
ri ji <
ε
4
.
One can show that
{lpn(ε); l ∈ Z} ⊆ T({ψk}, ε)
which completes the proof.
1.4 Application related to almost periodic diﬀerence
systems
Let m ∈ N be arbitrarily given. We analyse almost periodic systems of m homogeneous
linear diﬀerence equations of the form
xk+1 = Ak · xk, k ∈ Z (or k ∈ N0), (1.27)
where {Ak} is almost periodic. Let X denote the set of all systems (1.27).
An important characteristic property of linear diﬀerence systems, which makes them
simple to treat, is the well-known superposition principle (see, e.g., [1, 108, 115]). In
particular, since we study homogeneous systems, we obtain that every solution of a system
S ∈ X can be expressed as a right linear combination of m solutions of S; i.e., any
solution {xk} of S can be written as
xk = Pk · xl, k ∈ Z (or k ∈ N0), (1.28)
for some matrix valued sequence {Pk} and some l ∈ Z (l ∈ N0). Conversely, for any
considered r1, . . . , rm, the sequence {xk} deﬁned by the formula
xk := Pk ·



r1
...
rm


 , k ∈ Z (or k ∈ N0),
1.4 Application related to almost periodic diﬀerence systems 21
is a solution of S. For given S ∈ X determined by {Ak}, the sequence {Pk} is called the
principal fundamental matrix if P0 is the identity matrix. We immediately obtain
Pk =
k−1
i=0
Ak−i−1 for all k ∈ N;
Pk =
−1
i=k
A−1
i for all k ∈ Z N.
(1.29)
Our aim is to study the existence of S ∈ X which does not have any non-trivial almost
periodic solution. We treat this problem in a very general setting and this motivates our
requirements on the set of values of matrices Ak.
We need the set of entries of Ak to be a subset of a set R with two operations and
unit elements such that R with them is a ring, because the multiplication of matrices Ak
has to be associative (consider the natural expression of solutions of (1.27), i.e., consider
(1.28) and (1.29)). We also need the set of all considered Ak to form a set X which has
the below given properties (1.33); and we need that there exists at least one of the below
mentioned functions F1, F2 : [−1, 1] → X (see (1.34), (1.35)). The conditions (1.34) are
common. However, the main theorem of this chapter (the existence of the above system
S ∈ X) is true, e.g., for many subsets of the set of unitary or orthogonal matrices which
contain matrices having eigenvalue λ = 1. Thus, we also consider the existence of F2.
Let R = (R, ⊕, ) be an inﬁnite ring with a unit and a zero denoted as e1 and e0,
respectively. Symbol M(R, m) denotes the set of all m×m matrices with elements from R.
If we consider the i-th column of U ∈ M(R, m), then we write Ui; and Rm
if we consider
the set of all m × 1 vectors with entries attaining values from R. As usual, we deﬁne the
multiplication · of matrices from M(R, m) (and U ·v, U ∈ M(R, m), v ∈ Rm
) by ⊕ and .
Let d be a pseudometric on R and assume that
operations ⊕ and are continuous with respect to d. (1.30)
In particular, we have pseudometrics on Rm
and M(R, m), because M(R, m) can be
expressed as Rm×m
; i.e., d in Rm
and M(R, m) is the sum of m and m2
non-negative
numbers given by d in R, respectively. For simplicity, we also denote these pseudometrics
as d.
The vector v ∈ Rm
is called non-zero (or non-trivial) if d v, (e0, . . . , e0)T
> 0. We
say that a non-zero vector (r1, . . . , rm)T
, where r1, . . . , rm ∈ R, is an e1-eigenvector of
U ∈ M(R, m) if
d


U ·



r1
...
rm


 ,



r1
...
rm





 = 0,
and that V ∈ M(R, m) is regular for a non-zero vector (r1, . . . , rm)T
∈ Rm
if
d


V ·



r1
...
rm


 ,



e0
...
e0





 > 0. (1.31)
1.4 Application related to almost periodic diﬀerence systems 22
Next, we denote
I :=





e1 e0 . . . e0
e0 e1 . . . e0
...
...
...
...
e0 e0 . . . e1





∈ M(R, m).
If for given U ∈ M(R, m) and X ⊆ M(R, m), there exists the unique matrix V ∈ X (we
put V = W if d(V, W) = 0) for which
U · V = V · U = I,
then we deﬁne U−1
:= V and we say that V is the inverse matrix of U in X.
For any function H : [a, b] → X (a ≤ 0 < b, a, b ∈ R) and s ∈ R, we extend its domain
of deﬁnition by the formula
H(s) :=
H(σ) · (H(b))l
for s ≥ 0;
(H(a))l
· H(σ) for s < 0 if a < 0,
(1.32)
where s = lb + σ for l ∈ N0, σ ∈ [0, b) or s = la + σ for l ∈ N0, σ ∈ (a, 0]. Hereafter, we
restrict coeﬃcients Ak in (1.27) to be elements of an inﬁnite set X ⊆ M(R, m) such that
I ∈ X; U, V ∈ X =⇒ U · V ∈ X, U−1
exists in X; (1.33)
and either
there exists a continuous function F1 : [−1, 1] → X satisfying
F1(0) = I; F1(t) = F−1
1 (−t), t ∈ [0, 1]; (1.34)
and matrix F1(1) has no e1-eigenvector
or
there exist continuous F2 : [−1, 1] → X, t1, . . . , tq ∈ (0, 1], δ > 0 such that
F2(0) = I; F2
p
i=1
si =
p
i=1
F2 (si), s1, . . . , sp ∈ [−1, 1]; (1.35)
and, for any v ∈ Rm
, one can ﬁnd j ∈ {1, . . . , q} for which v is not an e1-eigenvector
of F2(t), t ∈ (max{0, tj − δ}, min{tj + δ, 1}).
Remark 1.30. A function F1 satisfying (1.34) exists, e.g., if the considered pseudometric
d : X × X → R+
0 is such that the map U → U−1
is continuous on X and if there
exists a continuous function G : [0, 1] → X which satisﬁes that at least one of matrices
G−1
(0) G(1), G(1) G−1
(0) does not have an e1-eigenvector.
Similarly, the conditions in (1.35) are realized if the map U → U−1
is continuous on X
and if there exist continuous functions G1, . . . , Gq : [0, 1] → X such that
G−1
j (0) · Gj
p
i=1
si =
p
i=1
G−1
j (0) · Gj (si)
1.4 Application related to almost periodic diﬀerence systems 23
and
Gj
p
i=1
si · Gj(0)
−1
=
p
i=1
G−1
j (0) · G−1
j (si)
or
Gj
p
i=1
si · G−1
j (0) =
p
i=1
Gj (si) · G−1
j (0)
and
Gj(0) · Gj
p
i=1
si
−1
=
p
i=1
G−1
j (si) · G−1
j (0),
where
j ∈ {1, . . . , q}, p ∈ N, s1, . . . , sp ∈ [0, 1] ;
for all j1, j2 ∈ {1, . . . , q}, one can ﬁnd r = r (j1, j2) ∈ (0, 1] with at least one property from
G−1
j1
(0) · Gj1 (1) = G−1
j2
(0) · Gj2 (r), G−1
j2
(0) · Gj2 (1) = G−1
j1
(0) · Gj1 (r)
or
Gj1 (1) · G−1
j1
(0) = Gj2 (r) · G−1
j2
(0), Gj2 (1) · G−1
j2
(0) = Gj1 (r) · G−1
j1
(0);
and the condition on arbitrary v ∈ Rm
is the same as in (1.35), where
F2(t), t ∈ (max{0, tj − δ}, min{tj + δ, 1}) ,
is replaced by
G−1
j (0) · Gj(1) or Gj(1) · G−1
j (0).
We recall that, for U1, . . . , Up ∈ X (p ∈ N), we deﬁne
p
i=1
Ui := U1 · U2 · · · Up,
1
i=p
Ui := Up · Up−1 · · · U1.
For the above function H, we also use the conventional notation
(H(s))0
:= I, H−1
(s) := (H(s))−1
for all considered s ∈ R.
Actually, a closer examination of our process reveals that the pseudometric d can be deﬁned
“only” on the set
{Fj(s1) · · · Fj(sn) · v; v ∈ Rm
, s1, . . . , sn ∈ [−1, 1]}
and the set {Fj(t); t ∈ [−1, 1]} can be countable for each j ∈ {1, 2}.
1.4 Application related to almost periodic diﬀerence systems 24
Remark 1.31. Now we comment our assumptions on R and X. Because of (1.30), the
requirement about the existence of δ > 0 (in (1.35)) can be omitted. Note that R does not
need to be commutative. Thus, the set of all solutions of (1.27) is not generally a modulus
over R with the scalar multiplication given by
r



x1
k
...
xm
k


 :=



r x1
k
...
r xm
k


 ,
where {(x1
k, . . . , xm
k )T
} is a solution of (1.27), r ∈ R, k ∈ Z (k ∈ N0). Indeed, the expression
Pk ·



x1
0
...
xm
0


 = x1
0 · (Pk)1 + · · · + xm
0 · (Pk)m
does not need to hold for considered k and a solution {xk} of (1.27) (see (1.28)).
For the main requirements, consider two results concerning the existence, the uniqueness
(and the uniform asymptotic stability) of an almost periodic solution of the almost periodic
real (non-)homogeneous linear system (1.27) for k ∈ Z in [188] or directly the following
simple example. Let R := R, m := 2, and
X1 :=
0 10l
10−l
0
; l ∈ Z ,
X2 :=
0 10l
10−l
0
; l ∈ Z ∪
10l
0
0 10−l ; l ∈ Z
with the usual metric on R. For X1, every S ∈ X has all solutions almost periodic; at the
same time, for X2, it is easy to ﬁnd a system from X which has only one almost periodic
solution—the trivial one.
To prove the announced result, we need a sequence {ak}k∈N0 of real numbers, which
has special properties (mentioned in the below given Lemmas 1.32–1.35). We deﬁne the
sequence {ak}k∈N0 by the recurrent formula
a0 := 1, a1 := 0, a2n+k := ak −
1
2n
, k = 0, . . . , 2n
− 1, n ∈ N. (1.36)
For this sequence, we have the following auxiliary results.
Lemma 1.32. The sequence {ak} is almost periodic.
Proof. The lemma follows from Theorem 1.23, where it suﬃces to put ϕk = ak (k ∈ N)
and
X = R, m = 0, j = 1, ϕ0 = 1, rn =
4
2n
, n ∈ N.
Lemma 1.33. The identity
a2n+2−1−i = −a2n+1+i (1.37)
holds for any n ∈ N0 and i ∈ {0, . . . , 2n
− 1}.
1.4 Application related to almost periodic diﬀerence systems 25
Before proving this lemma, observe that (1.37) is equivalent to
2n+2−1−i
k=2n+1+i
ak = 0, n ∈ N0, i ∈ {0, . . . , 2n
− 1};
i.e., to
2n+1−1+i
k=0
ak =
2n+2−1−i
k=0
ak, n ∈ N0, i ∈ {0, . . . , 2n
− 1}.
Proof of Lemma 1.33. Obviously, (1.37) is true for n ∈ {0, 1}, because
a2 = −a3 =
1
2
, a4 = −a7 =
3
4
, a5 = −a6 = −
1
4
;
i.e.,
1
k=0
ak =
3
k=0
ak =
7
k=0
ak = 1,
4
k=0
ak =
6
k=0
ak =
7
4
.
Suppose that (1.37) is true also for 2, . . . , n − 1. We choose i ∈ {0, . . . , 2n
− 1} arbitrarily.
(We have 2n+2
− 1 − i ≥ 2n+1
+ 2n
.) From (1.36) and the induction hypothesis, it follows
that
a2n+2−1−i + a2n+1+2n+i = −
1
2n
, a2n+1+i − a2n+1+2n+i =
1
2n
.
Summing up the above equalities, we get (1.37).
Lemma 1.34. We have n
k=0
ak ≥ 1, n ∈ N0. (1.38)
Proof. Evidently, a0 = a0 + a1 = 1. It means that (1.38) is true for n = 0 and n = 1 =
21
− 1. Let it be valid for arbitrarily given 2p
− 1 and all n < 2p
− 1, i.e., let
n
k=0
ak ≥ 1, n ≤ 2p
− 1, n ∈ N0.
Considering the deﬁnition of {ak}, we obtain
2p+j−1
k=0
ak =
2p−1
k=0
ak +
2p+j−1
k=2p
ak ≥ 1 +
j−1
k=0
ak − j
1
2p
≥ 1 + 1 − 1 = 1
for any j ∈ {1, . . . , 2p
}. Lemma 1.34 follows from the induction principle.
Lemma 1.35. We have
2n−1
k=0
ak = 1, (1.39)
2n+i+2n−1
k=0
ak = 2 −
1
2i
, (1.40)
where n ∈ N0, i ∈ N.
1.4 Application related to almost periodic diﬀerence systems 26
Proof. It is possible to prove this result by means of Lemma 1.33, but we prove it directly
using (1.36) and the induction principle. We have
a0 = 1, a0 + a1 = 1, a0 + a1 + a2 + a3 = 1.
If we assume that
2n−1−1
k=0
ak = 1,
then we get (see (1.36))
2n−1
k=0
ak =
2n−1−1
k=0
ak +
2n−1
k=2n−1
ak
=
2n−1−1
k=0
ak +
2n−1−1
k=0
ak −
1
2n−1
= 2
2n−1−1
k=0
ak − 1 = 1.
Therefore, (1.39) is proved. Analogously, applying (1.36) and (1.39), one can compute
2n+i+2n−1
k=0
ak =
2n+i−1
k=0
ak +
2n+i+2n−1
k=2n+i
ak
= 1 +
2n−1
k=0
ak −
1
2n+i
= 1 + 1 −
1
2i
which gives (1.40).
Applying matrix valued functions F1, F2, we obtain the next lemma.
Lemma 1.36. For each j ∈ {1, 2}, any n ∈ N0, and each i ∈ {0, . . . , 2n
− 1}, it holds
Fj(a2n+2−1−i) = F−1
j (a2n+1+i)
and, consequently,
2n+2−1−i
k=2n+1+i
Fj(ak) =
2n+1+i
k=2n+2−1−i
Fj(ak) = I.
Proof. Obviously, this lemma is a corollary of Lemma 1.33. Consider (1.34) and (1.35)
with the fact that the multiplication of matrices is associative.
From Lemma 1.35 (see (1.35)), we obtain the following formulas for F2.
Lemma 1.37. The equalities
2n−1
k=0
F2 (ak) = F2(1),
2n+i+2n−1
k=0
F2(ak) = F2 2 −
1
2i
hold for all n ∈ N0 and i ∈ N.
1.4 Application related to almost periodic diﬀerence systems 27
Now we can prove the main result of Chapter 1.
Theorem 1.38. There exists a system of the form (1.27) which does not possess a non-zero
almost periodic solution.
Proof. First we suppose that the coeﬃcients Ak belong to X so that there exists a function
F1 from (1.34). Using Theorem 1.6, we get the almost periodicity of the sequence
{F1(ak)}k∈N0 , where {ak} is given by (1.36). We want to show that all non-zero solutions
of the system S1 ∈ X determined by {F1(ak)} are not almost periodic.
By contradiction, suppose that there exist c1, . . . , cm ∈ R such that the vector valued
sequence
{fk} :=



Pk ·



c1
...
cm






, k ∈ N0, (1.41)
where {Pk}k∈N0 is the principal fundamental matrix of S1, is non-trivial and almost periodic;
i.e., suppose that S1 has a non-trivial almost periodic solution {fk}. Since {fk} is
almost periodic, (c1, . . . , cm)T
is non-zero, and it holds
fi = Ui · F1(1) ·



c1
...
cm


 for any i ∈ N and some Ui ∈ X, (1.42)
we know that (see (1.31))
F1(1) is regular for c := (c1, . . . , cm)T
. (1.43)
Considering (1.36), the uniform continuity of F1 and the continuity of the multiplication
of matrices (see (1.30)), Lemma 1.36, and (1.41), from the ﬁrst part of Theorem 1.25 (see
the proof of Corollary 1.26 and again Lemma 1.36), one can obtain that the sequence
{gk}k∈Z, where
gk := fk, k ∈ N0; gk := f−k, k ∈ Z N0, (1.44)
is almost periodic as well. Now we use Theorem 1.3 for {ϕk} ≡ {gk} and {hn}n∈N ≡
{2n
}n∈N (we can also consider directly {ϕk} ≡ {fk} and use Remark 1.2). Theorem 1.3
implies that, for any ε > 0, there exists an inﬁnite set N(ε) ⊆ N such that the inequality
d (gk+2n1 , gk+2n2 ) < ε, k ∈ Z, (1.45)
holds for all n1, n2 ∈ N(ε).
Using (1.34), we get d (c, F1(1) c) > 0 and (consider (1.43))
ϑ := d (F1(1) · c, F1(1) · F1(1) · c) > 0. (1.46)
From Lemma 1.36 (for i = 0), (1.41), and (1.44) (see also (1.42)), we have
g0 = c, g1 = F1(1) · c, . . . , g2n = F1(1) · c, (1.47)
1.4 Application related to almost periodic diﬀerence systems 28
where n ∈ N is arbitrary. Hence, considering (1.36), it holds
d (g2i+2n , F1(1) · F1(1) · c) → 0 as n → ∞ (1.48)
for every i ∈ N, because F1 is uniformly continuous and the multiplication of matrices is
continuous. We also have
d (g2n2 +2n1 , F1(1) · c) <
ϑ
2
(1.49)
for all n1, n2 ∈ N (ϑ/2). Indeed, put k = 2n2
in (1.45) and consider (1.47) for n = n2 + 1.
If we choose n1 ∈ N (ϑ/2) and put i = n1 in (1.48), then there exists n0 ∈ N such that, for
any n ≥ n0, it holds
d (g2n1 +2n , F1(1) · F1(1) · c) <
ϑ
2
.
Thus, for arbitrarily given n2 ≥ n0, n2 ∈ N (ϑ/2), we obtain
d (g2n2 +2n1 , F1(1) · F1(1) · c) <
ϑ
2
. (1.50)
Finally, applying (1.46), (1.49), and (1.50), we have
ϑ ≤ d (F1(1) · c, g2n2 +2n1 ) + d (g2n2 +2n1 , F1(1) · F1(1) · c) < ϑ.
This contradiction gives the proof when we consider (1.34) for k ∈ N0.
Let k ∈ Z. Then, we can consider the system S1 determined by the sequence
Bk := F1(ak), k ∈ N0; Bk := F1(−a−k−1), k ∈ Z N0. (1.51)
Since the sequence {| ak |}k∈N0 is almost periodic (see Theorem 1.6) and has the form of
{ϕk}k∈N0 from Theorem 1.23 and since it is valid (see (1.37))
| a2n+2−1−i | = | a2n+1+i |, n ∈ N0, i ∈ {0, . . . , 2n
− 1},
the fact that {Bk} is almost periodic follows from the second part of Corollary 1.26, from
Corollary 1.10, and Theorem 1.6. Next, the process is same as for k ∈ N0. Let {Pk}k∈Z be
the principal fundamental matrix of S1 and gk := fk, k ∈ Z. Also now we have (1.45) and,
consequently, we get the same contradiction.
Let the coeﬃcients Ak belong to X so that there exists a function F2 from (1.35).
Consider the numbers t1, . . . , tq ∈ (0, 1] and δ > 0 from (1.35). Without loss of generality,
we can assume that
δ < t1 < · · · < tq and tq < 1 − δ. (1.52)
Indeed, if tj = 1, then we can put tj := 1 − δ/2 and redeﬁne δ. We repeat that any
vector v ∈ Rm
determines some j ∈ {1, . . . , q} (see again (1.35)) such that v is not an
e1-eigenvector of F2(t) for t ∈ (tj − δ, tj + δ).
From Theorem 1.6 it follows that the sequence {F2(ak)}k∈N0 is almost periodic. Thus,
it determines a system of the form (1.27). We denote it as S2. Suppose that system S2 has
a non-trivial almost periodic solution {xk}k∈N0 . For the principal fundamental matrix {Pk}
of S2, we have
xk = Pk · x0, k ∈ N0,
1.4 Application related to almost periodic diﬀerence systems 29
where the vector x0 is non-zero. Using this fact and taking into account Lemma 1.34
and (1.35), we obtain
xn = F2(t) · Fi
2(1) · x0 for some i ∈ N, t ∈ [0, 1), (1.53)
and for arbitrary n ∈ N. From Lemma 1.37, we also get
x2n = F2(1) · x0 for all n ∈ N0 (1.54)
and
x2n+i+2n = F2 1 −
1
2i
· F2(1) · x0 for all n ∈ N0, i ∈ N. (1.55)
Analogously as for {fk}, one can extend {xk}k∈N0 by the formula
xk := x−k, k ∈ Z N0,
for all k ∈ Z so that the sequence {xk}k∈Z is almost periodic as well. Now we apply Theorem
1.3 for the sequences {xk}k∈Z and {2n
}n∈N. For any ε > 0, there exists an inﬁnite set
M(ε) ⊆ N such that, for any n1, n2 ∈ M(ε), we have
d (xk+2n1 , xk+2n2 ) < ε, k ∈ Z. (1.56)
Since F2 is uniformly continuous and the multiplication of matrices is continuous, for
arbitrary i ∈ N and ε > 0, we have from (1.36) and (1.54) that
d (x2i+2n , F2(1) · F2(1) · x0) < ε for suﬃciently large n ∈ N. (1.57)
Because of the almost periodicity of {xk} and (1.53), the matrix F2(1) has to be regular
for x0. Let ε > 0 be arbitrarily small and n1 ∈ M(ε) arbitrarily large. From (1.56) and
(1.57), where we choose k = 2n1−j
and i = 2n1−j
for j ∈ {0, . . . , n1}, it follows that, for
given n1, there exists suﬃciently large n2 ∈ M(ε) for which
d (x2n1−j+2n1 , F2(1) · F2(1) · x0) ≤ d (x2n1−j+2n1 , x2n1−j+2n2 )
+ d (x2n1−j+2n2 , F2(1) · F2(1) · x0) < 2ε.
(1.58)
Since ε (in (1.58)) is arbitrarily small, choosing j = 0, we get
d (x2n1+1 , F2(1) · F2(1) · x0) = 0
which gives (see (1.54)) that F2(1) x0 is an e1-eigenvector of F2(1), i.e., we have
d (F2(1) · x0, F2(1) · F2(1) · x0) = 0. (1.59)
If we choose j = 1, then we obtain (consider (1.55))
d F2
1
2
· F2(1) · x0, F2(1) · F2(1) · x0 = 0.
Analogously, for any j (the number n1 is arbitrarily large), we get
d F2 1 −
1
2j
· F2(1) · x0, F2(1) · F2(1) · x0 = 0.
1.4 Application related to almost periodic diﬀerence systems 30
Thus,
d F2 2 −
1
2j
· x0, F2(2) · x0 = 0, j ∈ N, (1.60)
and we have
d F2 2 −
1
2j
· x0, F2 2 −
1
2j−1
· x0 = 0, j ∈ N. (1.61)
Because of
F2 2 −
1
2j
= F2
1
2j
+ 2 −
1
2j−1
= F2
1
2j
· F2 2 −
1
2j−1
and (see (1.59) and (1.60))
d F2 2 −
1
2j−1
· x0, F2(1) · x0 = 0,
from (1.61) it follows
d F2
1
2j
· F2(1) · x0, F2(1) · x0 = 0 for all j ∈ N,
i.e., F2(1) x0 is an e1-eigenvector of F2(2−j
) for all j ∈ N.
Since any number t ∈ [0, 1] can be expressed in the form
∞
i=1
ai
2i
, where ai ∈ {0, 1},
for considered δ > 0, there exists n ∈ N such that, for every t ∈ [0, 1], there exist
a1, . . . , an ∈ {0, 1} satisfying
t −
n
i=1
ai
2i
< δ.
Thus, F2(1) x0 is an e1-eigenvector of F2(tj + sj) for some | sj | < δ and any j ∈ {1, . . . , q}
which cannot be true. This contradiction shows that {xk}k∈N0 is not almost periodic.
If one considers the system S2 obtained from S2 as in (1.51) (after replacing S1 by S2),
then, similarly as for F1 and k ∈ N0, one can prove that S2 ∈ X and that any its non-trivial
solution {xk}k∈Z is not almost periodic.
Remark 1.39. Let a non-zero F1(1) v ∈ Rm
not be an e1-eigenvector of matrix F1(1)
from (1.34); i.e., the condition (1.34) be weakened in this way. Then, from the ﬁrst part
of the proof of Theorem 1.38, we obtain that the sequence {fk}, given by (1.41), is not
almost periodic for (c1, . . . , cm)T
= v. It means that there exists a system S1
∈ X with
the principal fundamental matrix {P1
k } such that the sequence {P1
k v}k∈N0 or {P1
k v}k∈Z is
not almost periodic.
Analogously, if one requires in (1.35) only that, for a non-zero vector v ∈ Rm
, there
exists t ∈ (0, 1] for which F2(1) v is not an e1-eigenvector of F2(t), then there exists a system
S2
∈ X satisfying that the sequence {P2
k v}k∈Z (or {P2
k v}k∈N0 ), where {P2
k }k∈Z (or
{P2
k }k∈N0 ) is the principal fundamental matrix of S2
, is not almost periodic.
1.4 Application related to almost periodic diﬀerence systems 31
The condition
F2
p
i=1
si =
p
i=1
F2 (si), s1, . . . , sp ∈ [−1, 1], p ∈ N, (1.62)
in (1.35) is “strong”. For example, from it follows that the multiplication of matrices from
the set {F2(t); t ∈ R} is commutative. We point out that, for many subsets of unitary or
orthogonal matrices, it is not a limitation and that the method in the proof of Theorem 1.38
can be simpliﬁed in many cases. We show it in two important special cases.
Example 1.40. If for any non-trivial vector v ∈ Rm
, there exists ε(v) > 0 with the
property that
F2(t) · v /∈ Oε(v)(v) for all t ≥ 1 (see (1.32)),
then the fact, that the systems S2 and S2 from the proof of Theorem 1.38 do not have
non-trivial almost periodic solutions, follows directly from Lemma 1.34 and (1.62). Indeed,
the set T({xk}, ε(x0)) {0} is empty for any non-zero solution {xk}. ♦
Example 1.41. Let function F2, in addition to (1.35), satisfy
F2(s) = F2(0) = I (1.63)
for some positive irrational number s, (1.52) hold, and p ∈ N be arbitrary. Then, the
system S determined by the sequence {Ak} := {F2(k/p)}, where k ∈ N0 or k ∈ Z, has no
non-trivial almost periodic solution.
The function F2(t/p), t ∈ R, is continuous and periodic with period ps (see (1.62),
(1.63)). Using the compactness of the interval [0, ps], (1.62), and Theorem 1.3, we get that
{F2(k/p)}k∈Z is almost periodic. Then, the almost periodicity of {F2(k/p)}k∈N0 is obvious.
Suppose, by contradiction, that {xk} ≡ {Pk x0} is a non-trivial almost periodic solution
of S. We mention that there exists δ > 0 satisfying that, for any non-zero v ∈ Rm
, one
can ﬁnd j ∈ N such that there exists a positive number ϑ(v) for which
ϑ(v) ≤ d F2
j
p
+ t · v, v , t ∈ (−δ, δ), (1.64)
because
{F2(k/p); k ∈ N} is dense in {F2(t); t ∈ R}. (1.65)
Evidently, (1.65) gives that
{F2(k/p); k ∈ N} is dense in {F2(t); t ∈ R} (1.66)
for any set N which is relatively dense in N.
Since the multiplication of matrices is continuous, there exists ε > 0 which satisﬁes
that every vector u with the property d(u, x0) < ε determines the same j in (1.64) as x0
and one can ﬁnd
ϑ(u) ≥
ϑ(x0)
2
. (1.67)
1.4 Application related to almost periodic diﬀerence systems 32
From (1.62), we see that
xk = F2
k−1
i=0
i
p
· x0, k ∈ N. (1.68)
Let l be an arbitrary positive (ϑ(x0)/2)-translation number of {xk}. Thus, let
d(xk+l, xk) <
ϑ(x0)
2
for all k ∈ N (1.69)
and let N be the set of all positive ε-translation numbers of {xk}. Since
k+l−1
i=0
i
p
=
k−1
i=0
i
p
+
kl
p
+
l(l − 1)
2p
, k ∈ N,
for all k ∈ N, we have (see again (1.62))
d(xk+l, xk) = d F2
kl
p
+
l(l − 1)
2p
· F2
k−1
i=0
i
p
· x0, F2
k−1
i=0
i
p
· x0 . (1.70)
From (1.66), where we replace 1/p by l/p, we get the choice of k ∈ N such that
j
p
−
kl
p
−
l(l − 1)
2p
< δ (mod s) (1.71)
for j in (1.64) determined by x0. From (1.64), (1.67) (consider the deﬁnition of ε), (1.68),
(1.70), and (1.71), we have
d(xk+l, xk) ≥
ϑ(x0)
2
for at least one k ∈ N. But, at the same time, we have (1.69). This contradiction gives that
{xk} cannot be almost periodic. See also the proof of the ﬁrst part of [176, Proposition 2],
where almost periodic unitary systems are studied. ♦
Chapter 2
Solutions of almost periodic
diﬀerence systems
In this chapter, we study almost periodic solutions of the almost periodic homogeneous
linear diﬀerence systems
xk+1 = Ak · xk. (2.1)
Our aim is to analyse the systems (2.1) which have no non-trivial almost periodic solution.
We are motivated by paper [176], where unitary systems (determined by unitary matrices
Ak) are studied. One of the main results of [176] says that the systems whose solutions
are not almost periodic form an everywhere dense subset in the space of all considered
unitary systems. We remark that important partial cases of the theorem and the process
are mentioned in [169] and [63, 104, 172], respectively.
In the proof of this result, it is substantially used that the group of considered matrices
is not commutative. Thus, e.g., the dimension of the systems has to be at least two. We
use methods based on our general constructions, because we want to generalize the result
also for commutative groups of matrices (especially, for the scalar case). It implies that
we can treat the problem in a general setting. Scalar sequences attain values in a metric
space on an inﬁnite ﬁeld with continuous operations with respect to the metric similarly as
scalar discrete processes in [45], where the main results are proved only for real or complex
entries.
The almost periodicity of solutions of almost periodic linear diﬀerence equations is
studied in [5] and [188] (non-homogeneous systems). We can also refer again to [45].
Explicit almost periodic solutions are obtained for a class of these equations in [83]. For
other properties of (complex) almost periodic linear diﬀerence systems, see [15, 110, 148].
For diﬀerence systems of general forms, criteria of the existence of almost periodic solutions
are presented in [26, 162, 186, 189]. The existence of an almost periodic sequence of
solutions for an almost periodic diﬀerence equation is discussed in [93] (and [87] as in
[189]). Concerning the existence theorems for almost periodic solutions of almost periodic
delay diﬀerence systems, see [70] or [190] (methods and techniques from that paper are
similarly used and developed in [191]).
This chapter is organized as follows. We begin with notations which are used throughout
this chapter. Then we introduce general homogeneous linear diﬀerence systems and
33
2.1 Preliminaries 34
a metric in the space of all considered almost periodic systems. To formulate our results
in a simple and consistent form, we introduce the concepts of transformable and weakly
transformable groups of matrices. One of the conditions in the deﬁnition of transformable
groups means that it is possible to transform any matrix into any other using ﬁnitely
many arbitrarily small “jumps” in the complex case (for usual metrics on the group of
considered matrices). We show that the group of all unitary matrices, the group of all
orthogonal matrices of a dimension at least two with determinant 1, and some their subgroups
are transformable. In addition, examples of weakly transformable groups (which
are not transformable) are given.
In Section 2.3, we present a condition on weakly transformable groups ensuring that, in
any neighbourhood of every considered system, there exists a system which does not possess
an almost periodic solution other than the trivial one. Analogously, the corresponding
Cauchy problem is explicitly solved as well.
2.1 Preliminaries
We use the following notations: N0 for the set of positive integers including the zero;
R+
for the set of all positive reals; R+
0 for the set of all non-negative real numbers; and
symbol i for the imaginary unit. Let F = (F, ⊕, ) be an inﬁnite ﬁeld with a unit and
a zero denoted as e1 and e0, respectively; and let m ∈ N be arbitrarily given. Henceforth,
we consider m as the dimension of diﬀerence systems under consideration.
Symbol Mat(F, m) denotes the set of all m × m matrices with elements from F and
Fm
the set of all m × 1 vectors with entries attaining values from F. As usual, we deﬁne
the identity matrix I and the zero matrix O. Analogously, for the trivial vector, we put
o := [e0, e0, . . . , e0]T
∈ Fm
. Since F is a ﬁeld, we have the notion of the non-singular
matrices from Mat(F, m). For any invertible matrix U, we denote the inverse matrix
as U−1
. For arbitrary Uj, . . . , Uj+n ∈ Mat(F, m), j ∈ Z, n ∈ N, we deﬁne
j+n
i=j
Ui := Uj · Uj+1 · · · Uj+n,
j
i=j+n
Ui := Uj+n · Uj+n−1 · · · Uj.
Let be a metric on F and assume that the operations ⊕ and are continuous with
respect to and that the metric space (F, ) is complete. The metric induces the metric
in Fm
and Mat(F, m) as the sum of m and m2
non-negative numbers given by in F,
respectively. We also denote these metrics as . For any ε > 0 and α from a metric space,
the ε-neighbourhood of α is denoted by Oε (α). Note that the continuity of ⊕ and
implies that the multiplication · of matrices from Mat(F, m) (and U · v, U ∈ Mat(F, m),
v ∈ Fm
) is continuous. All considered sequences are deﬁned for k ∈ Z (or i, j ∈ N) and
attain values in one of the metric spaces F, Fm
, Mat(F, m) (or C).
2.2 General homogeneous linear diﬀerence systems 35
2.2 General homogeneous linear diﬀerence systems
We consider m-dimensional homogeneous linear diﬀerence equations of the form
xk+1 = Ak · xk, k ∈ Z, (2.2)
where {Ak} is an almost periodic sequence of non-singular matrices from a given inﬁnite
group X ⊂ Mat(F, m). We need the set of all considered Ak to form the group X which
has the below given properties (see Deﬁnitions 2.1 and 2.14 below). The set of all these
almost periodic systems is denoted by symbol AP (X).
We identify the sequence {Ak} with the system (2.2) which is determined by {Ak}. In
the space AP (X), we introduce the metric
σ ({Ak}, {Bk}) := sup
k∈Z
(Ak, Bk) , {Ak}, {Bk} ∈ AP (X).
For ε > 0, symbol Oσ
ε ({Ak}) denotes the ε-neighbourhood of {Ak} in AP (X).
2.2.1 Transformable groups
In this subsection, we introduce the concept of transformable groups (cf. [141, Deﬁnition
2.1]) and give illustrative examples of such groups.
Deﬁnition 2.1. We say that a group X ⊂ Mat(F, m) is transformable if the following
conditions are fulﬁlled.
(i) For any L ∈ R+
and ε > 0, there exists p = p(L, ε) ∈ N with the property that,
for any n ≥ p (n ∈ N) and any sequence {C0, C1, . . . , Cn} ⊂ X, L ≤ (Ci, O),
i ∈ {0, 1, . . . , n}, one can ﬁnd a sequence {D1, . . . , Dn} ⊂ X for which
Di ∈ Oε (Ci), i ∈ {1, . . . , n}, Dn · · · D2 · D1 = C0.
(ii) The multiplication of matrices is uniformly continuous on X and has the Lipschitz
property on a neighbourhood of I in X. Especially, for every ε > 0, there exists
η = η(ε) > 0 such that
C · D, D · C ∈ Oε (C) if C ∈ X, D ∈ Oη(I) ∩ X;
and there exist ζ > 0 and P ∈ R+
such that
C · D, D · C ∈ OεP (C) if C ∈ Oζ (I) ∩ X, D ∈ Oε (I) ∩ X, ε ∈ (0, ζ).
(iii) For any L ∈ R+
, there exists Q = Q(L) ∈ R+
with the property that, for every ε > 0
and C, D ∈ X OL(O) satisfying C ∈ Oε (D), it is valid that
C−1
· D, D · C−1
∈ OεQ(I).
2.2 General homogeneous linear diﬀerence systems 36
For simplicity, in the below mentioned examples, we consider only the complex or real
case and we speak about the classical case. Henceforth, we use known results of matrix
analysis which can be found, e.g., in [80, 82, 94, 106].
Let us consider
((F, ⊕, ), (·, ·)) = ((C, +, ·), | · − · |) or ((F, ⊕, ), (·, ·)) = ((R, +, ·), | · − · |)).
Evidently, the multiplication of matrices satisﬁes the Lipschitz condition on any set OK(O).
For an arbitrary matrix norm (especially, for the l1-norm) denoted by || · ||, we have
A−1
− (A + E)−1
≤
|| A−1
· E ||
1 − || A−1 · E ||
A−1
(2.3)
for any matrices A, E such that A is invertible and || A−1
E || < 1. If we have a bounded
group X ⊂ Mat(C, m), then from (2.3) it follows that the map C → C−1
, C ∈ X, has the
Lipschitz property as well. Hence, the condition (iii) is satisﬁed.
Thus, the conditions (ii) and (iii) are fulﬁlled for any bounded group X ⊂ Mat(C, m).
Further, for any bounded group X, there exists ε > 0 for which X ∩ Oε (O) = ∅. At the
same time, from the condition (i), we know that X ∩ O1(O) = ∅ for any transformable
group X ⊂ Mat(C, m). Indeed, it suﬃces to consider C0 = I and the constant sequence
{C1, . . . , Cn} given by a matrix C such that || C || < 1.
Let ε > 0, a bounded group X ⊂ Mat(C, m), and C0, C1, . . . , Cn ∈ X be arbitrarily
given. The uniform continuity of the multiplication of matrices on X implies the existence
of η = η(ε) > 0 such that C D, D C ∈ Oε (C) if D ∈ Oη(I) ∩ X, C ∈ X. We deﬁne the
maps H1, H2 on X × X by
H1 ((C, D)) := C · D · C−1
, H2 ((C, D)) := C−1
· D · C.
Since H1, H2 satisfy the Lipschitz condition, there exists R ∈ R+
such that the ranges of
{C} × Oη/R(I) ∩ X in both of H1 and H2 are subsets of Oη(I) for all C ∈ X.
If we replace
1
i1=n
Fi1 ·
n
i2=1
Ci2 by
n
i=1
Ei · Ci,
where
F1 = H1 ((I, E1)) , F2 = H1 ((E1 · C1, E2)) ,
. . . Fn = H1 ((E1 · C1 · · · En−1 · Cn−1, En)) ,
we see that Fi ∈ Oη/R(I) ∩ X, i ∈ {1, . . . , n}, implies Ei ∈ Oη(I), i ∈ {1, . . . , n}. Thus,
from the existence of matrices F1, . . . , Fn ∈ Oη/R(I) ∩ X for which
1
i1=n
Fi1 ·
n
i2=1
Ci2 = C0,
it follows the existence of matrices D1, . . . , Dn ∈ X satisfying
Di ∈ Oε (Ci), i ∈ {1, . . . , n}, Dn · · · D2 · D1 = C0.
2.2 General homogeneous linear diﬀerence systems 37
It means that a bounded group X ⊂ Mat(C, m) is transformable if, for any suﬃciently
small ε > 0, there exists p (ε) ∈ N such that, for all n ≥ p (ε) (n ∈ N), any matrix from X
can be expressed as a product of n matrices from Oε (I) ∩ X.
We point out that several processes in the proofs of the below given results can be
simpliﬁed in the classical case. For example, one can use that, for any ε > 0, K ∈ R+
, and
n ∈ N, there exists ξ = ξ(ε, K, n) > 0 for which
(M1 · M2 · · · Mn, O) < ε, M1, M2, . . . , Mn ∈ OK(O),
and
(M1 · · · Mn · u, o) < ε, M1, . . . , Mn ∈ OK(O), u ∈ OK(o),
if we have Mi ∈ Oξ (O) for at least one i ∈ {1, . . . , n} and u ∈ Oξ (o), respectively.
Now we mention important examples of transformable groups.
Example 2.2. The group of all unitary matrices is transformable. Obviously, it suﬃces to
show that, for every ε > 0, any unitary matrix can be obtained as the n-th power of some
unitary matrix from the ε-neighbourhood of I for all suﬃciently large n ∈ N. To show this,
let ε > 0, n ∈ N, and a m × m unitary matrix U with eigenvalues exp (iλ1) , . . . , exp (iλm),
where λ1, . . . , λm ∈ [−π, π), be arbitrarily given. We have
U = W · J · W∗
for some unitary matrix W = W(U),
where J = diag (exp (iλ1) , . . . , exp (iλm)) and W∗
denotes the conjugate transpose of W.
We ﬁnd a unitary matrix V for which V n
= U. By
W∗
· V n
· W = (W∗
· V · W)n
= J,
we obtain
V = W · diag (exp (iλ1/n) , . . . , exp (iλm/n)) · W∗
.
Since the multiplication of matrices is uniformly continuous on the set of all unitary matrices,
it remains to consider suﬃciently large n ∈ N. ♦
Example 2.3. Let m ≥ 2 and F = R. In this example, we show that the group SO(m)
of m × m orthogonal matrices with determinant 1 is transformable. Analogously as for
unitary matrices, it is enough to prove that any orthogonal matrix U for which det U = 1
is products of n ≥ p (ε), n ∈ N, orthogonal matrices from the ε-neighbourhood of I for
arbitrary ε > 0 and some p (ε) ∈ N. Indeed, it is seen that there exists a neighbourhood
of I which contains only orthogonal matrices with determinant 1.
Let m = 2. Observe that a two-dimensional orthogonal matrix has the form
cos α − sin α
sin α cos α
,
where α ∈ [−π, π), if and only if its determinant is 1. It can be easily computed that
cos α1 − sin α1
sin α1 cos α1
·
cos α2 − sin α2
sin α2 cos α2
=
cos (α1 + α2) − sin (α1 + α2)
sin (α1 + α2) cos (α1 + α2)
2.2 General homogeneous linear diﬀerence systems 38
for α1, α2 ∈ R and that, consequently, any this matrix (for some α ∈ [−π, π)) can be
obtained as the n-th power of the orthogonal matrix of this type given by the argument
α/n for all n ∈ N.
Now we use the induction principle with respect to m. Assume that the statement is
true for m − 1 ≥ 2 and prove it for m. Let U be an orthogonal m × m matrix which is not
in any one of the forms
1 oT
o V
,
V o
oT
1
, (2.4)
where V is an orthogonal matrix of dimension m − 1, o ∈ Fm−1
, and suppose that U has
the element on the position (1, m) diﬀerent from 0 (in the opposite case, we put U2 := U in
the below given process). We multiply U from the left by an orthogonal matrix U1 which
is in the second form from (2.4) and satisﬁes that U2 := U1 ·U has 0 on the position (1, m).
For U2, we deﬁne an orthogonal matrix U3 so that the m-th row of U3 is the last column
of U2 and so that the ﬁrst column and the ﬁrst row of U3 are zero except the number 1 on
the position (1, 1). Obviously, the product U4 := U3 · U2 is equal to a matrix which has the
second form from (2.4). Summarizing, we get U = UT
1 · UT
3 · U4. Thus, one can express any
orthogonal matrix U as a product of at most three matrices of the forms given in (2.4).
Further, the matrices of this product can be evidently chosen so that the determinant of
all of them is 1 if the determinant of the given matrix is 1 as well. Now the induction
hypothesis gives the validity of the above statement. ♦
Example 2.4. Let a unitary matrix S be given. Let XS be the set of the unitary matrices
which are simultaneously diagonalizable for the single similarity matrix S, i.e., let
XS = S−1
· diag (exp (iλ1) , . . . , exp (iλm)) · S; λ1, . . . , λm ∈ [−π, π) .
Obviously, XS is a subgroup of the m×m unitary group (diﬀerent from the group if m ≥ 2).
Since diagonalizable (normal) matrices are simultaneously (unitarily) diagonalizable if and
only if they commute under multiplication, XS is a commutative group. Analogously as
in Example 2.2, one can show that XS is transformable. Further, XS is transformable also
for arbitrary non-singular matrix S. Especially, a transformable set does not need to be
a subgroup of the m × m unitary group. ♦
Example 2.5. Now we consider the set of the unitary matrices with the determinant in the
form exp (ir), r ∈ Q or r ∈ Z. Evidently, these matrices form a group as well. Considering
diagonalizations of unitary matrices and the uniform continuity of the multiplication of
unitary matrices, we get that this group is dense in the group of all unitary matrices.
Thus (see Example 2.2), it satisﬁes (i). Finally, it is transformable. In general, any dense
subgroup of a transformable set is transformable as well. ♦
Example 2.6. Let a unitary matrix S be given. Analogously as in Examples 2.4 and 2.5,
we can show that the group
{S∗
· diag (exp (iλ1) , . . . , exp (iλm)) · S; λ1, . . . , λm ∈ Q}
is transformable. In general, the matrices with eigenvalues in the form exp (ir), where r ∈ Q
or r ∈ Z, from a given commutative transformable subgroup of the m × m unitary group
2.2 General homogeneous linear diﬀerence systems 39
form a transformable group if it is inﬁnite. Indeed, if complex matrices A, B commute and
have eigenvalues λ1, . . . , λm and µ1, . . . , µm, respectively, then the eigenvalues of A B are
λ1µj1 , λ2µj2 , . . . , λmµjm for some permutation j1, . . . , jm of the indices 1, . . . , m. ♦
Now we consider a bounded group X ⊂ Mat (C, m) or X ⊂ Mat (R, m) satisfying
X ∩ OL (O) = ∅ for some L > 0 and q(δ) ∈ N for any δ > 0 such that, for all C ∈ X and
l ≥ q(δ), l ∈ N, there exist matrices C1, . . . , Cl ∈ X with the property that
C1 ∈ Oδ (I), Cj ∈ Oδ (Cj+1) , j ∈ {1, . . . , l − 1}, Cl ∈ Oδ (C). (2.5)
This implies (see (iii))
C−1
j · Cj+1 ∈ OδQ(L)(I) ∩ X, j ∈ {1, . . . , l − 1}, C−1
l · C ∈ OδQ(L)(I) ∩ X.
For δ = ε/Q(L) > 0, we have
C1 ∈ Oδ (I) ⊆ Oε (I), C−1
j · Cj+1 ∈ Oε (I), j ∈ {1, . . . , l − 1}, C−1
l · C ∈ Oε (I).
Finally, since
C1 · C−1
1 · C2 · C−1
2 · C3 · · · C−1
l−1 · Cl · C−1
l · C = C,
the above mentioned condition is fulﬁlled (if one puts p(ε) = q(ε/Q(L)) + 1). Therefore,
the group X is transformable.
Example 2.7. Let us show that the special unitary group SU(m) (the group of all m×m
unitary matrices with determinant 1) is transformable for m ≥ 2, applying the implication
mentioned above (see (2.5)) for arbitrarily given C = I, C ∈ SU(m).
There exists an unitary matrix U such that
C = U∗
· D · U, D = diag (λ1, . . . , λm) ,
where λ1 · · · λm = 1. Let ϕ1, . . . , ϕm ∈ [0, 2π] be such that λj = eiϕj
, j ∈ {1, . . . , m}. Since
m
j=1
eiϕj
= e
i
m
j=1
ϕj
= 1,
we have ϕ1 +· · ·+ϕm ≡ 0 (mod 2π), i.e., ϕ1 +· · ·+ϕm = 2kπ for some k ∈ {1, . . . , m−1}.
Evidently, for an arbitrarily given δ > 0, there exists ε > 0 with the property that we can
change ϕj, j ∈ {1, . . . , m}, into ϕj(εj) := ϕj + εj, where
m
j=1
εj = 0, ε1, . . . , εk ∈ [0, ε), εk+1, . . . , εm ∈ (−ε, 0],
so that
U∗
· D · U, U∗
· diag eiϕ1(ε1)
, . . . , eiϕm(εm)
· U < δ. (2.6)
Indeed, it suﬃces to consider that the multiplication of matrices is uniformly continuous
on the m × m unitary group. We obtain the matrix
U∗
· diag eiϕ1(ε1)
, . . . , eiϕm(εm)
· U ∈ SU(m),
2.2 General homogeneous linear diﬀerence systems 40
because ϕ1(ε1) + · · · + ϕm(εm) = 2kπ.
Let n ∈ N be such that n > 2π/ε. We put
ε1
1 :=
1
n
(2π − ϕ1) , . . . ε1
k :=
1
n
(2π − ϕk) , ε1
k+1 := −
1
n
ϕk+1, . . . ε1
m := −
1
n
ϕm,
ε2
1 :=
2
n
(2π − ϕ1) , . . . ε2
k :=
2
n
(2π − ϕk) , ε2
k+1 := −
2
n
ϕk+1, . . . ε2
m := −
2
n
ϕm,
...
εn−1
1 :=
n − 1
n
(2π − ϕ1) , . . . εn−1
k :=
n − 1
n
(2π − ϕk) ,
εn−1
k+1 := −
n − 1
n
ϕk+1, . . . εn−1
m := −
n − 1
n
ϕm,
εn
1 := 2π − ϕ1, . . . εn
k := 2π − ϕk, εn
k+1 := −ϕk+1, . . . εn
m := −ϕm.
Let us consider the matrices
C1 := U∗
· diag eiϕ1(ε1
1), . . . , eiϕm(ε1
m) · U ∈ SU(m),
C2 := U∗
· diag eiϕ1(ε2
1), . . . , eiϕm(ε2
m) · U ∈ SU(m),
...
Cn−1 := U∗
· diag eiϕ1(εn−1
1 ), . . . , eiϕm(εn−1
m ) · U ∈ SU(m),
Cn := U∗
· diag eiϕ1(εn
1 ), . . . , eiϕm(εn
m)
· U = I.
We have (see (2.6))
C1 ∈ Oδ (C), C2 ∈ Oδ (C1) , . . . Cn−1 ∈ Oδ (I) ,
i.e., we can choose q(δ) > 2π/ε.
To show the transformability of SU(m), it suﬃces to consider that the group SU(m)
is connected (path-connected) and compact (totally bounded) for each m (see, e.g., [76]).
Nevertheless, using the above mentioned process, one can show the transformability of
other matrix groups. For example, the group of the unitary matrices with determinant eir
for some r ∈ Q is transformable as well. Indeed, for any δ > 0 and an arbitrary unitary
matrix C = I with determinant eir
, where r ∈ Q, there exists a matrix C0 satisfying
C0 ∈ Oδ (C), | C0 | = eir(C)
for some r(C) ∈ Q ∩ [0, 2π) and one can analogously ﬁnd n ∈ N
and unitary matrices C1, . . . , Cn such that
C1 ∈ Oδ (C0), C2 ∈ Oδ (C1) , . . . Cn ∈ Oδ (Cn−1)
and that
| C1 | = eir(C) n−1
n , | C2 | = eir(C) n−2
n , . . . | Cn | = 1.
♦
2.2 General homogeneous linear diﬀerence systems 41
Example 2.8. We consider Hermitian symplectic matrices of dimension m = 2n, n ∈ N.
At ﬁrst, we recall that a complex matrix S is said to be symplectic provided
S∗
· J · S = J, where J =
O I
−I O
.
The set Sp(n) of the Hermitian symplectic matrices is the intersection of the set of all
2n × 2n symplectic matrices and the set of all 2n × 2n unitary matrices. In fact (see [91]),
the Hermitian symplectic matrices are the matrices of the form
A −B
B A
with the property that
A∗
A + B∗
B = I, A∗
B = B∗
A.
It is known (see, e.g., [76]) that Sp(n) is a compact and simply connected group. This fact
implies (2.5), i.e., Sp(n) is a transformable group. ♦
Example 2.9. Let Xj ⊂ Mat (F, mj) for j ∈ {1, . . . , n} be transformable groups. The
direct sum n
j=1
Xj = X1 ⊕ X2 ⊕ · · · ⊕ Xn,
where M ∈ n
j=1 Xj if it is of the form
M =





M1 O · · · O
O M2 · · · O
...
...
...
...
O O · · · Mn





, M1 ∈ X1, M2 ∈ X2, . . . , Mn ∈ Xn,
is a transformable group as well. Condition (i) is fulﬁlled, because we can choose the same
p(L, ε) for all Xj; condition (ii) is obviously satisﬁed; to verify condition (iii), it suﬃces to
take into account that Xj ∩ O1 (O) = ∅, j ∈ {1, . . . , n}.
The importance of the direct sum of transformable groups lies, i.a., in many isomorphisms
of groups with applications in physics. For example, the spin group Spin(4) (see,
e.g., [178]) is isomorphic to SU(2)⊕SU(2), and H. Georgi and S. Glashow use the isomorphism
of SU(3) ⊕ SU(2) ⊕ U(1) to a subgroup of SU(5) for the Georgi–Glashow model
in [81]. ♦
Example 2.10. Now we show that the intersection of circulant matrices and unitary
matrices form a transformable group. A complex matrix is called circulant if it has the
form of 






a1 a2 a3 · · · am
am a1 a2 · · · am−1
am−1 am a1 · · · am−2
...
...
...
...
...
a2 a3 a4 · · · a1







.
2.2 General homogeneous linear diﬀerence systems 42
A matrix is circulant if and only if it can be written in the form of m
j=1 ajBj−1
, where the
permutation matrix B = (bkl) is given by b12 = b23 = · · · = b(m−1)m = bm1 = 1. Because
of Bm
= I, the product of two circulant matrices is a circulant matrix. The multiplication
of circulant matrices is commutative and the Hermitian adjoint of any circulant matrix
is circulant. Therefore, every circulant matrix is normal. A normal matrix is unitary if
and only if all its eigenvalues have the absolute value of 1. Hence, the set of the circulant
matrices, whose all eigenvalues have absolute value 1, is a group. We denote this group
by CU(m).
It is known (see, e.g., [3]) that all circulant matrices have the same eigenvectors. Thus,
there exists a unitary matrix U with the property that any circulant matrix can be expressed
as the product U∗
D U, where D is a diagonal matrix. It means that circulant
matrices are simultaneously diagonalizable for the single similarity matrix U. (In fact,
normal matrices are simultaneously diagonalizable if and only if they commute.) Furthermore,
for every diagonal matrix D, the matrix U∗
D U is circulant. Altogether, we have
that A ∈ CU(m) if and only if A = U∗
D U for some
D = diag (d1, d2, . . . , dm) , | dj | = 1, j ∈ {1, . . . , m}.
The fact that CU(m) is transformable comes from Example 2.4. Indeed, CU(m) = XU .
Evidently, CU(1) = U(1). The group CU(2) is just formed by symmetric unitary
matrices given by complex numbers a = a1 + a2i, b = b1 + b2i (a1, a2, b1, b2 ∈ R) in the ﬁrst
row for which |a|2
+ |b|2
= 1, a1b1 = −a2b2. Obviously, for any a ∈ C satisfying |a| ≤ 1,
one can ﬁnd b so that the above equalities are fulﬁlled. It is seen that b can be chosen, in
addition, so that the function a → b is continuous. Directly from this observation, we get
condition (i). We remark that, for m > 2, the symmetric unitary matrices do not form
a group, because the product of symmetric matrices is a symmetric matrix if and only if
they commute under multiplication. ♦
2.2.2 Strongly and weakly transformable groups
In this subsection, we deﬁne modiﬁcations of transformable groups—strongly and weakly
transformable groups. Before introducing strongly transformable groups, we mention an
auxiliary result.
Lemma 2.11. If X is transformable, then, for any {Li}i∈N ⊂ R+
and j ≥ 2, j ∈ N, one
can ﬁnd {εi}i∈N ≡ {εi({Li}, j)}i∈N ⊂ R+
satisfying
∞
i=1
εi < ∞,
ji
≥ p Lg(i), εi for inﬁnitely many i ∈ N, some g(i) ∈ N, (2.7)
where g(i) → ∞ as i → ∞.
Proof. The lemma follows directly from (i). Indeed, one can put
εi := 2−k
, i = f(k) for some k ∈ N;
εi := 2−i
, i /∈ {f(k); k ∈ N}, i ∈ N
2.2 General homogeneous linear diﬀerence systems 43
for arbitrarily given increasing discrete function f : N → N with the property that
jf(k)
≥ p Lk, 2−k
, k ∈ N,
whose inverse function is considered in (2.7) as g.
Of course, inequality (2.7) does not need to be true for all i ∈ N or for a set of i which
is relatively dense in N. This fact motivates the following deﬁnition.
Deﬁnition 2.12. A group X is strongly transformable if it is transformable and if for any
L ∈ R+
, there exist j = j(L) ∈ N and a sequence {εi}i∈N ≡ {εi(L)}i∈N ⊂ R+
such that
∞
i=1
εi < ∞, (2.8)
ji
≥ p (L, εi) for all i ∈ N. (2.9)
Example 2.13. Since we consider only maps which satisfy the Lipschitz condition in the
above examples (in the classical case) and since we can choose
p (L, ε, m + 1) ≤ 3p (L, ε, m)
in (i) when we use the induction principle with respect to m (see Example 2.3), all concrete
transformable groups of matrices mentioned in Examples 2.2–2.10 are actually strongly
transformable. ♦
Example 2.13 shows that several transformable groups used in applications are strongly
transformable. Nevertheless, if we change the metric in the examples above in a neighbourhood
of −1 (in R or C) so that the l−l
-neighbourhood of −1 becomes the l−1
-neighbourhood
of −1 for all suﬃciently large l ∈ N (and the rest remains unchanged), then all mentioned
groups are still transformable and none of them is strongly transformable.
Since certain matrix groups (important in applications) are not transformable, although
they possess transformable subgroups, we introduce the following generalization of the
transformability.
Deﬁnition 2.14. A matrix group X ⊂ Mat (F, m) is weakly transformable if there exist
a transformable group X0 ⊆ X, matrices X1, . . . , Xl ∈ X, and δX > 0 such that
(I) any U ∈ X can be expressed as U = C(U) · Xj for some C(U) ∈ X0, j ∈ {1, . . . , l},
and that
(II) (C · Xi, D · Xj) > δX for all C, D ∈ X0, i = j, i, j ∈ {1, . . . , l}.
Again, we give important examples of the considered type of matrix groups in the
complex and real case.
2.2 General homogeneous linear diﬀerence systems 44
Example 2.15. We use the fact that the group SO(m) of all real orthogonal matrices of
a dimension m ≥ 2 with determinant 1 is transformable to prove that the group O(m) of
all real orthogonal matrices is weakly transformable. We put
X0 := SO(m), X1 := I, X2 :=





−1 0 · · · 0
0 1 · · · 0
...
...
...
...
0 0 · · · 1





.
If U ∈ O(m) SO(m), then U = C(U) X2, where C(U) = U X−1
2 ∈ SO(m). Indeed,
| U | = −1, | X−1
2 | = −1. This gives us condition (I). The mapping C → | C | is continuous
on O(m). Thus, there exists δ > 0 such that
(C1, C2) > δ if C1, C2 ∈ O(m), | C1 | = 1, | C2 | = −1,
i.e., (C X1, D X2) > δ for any C, D ∈ X0 (consider | C X1 | = 1, | D X2 | = −1). ♦
Example 2.16. Let X1 be a weakly transformable group and let X2 be an arbitrary ﬁnite
matrix group. We can directly see that the direct sum X1 ⊕ X2 is weakly transformable.
Thus, the group O(2) ⊕ Ok(2), where
Ok(2) = {I, T, . . . , Tk−1
, S, ST, . . . , STk−1
}
and
T :=
cos 2π
k
sin 2π
k
− sin 2π
k
cos 2π
k
, S :=
0 1
1 0
,
is weakly transformable for all k ∈ N. For the representation of this group, see [76]; for
collections of ﬁnite groups, we refer to [16, 43]. ♦
Example 2.17. Evidently, any direct sum X1 ⊕X2 is weakly transformable if both X1 and
X2 are weakly transformable. Especially, the group O(m) ⊕ O(m) is weakly transformable
for m ≥ 2. This group is isomorphic to the perplectic orthogonal group which is denoted
by PO(2m) and deﬁned as
PO(2m) = {P ∈ O(2m); R · P = P · R} ,
where
R :=





0 · · · 0 1
0 · · · 1 0
... ... ...
...
1 · · · 0 0





.
Note that PO(2m) is exactly the set of all centro-symmetric orthogonal matrices.
Let us show that the group PO(3) is weakly transformable. In [58], there is shown that
PO(3) consists of {W(ϕ); ϕ ∈ [0, 2π]} and the components





0 0 1
0 −1 0
1 0 0

 · W(ϕ); ϕ ∈ [0, 2π]



,





1 0 0
0 −1 0
0 0 1

 · W(ϕ); ϕ ∈ [0, 2π]



,
2.2 General homogeneous linear diﬀerence systems 45





0 0 1
0 1 0
1 0 0

 · W(ϕ); ϕ ∈ [0, 2π]



,
where
W(ϕ) :=
1
2


cos ϕ + 1
√
2 sin ϕ cos ϕ − 1
−
√
2 sin ϕ 2 cos ϕ −
√
2 sin ϕ
cos ϕ − 1
√
2 sin ϕ cos ϕ + 1

 .
Since (W(ϕ))−1
= W(−ϕ), ϕ ∈ [0, 2π], we can express PO(3) by the following compo-
nents
PO1(3) = {W(ϕ); ϕ ∈ [0, 2π]} ,
PO2(3) =



W(ϕ) ·


0 0 1
0 −1 0
1 0 0


−1
; ϕ ∈ [0, 2π]



,
PO3(3) =



W(ϕ) ·


1 0 0
0 −1 0
0 0 1


−1
; ϕ ∈ [0, 2π]



,
PO4(3) =



W(ϕ) ·


0 0 1
0 1 0
1 0 0


−1
; ϕ ∈ [0, 2π]



.
We put (see Deﬁnition 2.14)
X0 := PO1(3), X1 := I, X2 :=


0 0 1
0 −1 0
1 0 0

 = X−1
2 ,
X3 :=


1 0 0
0 −1 0
0 0 1

 = X−1
3 , X4 :=


0 0 1
0 1 0
1 0 0

 = X−1
4 .
The transformability of PO1(3) follows from the fact that PO1(3) is a connected matrix
subgroup of O(3) (see [58]). Let us explain it in detail. For any ϕ ∈ [0, 2π] and suﬃciently
large n ∈ N, we consider the sequence
ϕ
n
, 2
ϕ
n
, . . . (n − 1)
ϕ
n
, ϕ.
Obviously, for every δ > 0, there exists q ∈ N satisfying
W
ϕ
l
∈ Oδ (I), W 2
ϕ
l
∈ Oδ W
ϕ
l
, . . .
W (l − 1)
ϕ
l
∈ Oδ W (l − 2)
ϕ
l
, W (l − 1)
ϕ
l
∈ Oδ (W (ϕ))
for all ϕ ∈ [0, 2π] and l ≥ q. Now it suﬃces to realize (2.5).
It remains to show that the components have a positive distance. Let ϕ, ψ ∈ [0, 2π]
and Wj(·) ∈ POj(3), j ∈ {1, 2, 3, 4}. Since the entries of these matrices are continuous
functions deﬁned on the compact set [0, 2π], the sets POj(3), j ∈ {1, 2, 3, 4}, are also
compact. Therefore, it suﬃces to show that these components are disjoint.
2.3 Systems without almost periodic solutions 46
(1 & 2) Suppose that there exist ϕ, ψ ∈ [0, 2π] such that W1(ϕ) = W2(ψ), i.e.,


cos ϕ + 1
√
2 sin ϕ cos ϕ − 1
−
√
2 sin ϕ 2 cos ϕ −
√
2 sin ϕ
cos ϕ − 1
√
2 sin ϕ cos ϕ + 1

 =


cos ψ − 1 −
√
2 sin ψ cos ψ + 1
−
√
2 sin ψ −2 cos ψ −
√
2 sin ψ
cos ψ + 1 −
√
2 sin ψ cos ψ − 1

 .
Considering the entries in the ﬁrst row and the ﬁrst column, we have ϕ = π
and ψ ∈ {0, 2π}. On the other hand, from entries in the ﬁrst row and the third
column, we obtain ϕ ∈ {0, 2π} and ψ = π. This gives a contradiction.
(1 & 3) Again by contradiction, we suppose that there exist ϕ, ψ ∈ [0, 2π] such that
W1(ϕ) = W3(ψ), i.e.,


cos ϕ + 1
√
2 sin ϕ cos ϕ − 1
−
√
2 sin ϕ 2 cos ϕ −
√
2 sin ϕ
cos ϕ − 1
√
2 sin ϕ cos ϕ + 1

 =


cos ψ + 1 −
√
2 sin ψ cos ψ − 1
−
√
2 sin ψ −2 cos ψ −
√
2 sin ψ
cos ψ − 1 −
√
2 sin ψ cos ψ + 1

 .
At ﬁrst, we pay our attention to the entries on the positions (1, 2) and (2, 1). We
obtain
sin ϕ = − sin ψ = − sin ϕ, i.e., ϕ, ψ ∈ {0, π, 2π}.
Next, the entries (3, 1) and (2, 2) give
cos ϕ = cos ψ = − cos ϕ, i.e., ϕ, ψ ∈ {π/2, 3π/2} .
Hence, the components PO1(3) and PO3(3) are disjoint as well.
Finally, the cases (1 & 4), (2 & 3), and (3 & 4) are analogous to (1 & 2) and (2 & 4) is
analogous to (1 & 3). ♦
2.3 Systems without almost periodic solutions
Now we can consider the gist of this chapter. At ﬁrst, we prove an auxiliary result and
a result concerning strongly transformable groups which is generalized later in this work.
Lemma 2.18. If an almost periodic sequence of non-singular Ak ∈ Mat(F, m) is such
that, for any ε > 0, there exists i = i(ε) ∈ Z for which (O, Ai) < ε, then the system
xk+1 = Ak xk, k ∈ Z, does not have a non-trivial almost periodic solution.
Proof. By contradiction, suppose that we have an almost periodic sequence {Ak}, a sequence
{hi}i∈N ⊆ Z satisfying
(O, Ahi
) <
1
i
, i ∈ N, (2.10)
and a non-trivial almost periodic solution {xk} of the system xk+1 = Ak xk. Using Corollary
1.5, we obtain uniformly convergent common subsequences
{{Ak+˜hi
}k∈Z}i∈N and {{xk+˜hi
}k∈Z}i∈N
2.3 Systems without almost periodic solutions 47
of the sequences {{Ak+hi
}k∈Z}i∈N and {{xk+hi
}k∈Z}i∈N. The limits are denoted as {Bk}
and {yk}.
We put ε := (x0, o) /2 > 0. Because of the almost periodicity of {xk}, there exists
some p (ε) from Deﬁnition 1.1. We consider the sets Ni := {i + 1, i + 2, . . . , i + p (ε)} for
i ∈ Z. Any one of the sets Ni contains a number l ∈ T({xk}, ε). Thus,
xl /∈ Oε (o). (2.11)
From (2.10) it follows that B0 = O. Since the multiplication of matrices is continuous, one
can ﬁnd ϑ > 0 for which
Cj · · · C0 · y ∈ Oε (o), j = 0, 1, . . . , p (ε) − 1,
if y ∈ Oϑ(y0) and Ci ∈ Oϑ(Bi), i ∈ {0, . . . , j}. There exists i ∈ N such that
Ak+˜hi
, Bk < ϑ, xk+˜hi
, yk < ϑ, k ∈ Z.
Therefore,
xj+˜hi
∈ Oε (o), j = 1, . . . , p (ε) . (2.12)
Indeed, it is valid
xj+˜hi
= Aj+˜hi−1 · · · A˜hi
· x˜hi
, j = 1, . . . , p (ε) .
This contradiction (compare (2.11) with (2.12)) gives the proof.
Theorem 2.19. Let X be strongly transformable. Let {Ak} ∈ AP(X) and ε > 0 be
arbitrarily given. If there exist L ∈ R+
and {Mi}i∈N such that
Mi, M−1
i ∈ X OL(O), i ∈ N, (2.13)
and that, for any non-zero vector u ∈ Fm
, one can ﬁnd i = i(u) ∈ N with the property that
Mi u = u, then there exists {Bk} ∈ Oσ
ε ({Ak}) which does not possess a non-trivial almost
periodic solution.
Proof. If {Ak} has a non-trivial almost periodic solution, then there exists K ∈ R+
such
that (Ak, O) > K for all k. Indeed, it follows from Lemma 2.18. Since it suﬃces to
consider only very small ε > 0, we can assume without loss of generality that
L + ε < (I, O) , L + ε < (Ak, O) , k ∈ Z; (2.14)
otherwise we can put Bk := Ak, k ∈ Z.
Let η = η(ε/2), ζ, P and Q = Q(L) be from (ii) and (iii), respectively. Further, let
η < ε < ζ and let {εi}i∈N ⊂ R+
, n ∈ N, and j ≥ 2 (j ∈ N) satisfy
∞
i=1
εi <
η
PQ
, (2.15)
ji
(n + 1) ≥ p (L, εi) for all i ∈ N. (2.16)
2.3 Systems without almost periodic solutions 48
The inequality (2.15) follows from (2.8) if we omit ﬁnitely many values of εi, and (2.16)
from the fact that n and j can be arbitrarily large and from (2.9). We remark that P, Q ≥ 1.
We put
Bk := Ak, Ck := I for k ∈ {0, 1, . . . , n}
and we choose
Bk = Ak · Ck for some Ck ∈ Oε2Q Ck−(n+1) ∩ X, k ∈ {n + 1, . . . , 2(n + 1) − 1},
...
Bk = Ak · Ck, Ck ∈ Oε2Q Ck−j4(n+1) ∩ X, k ∈ {j4
(n + 1), . . . , (j4
+ 1)(n + 1) − 1},
arbitrarily such that
n+1
k=(j2+1)(n+1)−1
Bk = M1,
n+1
k=(j3+j2)(n+1)−1
Bk =
n+1
k=(j4+1)(n+1)−1
Bk = I.
For
C1, . . . , Cn, D1, . . . , Dn ∈ X, Di ∈ Oϑ(Ci),
where ϑ > 0, L + ϑ ≤ (Ci, O), i ∈ {1, . . . , n}, and n ≥ p (L, ϑ), we can express
Dn · · · D2 · D1 = Cn · C−1
n · Dn · · · C1 · C−1
1 · D1 ,
where
(C−1
i · Di) ∈ OϑQ (I) , i ∈ {1, . . . , n}.
Using this fact and considering (2.13), (2.14), and (2.16), we get the existence of the above
matrices Ck.
In the second step, we put
Bk := Ak · Ck+(j4+1)(n+1), k ∈ {−(j4
+ 1)(n + 1), . . . , −1},
...
Bk := Ak · Ck+j4(j4+1)(n+1), k ∈ {−j4
(j4
+ 1)(n + 1), . . . , −(j4
− 1)(j4
+ 1)(n + 1) − 1},
and we denote
Ck := A−1
k · Bk, k ∈ {−j4
(j4
+ 1)(n + 1), . . . , −1}.
Now we choose
Bk = Ak · Ck, Ck ∈ Oε4PQ Ck−(j4+1)2(n+1) ∩ X,
k ∈ {(j4
+ 1)(n + 1), . . . , (j4
+ 1)(n + 1) + (j4
+ 1)2
(n + 1) − 1},
...
2.3 Systems without almost periodic solutions 49
Bk = Ak · Ck, Ck ∈ Oε4PQ Ck−j4(j4+1)2(n+1) ∩ X,
k ∈ {(j4
+ 1 + (j4
− 1)(j4
+ 1)2
)(n + 1), . . . , (j4
+ 1 + j4
(j4
+ 1)2
)(n + 1) − 1},
arbitrarily such that
(j4+1)(n+1)
k=j7(n+1)−1
Bk = M1,
(j4+1)(n+1)
k=(j7+j6−j0)(n+1)−1
Bk = I,
(j4+1)(n+1)
k=j8 (n+1)−1
Bk = M1,
(j4+1)(n+1)
k=(j8+j6−j3)(n+1)−1
Bk = I,
(j4+1)(n+1)
k=j9 (n+1)−1
Bk = M2,
(j4+1)(n+1)
k=(j9+j6−j0)(n+1)−1
Bk = I,
(j4+1)(n+1)
k=j10(n+1)−1
Bk = M2,
(j4+1)(n+1)
k=(j10+j6−j3)(n+1)−1
Bk = I,
(j4+1)(n+1)
k=(j4+1)(n+1)+j4(j4+1)2(n+1)−1
Bk = I.
Such matrices Ck exist. Indeed, we can transform
˜Bk := Ak · Ck−(j4+1)2(n+1),
k ∈ {(j4
+ 1)(n + 1), . . . , (j4
+ 1)(n + 1) + (j4
+ 1)2
(n + 1) − 1},
...
˜Bk := Ak · Ck−j4(j4+1)2(n+1),
k ∈ {(j4
+ 1 + (j4
− 1)(j4
+ 1)2
)(n + 1), . . . , (j4
+ 1 + j4
(j4
+ 1)2
)(n + 1) − 1},
into Bk by
Bk = Ak · Ck−(j4+1)2(n+1) · ˜Ck,
k ∈ {(j4
+ 1)(n + 1), . . . , (j4
+ 1)(n + 1) + (j4
+ 1)2
(n + 1) − 1},
...
Bk = Ak · Ck−j4(j4+1)2(n+1) · ˜Ck,
k ∈ {(j4
+ 1 + (j4
− 1)(j4
+ 1)2
)(n + 1), . . . , (j4
+ 1 + j4
(j4
+ 1)2
)(n + 1) − 1},
where ˜Ck ∈ Oε4Q (I) for all considered k (see (iii)). Hence, we have (see (ii) and also the
below given (2.20) and (2.21))
Ck−(j4+1)2(n+1) · ˜Ck ∈ Oε4PQ Ck−(j4+1)2(n+1) ,
k ∈ {(j4
+ 1)(n + 1), . . . , (j4
+ 1)(n + 1) + (j4
+ 1)2
(n + 1) − 1},
2.3 Systems without almost periodic solutions 50
...
Ck−j4(j4+1)2(n+1) · ˜Ck ∈ Oε4PQ Ck−j4(j4+1)2(n+1) ,
k ∈ {(j4
+ 1 + (j4
− 1)(j4
+ 1)2
)(n + 1), . . . , (j4
+ 1 + j4
(j4
+ 1)2
)(n + 1) − 1}.
Thus, we can obtain Ck from the previous step and from the above ˜Ck.
We put
Bk := Ak · Ck+(j4+1)3(n+1),
k ∈ {−(j4
+ 1)3
(n + 1) − j4
(j4
+ 1)(n + 1), . . . , −j4
(j4
+ 1)(n + 1) − 1},
...
Bk := Ak · Ck+j4(j4+1)3(n+1),
k ∈ {−j4
(j4
+ 1)3
(n + 1) − j4
(j4
+ 1)(n + 1), . . . ,
− (j4
− 1)(j4
+ 1)3
(n + 1) − j4
(j4
+ 1)(n + 1) − 1},
and we denote
Ck := A−1
k · Bk,
k ∈ {−j4
(j4
+ 1)3
(n + 1) − j4
(j4
+ 1)(n + 1), . . . , −j4
(j4
+ 1)(n + 1) − 1}.
We proceed further in the same way. In the (2i − 1)-th step, we choose
Bk = Ak · Ck, Ck ∈ Oε2iPQ Ck−(j4+1)2i−2(n+1) ∩ X,
k ∈ {(j4
+ 1)(n + 1) + j4
(j4
+ 1)2
(n + 1) + · · · + j4
(j4
+ 1)2i−4
(n + 1),
. . . , (j4
+ 1)(n + 1) + j4
(j4
+ 1)2
(n + 1) + · · ·
+ · · · + j4
(j4
+ 1)2i−4
(n + 1) + (j4
+ 1)2i−2
(n + 1) − 1},
...
Bk = Ak · Ck, Ck ∈ Oε2iPQ Ck−j4(j4+1)2i−2(n+1) ∩ X,
k ∈ {(j4
+ 1)(n + 1) + j4
(j4
+ 1)2
(n + 1) + · · · + j4
(j4
+ 1)2i−4
(n + 1)
+ (j4
− 1)(j4
+ 1)2i−2
(n + 1), . . . , (j4
+ 1)(n + 1) + j4
(j4
+ 1)2
(n + 1)
+ · · · + j4
(j4
+ 1)2i−4
(n + 1) + j4
(j4
+ 1)2i−2
(n + 1) − 1},
such that
q(i)
k=p0
1−1
Bk = M1,
q(i)
k=p0
1+(j3i−j0)(n+1)−1
Bk = I,
q(i)
k=p1
1−1
Bk = M1,
q(i)
k=p1
1+(j3i−j3)(n+1)−1
Bk = I,
...
2.3 Systems without almost periodic solutions 51
q(i)
k=pi−1
1 −1
Bk = M1,
q(i)
k=pi−1
1 +(j3i−j3(i−1))(n+1)−1
Bk = I,
...
q(i)
k=p0
i −1
Bk = Mi,
q(i)
k=p0
i +(j3i−j0)(n+1)−1
Bk = I,
q(i)
k=p1
i −1
Bk = Mi,
q(i)
k=p1
i +(j3i−j3)(n+1)−1
Bk = I,
...
q(i)
k=pi−1
i −1
Bk = Mi,
q(i)
k=pi−1
i +(j3i−j3(i−1))(n+1)−1
Bk = I,
and
q(i)
k=p(i)
Bk = I,
where p0
1, . . . , pi−1
1 , . . . , p0
i , . . . , pi−1
i are arbitrary positive integers for which
q(i) + j2i
(n + 1) ≤ p0
1
and
p0
1 + (j2i
+ j3i
)(n + 1) ≤ p1
1, . . . pi−2
1 + (j2i
+ j3i
)(n + 1) ≤ pi−1
1 ,
pi−1
1 + (j2i
+ j3i
)(n + 1) ≤ p0
2,
...
p0
i + (j2i
+ j3i
)(n + 1) ≤ p1
i , . . . pi−2
i + (j2i
+ j3i
)(n + 1) ≤ pi−1
i ,
pi−1
i + (j2i
+ j3i
)(n + 1) ≤ p(i)
if
q(i) = (j4
+ 1)(n + 1) + j4
(j4
+ 1)2
(n + 1) + · · · + j4
(j4
+ 1)2i−4
(n + 1),
p(i) = (j4
+ 1)(n + 1) + j4
(j4
+ 1)2
(n + 1) + · · · + j4
(j4
+ 1)2i−2
(n + 1) − 1.
The existence of these numbers follows from
p(i) − q(i) = j4
(j4
+ 1)2i−2
(n + 1) − 1 ≥ (j2i
+ j3i
)(i2
+ 1)(n + 1),
i, j ≥ 2 (i, j ∈ N), n ∈ N,
and the existence of the above matrices Bk follows from (2.16) and from
j3i
− j3(i−k)
≥ j2i
, k ∈ {1, . . . , i}, i ∈ N, j ≥ 2 (j ∈ N) .
2.3 Systems without almost periodic solutions 52
In the 2i-th step, we put
Bk := Ak · Ck+(j4+1)2i−1(n+1),
k ∈ {−((j4
+ 1)2i−1
+ · · · + j4
(j4
+ 1)3
+ j4
(j4
+ 1))(n + 1),
. . . , −(j4
(j4
+ 1)2i−3
+ · · · + j4
(j4
+ 1)3
+ j4
(j4
+ 1))(n + 1) − 1},
...
Bk := Ak · Ck+j4(j4+1)2i−1(n+1),
k ∈ {−j4
((j4
+ 1)2i−1
+ · · · + (j4
+ 1)3
+ (j4
+ 1))(n + 1),
. . . , −((j4
− 1)(j4
+ 1)2i−1
+ · · · + j4
(j4
+ 1)3
+ j4
(j4
+ 1))(n + 1) − 1},
and we denote
Ck := A−1
k · Bk, k ∈ {−(j4
(j4
+ 1)2i−1
+ · · · + j4
(j4
+ 1)3
+ j4
(j4
+ 1))(n + 1),
. . . , −(j4
(j4
+ 1)2i−3
+ · · · + j4
(j4
+ 1)3
+ j4
(j4
+ 1))(n + 1) − 1}.
Using this construction, we obtain the sequence {Bk}k∈Z ⊆ X.
We consider the system
xk+1 = Bk · xk, k ∈ Z. (2.17)
Suppose that there exists a non-zero vector u ∈ Fm
for which the solution {xk} of (2.17)
satisfying xn+1 = u is almost periodic. We know that
xk =
n+1
i=k−1
Bi · u for k > n + 1, k ∈ N. (2.18)
If we choose {hi}i∈N ≡ {j3(i−1)
(n + 1)}i∈N for {ϕk} ≡ {xk} in Corollary 1.5 (see also Theorem
1.3), then, for any ϑ > 0, we get the existence of an inﬁnite set N = N(ϑ) ⊆ N0 such
that
xk+j3i1 (n+1), xk+j3i2 (n+1) < ϑ, k ∈ Z, i1, i2 ∈ N. (2.19)
Thus, for every ϑ > 0, there exist inﬁnitely many ϑ-translation numbers in the form
(j3i1
− j3i2
)(n + 1), where i1 > i2 (i1, i2 ∈ N). For some i ∈ N with the property that
Mi u = u, we choose ϑ < (Mi u, u) and the above i1 > i2 > i (i1, i2 ∈ N) arbitrarily. We
have (see (2.19))
xk+(j3i1 −j3i2 )(n+1), xk < ϑ, k ∈ Z.
From (2.18) and the construction of {Bk}, we obtain
xk+(j3i1 −j3i2 )(n+1), xk > ϑ for at least one k ∈ N.
This contradiction gives that {xk} cannot be almost periodic. It means that system (2.17)
does not have a non-trivial almost periodic solution.
Now it suﬃces to show that {Bk} ∈ Oσ
ε ({Ak}); i.e., that Bk ∈ O˜ε (Ak) for all k and
some ˜ε ∈ (0, ε) and that {Bk} is almost periodic. It is seen that
Ck ∈ Oε2Q(I), k ∈ {−j4
(j4
+ 1)(n + 1), . . . , 0, . . . , (j4
+ 1)(n + 1) − 1},
2.3 Systems without almost periodic solutions 53
Ck ∈ O(ε2+ε4)PQ(I),
k ∈ {(j4
+ 1)(n + 1), . . . , (j4
+ 1 + j4
(j4
+ 1)2
)(n + 1) − 1},
Ck ∈ O(ε2+ε4)PQ(I),
k ∈ {−j4
(j4
+ 1)3
(n + 1) − j4
(j4
+ 1)(n + 1), . . . , −j4
(j4
+ 1)(n + 1) − 1},
and that, for all i ≥ 3 (i ∈ N), it is valid
Ck ∈ O(ε2+ε4+···+ε2i)PQ(I),
k ∈ {(j4
+ 1)(n + 1) + j4
(j4
+ 1)2
(n + 1) + · · · + j4
(j4
+ 1)2i−4
(n + 1), . . . ,
(j4
+ 1)(n + 1) + j4
(j4
+ 1)2
(n + 1) + · · · + j4
(j4
+ 1)2i−2
(n + 1) − 1},
Ck ∈ O(ε2+ε4+···+ε2i)PQ(I),
k ∈ {−(j4
(j4
+ 1)2i−1
+ · · · + j4
(j4
+ 1)3
+ j4
(j4
+ 1))(n + 1), . . . ,
− (j4
(j4
+ 1)2i−3
+ · · · + j4
(j4
+ 1)3
+ j4
(j4
+ 1))(n + 1) − 1}.
Thus, we have (see (2.15))
Ck ∈ Oη(I), k ∈ Z, (2.20)
and (see (iii))
Bk ∈ Oε/2(Ak), k ∈ Z. (2.21)
Indeed,
Bk = Ak · Ck for all k ∈ Z. (2.22)
From Theorem 1.27 it follows that the sequence {Ck} is almost periodic. Using Corollary
1.12 and the almost periodicity of {Ak}, we see that the set
T({Ak}, δ) ∩ T({Ck}, δ) is relatively dense in Z (2.23)
for any δ > 0. Since the multiplication of matrices is uniformly continuous on X, considering
(2.22), we have
T({Ak}, δ(ϑ)) ∩ T({Ck}, δ(ϑ)) ⊆ T({Bk}, ϑ) (2.24)
for arbitrary ϑ > 0, where δ(ϑ) > 0 is the number corresponding to ϑ from the deﬁnition
of the uniform continuity of the matrix multiplication. Finally, (2.23) and (2.24) give the
almost periodicity of {Bk} which completes the proof.
Note that, in Theorem 2.19, condition (2.13) can be omitted (i.e., one can put L = 0).
This fact follows from the below given Lemma 2.27. Now, for the later comparison, let us
recall a result from [45] (see also [1, Theorem 2.10.1]) and one of its consequences.
Theorem 2.20. Let (F, (·, ·)) = (C, | · − · |). If a vector valued sequence {bk} is almost
periodic and a matrix A ∈ Mat(C, m) non-singular, then a solution of
xk+1 = A · xk + bk, k ∈ Z,
is almost periodic if and only if it is bounded.
2.3 Systems without almost periodic solutions 54
Remark 2.21. Several modiﬁcations and generalizations of Theorem 2.20 are known. The
ﬁrst theorem of the type as Theorem 2.20 was established by E. Esclangon (in [64]) for
quasiperiodic solutions of linear diﬀerential equations of higher orders. It was extended
by H. Bohr and O. Neugebauer (in [29]) to the form mentioned in Remark 2.23 below.
In [152], Theorem 2.20 is proved if A ∈ Mat(R, m) and {bk} is almost periodic in various
metrics.
Corollary 2.22. Let (F, (·, ·)) = (C, | · − · |). Let a periodic sequence {Ak} of m × m
non-singular matrices with complex elements be given. Then, a solution of the system
xk+1 = Ak xk, k ∈ Z, is almost periodic if and only if it is bounded.
Proof. Every almost periodic sequence is bounded. Hence, we need to show only that the
boundedness of a solution implies its almost periodicity. Assume that we have a periodic
system xk+1 = Ak xk, k ∈ Z, and its bounded solution {xk}. Let n ∈ N be a period
of {Ak}. Applying Theorem 2.20, we get that the sequence {y1
k} ≡ {xnk}; i.e.,
y1
0 = x0, y1
k =
0
i=nk−1
Ai · x0, k ∈ N, y1
k =
−1
i=nk
A−1
i · x0, k ∈ Z N;
is almost periodic. Indeed, {y1
k} is a bounded solution of the constant system
yk+1 = An−1 · · · A1 · A0 · yk, k ∈ Z.
Analogously, one can show that the sequences {yj
k} ≡ {xnk+j−1}, j ∈ {2, 3, . . . , n}, are
almost periodic as well. The almost periodicity of {xk} follows from Corollary 1.10.
Remark 2.23. We add that it is possible to obtain several modiﬁcations of Corollary 2.22
for non-homogeneous systems if the non-homogeneousness is almost periodic. We mention
at least the most important one—the continuous version for diﬀerential systems. If
a complex matrix valued function A(t), t ∈ R, is periodic and a complex vector valued
function b(t), t ∈ R, is almost periodic, then any solution of x (t) = A(t) x(t) + b(t), t ∈ R,
is almost periodic if and only if it is bounded. See the introduction of Chapter 6 or, e.g.,
[72, Corollary 6.5]; for generalizations and supplements, see [129].
Example 2.24. Consider again (F, (·, ·)) = (C, | · − · |). We want to document that
Corollary 2.22 is no longer true if {Ak} is only almost periodic. We know (see Lemma 1.32
and consider the second part of Corollary 1.26) that the real sequence {ak} deﬁned by the
recurrent formula (1.36) on N0 and by the prescription
ak := −a−k−1 for k ∈ Z N0 (2.25)
is almost periodic and that it satisﬁes (see Lemma 1.35)
2n−1
k=0
ak = 1,
2n+j+2n−1
k=0
ak = 2 −
1
2j
, n ∈ N0, j ∈ N. (2.26)
Let X be the set of all m × m diagonal matrices with numbers on the diagonal which
has absolute value 1. (It is easily seen that, in this case, X is strongly transformable.
2.3 Systems without almost periodic solutions 55
See also Examples 2.4 and 2.13.) All solutions of the system of the form (2.2) given by
the almost periodic (see Theorem 1.6) sequence {Ak} ≡ diag (exp (iak) , . . . , exp (iak)) are
obviously bounded, but we will show that they are not almost periodic (except the trivial
one).
It suﬃces to consider the scalar case, i.e., m = 1. We suppose that the system has an
almost periodic solution {xk}. We have
xk = exp i
k−1
j=0
aj · x0, k ∈ N.
Especially (see (2.26)),
x2n = exp (i) · x0, x2n+j+2n = exp 2i − 2−j
i · x0, n ∈ N0, j ∈ N. (2.27)
Using Corollary 1.5 (or Theorem 1.3) for the sequence {2j
}j∈N, for any ε > 0, we get an
inﬁnite set N = N(ε) ⊆ N such that
| xk+2j(1) − xk+2j(2) | < ε for all k ∈ Z, j(1), j(2) ∈ N.
For some j(1) ∈ N, the choice k = 2j(1)
and (2.27) give
exp (i) − exp 2i − 2j(1)−j(2)
i · | x0 | < ε, j(1) < j(2), j(1), j(2) ∈ N.
Since ε can be arbitrarily small and j(2) > j(1) can be found for every ε > 0, we obtain
x0 = 0. Thus, the system does not have a non-trivial almost periodic solution. ♦
In the above example, we see that the boundedness is necessary to the almost periodicity
of solutions but not suﬃcient. Now we prove a more important necessary (also
not suﬃcient, see again Example 2.24) condition about the limitation of almost periodic
solutions.
Lemma 2.25. Let an almost periodic sequence of non-singular Ak ∈ Mat(F, m) be given.
Let {xk} be an almost periodic solution of the system xk+1 = Ak xk, k ∈ Z. Then, it is
valid either xk = o, k ∈ Z, or
inf
k∈Z
(xk, o) > 0.
Proof. Suppose that an almost periodic solution {xk} of a system satisﬁes
inf
k∈Z
(xk, o) = 0.
Let {hi}i∈N ⊆ Z be such that
lim
i→∞
(xhi
, o) = 0. (2.28)
Considering Corollary 1.5 and Theorems 1.7 and 1.8, we get a subsequence {˜hi} of {hi}
for which there exist almost periodic sequences {Bk}, {yk} satisfying
lim
i→∞
Ak+˜hi
= Bk, lim
i→∞
Bk−˜hi
= Ak, lim
i→∞
xk+˜hi
= yk, lim
i→∞
yk−˜hi
= xk,
where the convergences can be uniform with respect to k ∈ Z (see Remark 1.9). We
have yk+1 = Bk yk, k ∈ Z, and y0 = o (see (2.28)). Thus, {yk} ≡ {o}. Consequently,
xk = limi→∞ yk−˜hi
= o for k ∈ Z.
2.3 Systems without almost periodic solutions 56
Example 2.26. Applying Theorem 1.27 for n = 0, ϕ0 = 2, j = 1, and ri = 3/2i
, i ∈ N,
we construct the everywhere non-zero almost periodic sequence
b0 := 2, b1 := 2 − 1, b−2 := 2 −
1
2
, b−1 := 1 −
1
2
,
...
bk := bk+22i−1 −
1
22i−1
, k ∈ {−22i−1
− · · · − 23
− 2, . . . , −22i−3
− · · · − 23
− 2 − 1},
bk := bk−22i −
1
22i
, k ∈ {2 + 22
+ · · · + 22i−2
, . . . , 2 + 22
+ · · · + 22i−2
+ 22i
− 1},
...
in the space (R, | · − · |). Since
lim
i→∞
b20−21+22−23+···+(−2)i = 0,
the equation xk+1 = bk xk, k ∈ Z, does not have a non-trivial almost periodic solution (see
Lemma 2.18) and the vector valued sequence {bk u}, where u = o, u ∈ Rm
, is not a solution
of an almost periodic homogeneous linear diﬀerence system (see Lemma 2.25).
Moreover, for any bounded countable set of real numbers, it is shown in [73] that there
exists an almost periodic sequence whose range is the set. (For details, we refer to the
next chapter.) It means that there exists a large class of almost periodic sequences which
cannot be solutions of any almost periodic system (2.2). ♦
To improve Theorem 2.19, we need also the next lemma.
Lemma 2.27. Let X be transformable and let {Ak} ∈ AP(X) and ε > 0 be arbitrary. If
there exists a matrix M(ϑ) ∈ Oϑ(O)∩X for any ϑ > 0, then there exists {Bk} ∈ Oσ
ε ({Ak})
which does not have an almost periodic solution other than the trivial one.
Proof. We put Li := (Mi, O) for matrices Mi ∈ X, i ∈ N, such that
lim
i→∞
Li = 0, Li+1 < Li, i ∈ N. (2.29)
Let η = η(ε/2), ζ, and P and Q = Q(L1) be from (ii) and (iii), respectively. We can
assume (or choose {Bk} ≡ {Ak}) that
η < ε < ζ, L1 + ε < (Ak, O) , k ∈ Z,
and that (see also Lemma 2.11) we have {εi}i∈N ⊂ R+
, j ≥ 2 (j ∈ N), and n ∈ N satisfying
∞
i=1
εi <
η
PQ
,
ji
(n + 1) ≥ p Lg(i), εi for inﬁnitely many odd i ∈ N, (2.30)
2.3 Systems without almost periodic solutions 57
where g(i) is from Lemma 2.11 as well.
The set of all i = 1 (i ∈ N), which are not divisible by 2 and for which (2.30) is valid, is
denoted by N. Let N = {i1, i2, . . . , il, . . . }, where il < il+1, l ∈ N. Since we can redeﬁne Li
(choose other Mi), we can also assume that g(il) ≥ l, l ∈ N. We will construct sequences
{Bk} and {Ck} as in the proof of Theorem 2.19 for j4
replaced by j. First of all we put
pl := (j + 1 + j(j + 1)2
+ · · · + j(j + 1)il−3
)(n + 1), l ∈ N,
ql := (j + 1 + j(j + 1)2
+ · · · + j(j + 1)il−1
)(n + 1) − 1, l ∈ N.
Before the i1-th step (for k ≤ p1 − 1), we choose the matrices Ck (consequently Bk)
arbitrarily. We will obtain Bk and deﬁne
J1 :=
0
k=p1
Bk, J2 :=
0
k=p2
Bk, . . .
In the i1-th step, we choose the matrices Ck arbitrarily if J1 ∈ OL1
(O), and so that
0
k=q1
Bk = M1 if J1 /∈ OL1
(O).
Between the i1-th step and the i2-th step, we choose them again arbitrarily. In the i2-th
step, we choose them arbitrarily if J2 ∈ OL2
(O), and so that
0
k=q2
Bk = M2 if J2 /∈ OL2
(O).
If we proceed further in the same way, then we get matrices Ck, Bk for all k ∈ Z.
Analogously as in the proof of Theorem 2.19, we can prove that {Bk} ∈ Oσ
ε ({Ak}). We
have
0
k=rl
Bk, O ≤ Ll for rl ∈ {pl, ql}, l ∈ N. (2.31)
Evidently, for any u ∈ Fm
and µ > 0, there exists δ = δ (u, µ) > 0 with the property
that (C u, o) < µ if C ∈ Oδ (O). Using this, from (2.29), (2.31), and Lemma 2.25, we get
that all non-trivial solutions of the system of the form (2.2) given by {Bk} are not almost
periodic.
Now we can generalize Theorem 2.19 into the case of general transformable groups.
Theorem 2.28. Let X be transformable. Let {Ak} ∈ AP (X) and ε > 0 be arbitrarily
given. If there exists a sequence {Mi}i∈N ⊆ X such that, for any non-zero vector u ∈ Fm
,
one can ﬁnd i = i(u) ∈ N with the property that Mi u = u, then there exists {Bk} ∈
Oσ
ε ({Ak}) which does not possess a non-trivial almost periodic solution.
Proof. We obtain from Lemma 2.27 that it suﬃces to consider the case when
Oε (O) ∩ X = ∅, (2.32)
2.3 Systems without almost periodic solutions 58
because we can assume that the number ε is small enough. Let η = η(ε/2), ζ, P, and
Q = Q(ε) be from Deﬁnition 2.1. Note that Q does not depend on ε (consider (2.32)) and
that P, Q ≥ 1, η ≤ ε/2 (consider C = I in Deﬁnition 2.1). For simplicity, we also assume
that ε ≤ ζ (the number ζ is given for X). Let n ∈ N be arbitrary such that
PQ
22n
≤ η. (2.33)
We will construct an almost periodic sequence of matrices Ck ∈ X, k ∈ Z, applying
Corollary 1.28. Let us denote (see Deﬁnition 2.1 and also (2.32))
pi := p ε,
1
22n+i
, i ∈ N, (2.34)
and
si := −22i−1
− 22i−3
− · · · − 23
− 2, i ∈ N,
ti := 2 + 22
+ · · · + 22i−2
+ 22i
, i ∈ N.
Evidently, there exists an increasing sequence {l(i)}i∈N ⊆ N for which
i2
≤ 2l(i)−1
, pi ≤ 2l(i)−1
, i ∈ N. (2.35)
In the ﬁrst step of the construction of {Ck}, we put C0 := I, C1 := I,
Ck := I, k ∈ {−2, −1}, Ck := I, k ∈ {2, . . . , 2 + 22
− 1},
...
Ck := I, k ∈ {tl(1)−2, . . . , tl(1)−1 − 1}, Ck := I, k ∈ {sl(1), . . . , sl(1)−1 − 1},
and deﬁne
Bk := Ak, k ∈ {sl(1), . . . , tl(1)−1 − 1}.
In the second step, we choose matrices
Bk ∈ O2−2n−1 (Ak) , k ∈ {tl(1)−1, . . . , tl(1) − 1}, (2.36)
arbitrarily so that
0
k=tl(1)−1+p1−1
Bk = I,
tl(1)−1+p1
k=tl(1)−1+p1+(2l(1)−1)−1
Bk = M1.
The existence of such matrices Bk follows directly from Deﬁnition 2.1 (see (2.34), (2.35)).
We deﬁne
Ck := A−1
k · Bk, k ∈ {tl(1)−1, . . . , tl(1) − 1}.
From (iii) in Deﬁnition 2.1 and from (2.36), it follows that
Ck ∈ O2−2n−1Q (I) , k ∈ {tl(1)−1, . . . , tl(1) − 1},
2.3 Systems without almost periodic solutions 59
i.e.,
Ck ∈ O2−2n−1Q (Ck−22l(1) ) ∩ X, k ∈ {tl(1)−1, . . . , tl(1) − 1}.
In the third step, we deﬁne
Ck := Ck+22l(1)+1 , k ∈ {sl(1)+1, . . . , sl(1) − 1},
Ck := Ck−22l(1)+2 , k ∈ {tl(1), . . . , tl(1)+1 − 1},
...
Ck := Ck−22l(2)−2 , k ∈ {tl(2)−2, . . . , tl(2)−1 − 1},
Ck := Ck+22l(2)−1 , k ∈ {sl(2), . . . , sl(2)−1 − 1},
and put
Bk := Ak · Ck, k ∈ {sl(2), . . . , tl(2)−1 − 1}.
We remark that, before the third step, we have
Bk = Ak · Ck, k ∈ {sl(1), . . . , tl(1) − 1}.
In the fourth step, we choose
Bk = Ak · Ck, Ck ∈ O2−2n−2PQ (Ck−22l(2) ) ∩ X, k ∈ {tl(2)−1, . . . , tl(2) − 1},
arbitrarily so that
0
k=tl(2)−1+p2−1
Bk = I,
tl(2)−1+p2
k=tl(2)−1+p2+(2l(2)−1)−1
Bk = M1,
0
k=tl(2)−1+2p2+2l(2)−1
Bk = I,
tl(2)−1+2p2+2l(2)
k=tl(2)−1+2p2+2l(2)+(2l(2)−2l(1))−1
Bk = M1,
0
k=tl(2)−1+3p2+2·2l(2)−1
Bk = I,
tl(2)−1+3p2+2·2l(2)
k=tl(2)−1+3p2+2·2l(2)+(2l(2)−1)−1
Bk = M2,
0
k=tl(2)−1+4p2+3·2l(2)−1
Bk = I,
tl(2)−1+4p2+3·2l(2)
k=tl(2)−1+4p2+3·2l(2)+(2l(2)−2l(1))−1
Bk = M2.
Note that it is valid
tl(2)−1 + 4p2 + 3 · 2l(2)
+ 2l(2)
− 2l(1)
< tl(2),
because tl(2) − tl(2)−1 = 22l(2)
and (see (2.35))
4p2 + 3 · 2l(2)
+ 2l(2)
− 2l(1)
< 22
2l(2)−1
+ 22
2l(2)
< 2 · 22
· 2l(2)
≤ 22l(2)
.
2.3 Systems without almost periodic solutions 60
Now we show that the above matrices Bk, Ck for k ∈ {tl(2)−1, . . . , tl(2) − 1} exist. First,
we put
˜Bk := Ak · Ck−22l(2) , k ∈ {tl(2)−1, . . . , tl(2) − 1}. (2.37)
We use (i) in Deﬁnition 2.1 to transform ˜Bk into Bk ∈ O2−2n−2
˜Bk ∩ X arbitrarily as
U1, . . . , Ul into V1, . . . , Vl for
l = p2, l = 2l(2)
− 1, l = p2 + 2l(2)
− 2l(2)
− 1 , l = 2l(2)
− 2l(1)
,
l = p2 + 2l(2)
− 2l(2)
− 2l(1)
, l = 2l(2)
− 1,
l = p2 + 2l(2)
− 2l(2)
− 1 , l = 2l(2)
− 2l(1)
and
U0 =
tl(2)−1
k=tl(2)−1+p2−1
Bk =


0
k=tl(2)−1−1
Bk


−1
, U0 =
tl(2)−1+p2
k=tl(2)−1+p2+(2l(2)−1)−1
Bk = M1,
U0 =
tl(2)−1+p2+(2l(2)−1)
k=tl(2)−1+2p2+2l(2)−1
Bk = M−1
1 , U0 =
tl(2)−1+2p2+2l(2)
k=tl(2)−1+2p2+2l(2)+(2l(2)−2l(1))−1
Bk = M1,
U0 =
tl(2)−1+2p2+2l(2)+(2l(2)−2l(1))
k=tl(2)−1+3p2+2·2l(2)−1
Bk = M−1
1 ,
U0 =
tl(2)−1+3p2+2·2l(2)
k=tl(2)−1+3p2+2·2l(2)+(2l(2)−1)−1
Bk = M2,
U0 =
tl(2)−1+3p2+2·2l(2)+(2l(2)−1)
k=tl(2)−1+4p2+3·2l(2)−1
Bk = M−1
2 ,
U0 =
tl(2)−1+4p2+3·2l(2)
k=tl(2)−1+4p2+3·2l(2)+(2l(2)−2l(1))−1
Bk = M2,
respectively. It suﬃces to consider (2.32), (2.34), and the inequalities (the sequence {l(i)}
is increasing)
p2 ≤ 2l(2)−1
≤ 2l(2)
− 2l(1)
≤ 2l(2)
− 1,
p2 ≤ p2 + 2l(2)
− 2l(2)
− 1 ≤ p2 + 2l(2)
− 2l(2)
− 2l(1)
.
To prove the existence of the above Bk, Ck = A−1
k Bk, it remains to show that
Ck ∈ O2−2n−2PQ (Ck−22l(2) ) ∩ X, k ∈ {tl(2)−1, . . . , tl(2) − 1}. (2.38)
2.3 Systems without almost periodic solutions 61
We can express
Bk = Ak · Ck−22l(2) · ˜Ck, k ∈ {tl(2)−1, . . . , tl(2) − 1}, (2.39)
where ˜Ck = C−1
k−22l(2) A−1
k Bk ∈ X, i.e., it is valid (consider Bk = Ak Ck) that
Ck = Ck−22l(2) · ˜Ck ∈ X, k ∈ {tl(2)−1, . . . , tl(2) − 1}. (2.40)
Further, we have (see (2.37) and (2.39))
˜Ck = ˜B−1
k · Bk, k ∈ {tl(2)−1, . . . , tl(2) − 1}.
We repeat that
Bk ∈ O2−2n−2
˜Bk , k ∈ {tl(2)−1, . . . , tl(2) − 1}.
Using (iii) in Deﬁnition 2.1, we obtain
˜Ck ∈ O2−2n−2Q (I) , k ∈ {tl(2)−1, . . . , tl(2) − 1},
and, using condition (ii) (consider (2.33) and η ≤ ε/2 < ζ), we obtain
Ck−22l(2) · ˜Ck ∈ O2−2n−2PQ (Ck−22l(2) ) , k ∈ {tl(2)−1, . . . , tl(2) − 1}. (2.41)
Now (2.40) and (2.41) give (2.38).
Continuing in the same manner, in the 2i-th step, we choose arbitrary matrices
Bk = Ak · Ck, Ck ∈ O2−2n−iPQ
(Ck−22l(i) ) ∩ X, k ∈ {tl(i)−1, . . . , tl(i) − 1},
for which
0
k=tl(i)−1+pi−1
Bk = I,
tl(i)−1+pi
k=tl(i)−1+pi+(2l(i)−1)−1
Bk = M1,
...
0
k=tl(i)−1+ipi+(i−1)2l(i)−1
Bk = I,
tl(i)−1+ipi+(i−1)2l(i)
k=tl(i)−1+ipi+(i−1)2l(i)+(2l(i)−2l(i−1))−1
Bk = M1,
0
k=tl(i)−1+(i+1)pi+i2l(i)−1
Bk = I,
tl(i)−1+(i+1)pi+i2l(i)
k=tl(i)−1+(i+1)pi+i2l(i)+(2l(i)−1)−1
Bk = M2,
...
0
k=tl(i)−1+2ipi+(2i−1)2l(i)−1
Bk = I,
tl(i)−1+2ipi+(2i−1)2l(i)
k=tl(i)−1+2ipi+(2i−1)2l(i)+(2l(i)−2l(i−1))−1
Bk = M2,
...
2.3 Systems without almost periodic solutions 62
0
k=tl(i)−1+((i−1)i+1)pi+(i−1)i2l(i)−1
Bk = I,
tl(i)−1+((i−1)i+1)pi+(i−1)i2l(i)
k=tl(i)−1+((i−1)i+1)pi+(i−1)i2l(i)+(2l(i)−1)−1
Bk = Mi,
...
0
k=tl(i)−1+i2pi+(i2−1)2l(i)−1
Bk = I,
tl(i)−1+i2pi+(i2−1)2l(i)
k=tl(i)−1+i2pi+(i2−1)2l(i)+(2l(i)−2l(i−1))−1
Bk = Mi.
We remark again that it is true (see (2.35))
i2
pi + i2
− 1 2l(i)
+ 2l(i)
− 2l(i−1)
< i2
2l(i)−1
+ i2
2l(i)
< 22l(i)
.
Thus, we have
tl(i)−1 + i2
pi + i2
− 1 2l(i)
+ 2l(i)
− 2l(i−1)
< tl(i).
The existence of the above matrices Bk, Ck ∈ X can be shown analogously to the fourth
step.
In the (2i + 1)-th step, we deﬁne
Ck := Ck+22l(i)+1 , k ∈ {sl(i)+1, . . . , sl(i) − 1},
Ck := Ck−22l(i)+2 , k ∈ {tl(i), . . . , tl(i)+1 − 1},
...
Ck := Ck−22l(i+1)−2 , k ∈ {tl(i+1)−2, . . . , tl(i+1)−1 − 1},
Ck := Ck+22l(i+1)−1 , k ∈ {sl(i+1), . . . , sl(i+1)−1 − 1},
and put
Bk := Ak · Ck, k ∈ {sl(i+1), . . . , tl(i+1)−1 − 1}.
We can construct {Ck}k∈Z ⊆ X and {Bk}k∈Z ⊆ X so that
Bk = Ak · Ck, k ∈ Z. (2.42)
We know that
Ck = I, k ∈ {sl(1), . . . , tl(1)−1 − 1},
Ck ∈ O2−2n−1Q (Ck−22l(1) ) ⊆ O2−2n−1PQ (I) , k ∈ {tl(1)−1, . . . , tl(1) − 1},
Ck = Ck+22l(1)+1 ∈ O2−2n−1PQ (I) , k ∈ {sl(1)+1, . . . , sl(1) − 1},
Ck = Ck−22l(1)+2 ∈ O2−2n−1PQ (I) , k ∈ {tl(1), . . . , tl(1)+1 − 1},
2.3 Systems without almost periodic solutions 63
...
Ck = Ck−22l(2)−2 ∈ O2−2n−1PQ (I) , k ∈ {tl(2)−2, . . . , tl(2)−1 − 1},
Ck = Ck+22l(2)−1 ∈ O2−2n−1PQ (I) , k ∈ {sl(2), . . . , sl(2)−1 − 1},
Ck ∈ O2−2n−2PQ (Ck−22l(2) ) ⊆ O(2−2n−1+2−2n−2)PQ (I) , k ∈ {tl(2)−1, . . . , tl(2) − 1},
...
Ck ∈ O2−2n−iPQ
(Ck−22l(i) ) ⊆ O(2−2n−1+···+2−2n−i)PQ
(I) , k ∈ {tl(i)−1, . . . , tl(i) − 1},
Ck = Ck+22l(i)+1 ∈ O(2−2n−1+···+2−2n−i)PQ
(I) , k ∈ {sl(i)+1, . . . , sl(i) − 1},
...
Especially, we see that we can construct {Ck} as in Corollary 1.28 for the sequence of
numbers
εj := 2−i
PQ if j = f(i) for some i ∈ N {1, . . . , 2n};
εj := 0 if j ∈ N {f(i); i ∈ N {1, . . . , 2n}},
where f : N {1, . . . , 2n} → N {1, . . . , 2n} is a given increasing discrete function. Since
the sequence {εj}j∈N satisﬁes (1.25), {Ck} is almost periodic.
Further, we see that
Ck ∈ O(2−2n−1+2−2n−2+···+2−2n−j(k))PQ
(I)
for all k ∈ Z and some j(k) ∈ N (which depends on k); i.e., we have
Ck ∈ O2−2nPQ (I) , k ∈ Z.
From (2.33) it follows Ck ∈ Oη (I) for k ∈ Z and, consequently, from condition (ii)
and (2.42) it follows
Bk ∈ Oε/2 (Ak) , k ∈ Z, i.e., sup
k∈Z
(Ak, Bk) ≤
ε
2
< ε. (2.43)
Now we prove the almost periodicity of {Bk}k∈Z applying the almost periodicity of
{Ak} and {Ck}. Let ϑ > 0 be arbitrarily small and let δ = δ(ϑ) > 0 be such that (see (ii)
in Deﬁnition 2.1)
U2 · V2 ∈ Oϑ (U1 · V1) if U1, V1, U2, V2 ∈ X and U1 ∈ Oδ (U2) , V1 ∈ Oδ (V2) . (2.44)
Corollary 1.11 implies that the sequence {[Ak, Ck]}k∈Z in Mat (F, m) × Mat (F, m) is
almost periodic. Thus (or see directly Corollary 1.12), the set T({Ak}, δ) ∩ T({Ck}, δ) is
relatively dense in Z and, from (2.42) and (2.44), we have
T({Ak}, δ) ∩ T({Ck}, δ) ⊆ T({Bk}, ϑ).
Because of the arbitrariness of ϑ > 0, we obtain the fact that {Bk} is almost periodic.
2.3 Systems without almost periodic solutions 64
The almost periodicity of {Bk} together with (2.43) give that {Bk} ∈ Oσ
ε ({Ak}). Suppose
that there exists a vector u = o, u ∈ Fm
, for which the solution {xk}k∈Z of
xk+1 = Bk · xk, k ∈ Z, x0 = u
is almost periodic. Let i(u) ∈ N and ϑ = ϑ(u) > 0 satisfy ϑ < Mi(u) u, u . From
Theorem 1.3 for h1 = 1 and hi = 2l(i−1)
, i ≥ 2, i ∈ N, it follows that the inequality
(xk+2l(i(1)) , xk+2l(i(2)) ) < ϑ, k ∈ Z, (2.45)
is satisﬁed for inﬁnitely many i(1) < i(2), i(1), i(2) ∈ N.
It is easily seen that
xk =
0
i=k−1
Bi · u, k ∈ N, (2.46)
and that
xk+2l(i(2))−2l(i(1)) =
k
i=k+2l(i(2))−2l(i(1))−1
Bi · xk, i(2) > i(1) ≥ i(u), k, i(1), i(2) ∈ N. (2.47)
If we choose
k = tl(i(2))−1 + ((i(u) − 1)i(2) + i(1) + 1) pi(2) + ((i(u) − 1)i(2) + i(1)) 2l(i(2))
, (2.48)
we obtain (from (2.46) and from the construction of {Bk})
xk =
0
i=k−1
Bi · u = I · u = u
and (from (2.47) and the construction)
xk+2l(i(2))−2l(i(1)) =
k
i=k+2l(i(2))−2l(i(1))−1
Bi · xk = Mi(u) · xk = Mi(u) · u
for i(2) > i(1) ≥ i(u), i(1), i(2) ∈ N.
Finally, we have
(xk, xk+2l(i(2))−2l(i(1)) ) = u, Mi(u) · u > ϑ, i(2) > i(1) ≥ i(u), i(1), i(2) ∈ N,
where k is given in (2.48). Of course, we can rewrite (2.45) into the form (k is replaced by
k − 2l(i(1))
) of
(xk, xk+2l(i(2))−2l(i(1)) ) < ϑ, k ∈ Z,
which is valid for inﬁnitely many i(1), i(2) ∈ N such that i(1) < i(2). This contradiction
proves that the system {Bk} ∈ AP (X) does not have any non-trivial almost periodic
solution.
Now we mention preliminary results, which we need to prove our main result concerning
weakly transformable groups (the main generalization of Theorem 2.19).
2.3 Systems without almost periodic solutions 65
Lemma 2.29. Let X be transformable. For any ε > 0 and C, D ∈ X, there exist matrices
C1, . . . , Cj ∈ X satisfying
C1 ∈ Oε (C) , Ci ∈ Oε (Ci+1) for i ∈ {1, . . . , j − 1}, Cj ∈ Oε (D) .
Proof. Let ε > 0 and C, D ∈ X be given. To prove the lemma, it suﬃces to use (i) in
Deﬁnition 2.1 for η = η(ε) ≤ ε mentioned in (ii). Let L > 0 be such that L < (I, O),
L < (C, O), and L < (D, O) and let p = p (L, η) have the property mentioned in (i). Let
us choose U0 = C and Ui = I, i ∈ {1, . . . , p}, in (i). There exist matrices Vi ∈ Oη(I) ∩ X,
i ∈ {1, . . . , p}, for which Vp · · · V2 · V1 = C. We put
M1 := V1, M2 := V2 · V1, . . . Mp := Vp · · · V2 · V1.
Since Vi ∈ Oη(I) ∩ X, i ∈ {1, . . . , p}, and
M1 = V1 ∈ X, M2 = V2 · M1 ∈ X, . . . Mp = Vp · Mp−1 = C ∈ X,
we have
M1 ∈ Oε (I) , Mi ∈ Oε (Mi+1) for i ∈ {1, . . . , p − 1}.
Especially, Mp−1 ∈ Oε (C). Analogously, we can ﬁnd matrices Mp+1, . . . , M2p−1 ∈ X with
the property that
Mp+1 ∈ Oε (I) , Mp+i ∈ Oε (Mp+i+1) for i ∈ {1, . . . , p − 2}, M2p−1 ∈ Oε (D) .
For the sequence Mp−1, . . . , M1, I, Mp+1, . . . , M2p−1 as C1, . . . , Cj, we obtain the statement
of the lemma.
Lemma 2.30. Let X be weakly transformable and let X0 ⊆ X, X1, . . . , Xl ∈ X be
from Deﬁnition 2.14. If for some C0, D0 ∈ X0 and i(1), i(2) ∈ {1, . . . , l}, the product
C0 Xi(1) D0 Xi(2) can be expressed as M0 Xj for some M0 ∈ X0 and j ∈ {1, . . . , l}, then, for
all C, D ∈ X0, matrix C Xi(1) D Xi(2) can be expressed in the form M Xj for some M ∈ X0.
Proof. Since the multiplication of matrices is uniformly continuous on X0 (see (ii) in Deﬁnition
2.1) and since the multiplication of matrices is continuous on Mat (F, m), the
multiplication of matrices is uniformly continuous on X (the set of Xj is ﬁnite). Hence,
there exists δ > 0 such that
C1 Xi(1) D1 Xi(2) ∈ OδX
(C0 Xi(1) D0 Xi(2)) if C1 ∈ Oδ (C0) ∩ X, D1 ∈ Oδ (D0) ∩ X,
for each i(1), i(2) ∈ {1, . . . , l}, where δX is from Deﬁnition 2.14. This fact implies the
lemma for matrices C, D (as C1, D1) from neighbourhoods of C0, D0. The radius δ of the
neighbourhoods does not depend on the choices of C0, D0. Thus, it is valid
C2 Xi(1) D2 Xi(2) ∈ OδX
(C1 Xi(1) D1 Xi(2)) if C2 ∈ Oδ (C1) ∩ X, D2 ∈ Oδ (D1) ∩ X,
...
Cj Xi(1) Dj Xi(2) ∈ OδX
(Cj−1 Xi(1) Dj−1 Xi(2)) if Cj ∈ Oδ (Cj−1) ∩ X, Dj ∈ Oδ (Dj−1) ∩ X.
Now, from Deﬁnition 2.1 and Lemma 2.29, it follows the statement for all C, D ∈ X0.
2.3 Systems without almost periodic solutions 66
Lemma 2.31. Let X be weakly transformable and let X0 ⊆ X, X1, . . . , Xl ∈ X be from
Deﬁnition 2.14. There exists r ∈ N with the property that
C1 · Xi · C2 · Xi · · · Cr · Xi ∈ X0 for all C1, C2, . . . , Cr ∈ X0, i ∈ {1, . . . , l}.
Proof. Let i ∈ {1, . . . , l} be arbitrarily given. Using Lemma 2.30 (j−1)-times (for arbitrary
j ∈ N), we obtain that
C1 · Xi · C2 · Xi · · · Cj · Xi ∈ X0 for all C1, C2, . . . , Cj ∈ X0
if and only if Xj
i ∈ X0. Indeed, it suﬃces to replace C1, . . . , Cj by I, . . . , I. For all j ∈ N,
one can express Xj
i = C(j) · Xl(j), where l(j) ∈ {1, . . . , l} and C(j) ∈ X0. Evidently, there
exist j(1) > j(2) (j(1), j(2) ∈ N) for which l(j(1)) = l(j(2)), i.e.,
X
j(1)
i = C(j(1)) · Xl(j(1)), X
j(2)
i = C(j(2)) · Xl(j(1)).
Hence, we have
X
j(1)−j(2)
i = C(j(1)) · (C(j(2)))−1
∈ X0.
We see that, for each i ∈ {1, . . . , l}, there exists r(i) such that X
r(i)
i ∈ X0. If r ∈ N is
divisible by all r(i), i ∈ {1, . . . , l}, then Xr
i ∈ X0, i ∈ {1, . . . , l}. The above equivalence
completes the proof.
Theorem 2.32. Let X be weakly transformable and let {Ak} ∈ AP (X), ε > 0 be arbitrarily
given. If there exists a sequence {Mi}i∈N ⊆ X0 such that, for any non-zero vector
u ∈ Fm
, one can ﬁnd i = i(u) ∈ N with the property that Mi u = u, then there exists
{Bk} ∈ Oσ
ε ({Ak}) which does not have an almost periodic solution other than the trivial
one.
Proof. Let ε < δX (see Deﬁnition 2.14) and let n ∈ N be an ε-translation number of {Ak}.
Let us express (see again Deﬁnition 2.14)
Ak = C(Ak) · Xi(k), C(Ak) ∈ X0, i(k) ∈ {1, . . . , l},
where k ∈ Z. An arbitrary system {Bk} ∈ Oσ
ε ({Ak}) has the feature of
Bk = C(Bk) · Xi(k), C(Bk) ∈ X0, k ∈ Z, (2.49)
because (Ak, Bk) < ε < δX , k ∈ Z. We also know that i(k) = i(k + jn) for k, j ∈ Z.
Indeed, we have
(Ak, Ak+n) < ε < δX , k ∈ Z,
i.e.,
Ak+jn, Ak+(j+1)n < ε < δX , k, j ∈ Z.
Thus, Lemma 2.30 and (2.49) imply that there exists i ∈ {1, . . . , l} for which
B(j+1)n−1 · · · Bjn = C(B(j+1)n−1 · · · Bjn) · Xi, C(B(j+1)n−1 · · · Bjn) ∈ X0,
where j ∈ Z and {Bk} ∈ Oσ
ε ({Ak}). Furthermore, Lemma 2.31 gives r ∈ N with the
property that
B(j+1)rn−1 · · · Bjrn ∈ X0, j ∈ Z, {Bk} ∈ Oσ
ε ({Ak}). (2.50)
2.3 Systems without almost periodic solutions 67
The multiplication of matrices is uniformly continuous on X (see also the beginning of
the proof of Lemma 2.30). Hence, for any ϑ > 0, there exists δ(ϑ) > 0 such that
(C1 · · · Crn, D1 · · · Drn) < ϑ if Ci, Di ∈ X, Ci ∈ Oδ(ϑ) (Di) , i ∈ {1, . . . , rn}. (2.51)
Using (2.51), we obtain the inclusion
T({Ak}, δ(ϑ)) ⊆ T({Ak+rn−1 · · · Ak}, ϑ), ϑ > 0,
which proves that {Ak+rn−1 · · · Ak}k∈Z is almost periodic. Thus, {A(j+1)rn−1 · · · Ajrn}j∈Z is
almost periodic as well. From (2.50) it follows that {A(j+1)rn−1 · · · Ajrn} ∈ AP (X0).
Theorem 2.28 says that, in any neighbourhood of {A(j+1)rn−1 · · · Ajrn}, there exists
a system {B(j+1)rn−1 · · · Bjrn}j∈Z ∈ AP (X0) which does not possess non-trivial almost
periodic solutions. Let
{B(j+1)rn−1 · · · Bjrn} ∈ Oσ
δ(ε/2)/Q {A(j+1)rn−1 · · · Ajrn} ,
where δ(ε/2) > 0 is from (2.51) for rn = 2 and Q > 0 is from Deﬁnition 2.1 for X0.
Especially,
B(j+1)rn−1 · · · Bjrn ∈ Oδ(ε/2)/Q A(j+1)rn−1 · · · Ajrn ∩ X0, j ∈ Z. (2.52)
For all j ∈ Z, we can express
B(j+1)rn−1 · · · Bjrn = A(j+1)rn−1 · · · Ajrn · ˜Aj,
i.e.,
˜Aj = A(j+1)rn−1 · · · Ajrn
−1
· B(j+1)rn−1 · · · Bjrn ∈ X0. (2.53)
We have (see also (2.52) and (iii) in Deﬁnition 2.1)
˜Aj ∈ Oδ(ε/2)(I), i.e., Ajrn · ˜Aj ∈ Oε/2 (Ajrn) , j ∈ Z. (2.54)
Now we show that the sequence {Bk}k∈Z, given by
Bjrn = Ajrn · ˜Aj, Bjrn+i = Ajrn+i, j ∈ Z, i ∈ {1, . . . , rn − 1}, (2.55)
is almost periodic. Applying Corollary 1.10, we get that {Bk} is almost periodic if and
only if all sequences {Ajrn · ˜Aj}j∈Z, {Ajrn+i}j∈Z, i ∈ {1, . . . , rn − 1}, are almost periodic.
Of course, the almost periodicity of each {Ajrn+i} and {Ajrn} is obvious. The sequence
{[Ajrn, ˜Aj]}j∈Z in Mat (F, m) × Mat (F, m) is almost periodic (consider Corollary 1.11) if
{ ˜Aj} is almost periodic. Since we have
T({[Ajrn, ˜Aj]}, δ(ϑ)) ⊆ T({Ajrn · ˜Aj}, ϑ) for ϑ > 0
from (2.51) if rn ≥ 2, the sequence {Ajrn · ˜Aj} is almost periodic if { ˜Aj} is almost periodic.
The fact that we can assume the almost periodicity of { ˜Aj}j∈Z follows from the proof of
Theorem 2.28. There is constructed {Bk}k∈Z as Bk = Ak ·Ck, k ∈ Z, for an almost periodic
sequence {Ck}k∈Z. If (2.32) is fulﬁlled, then it suﬃces to consider (2.53) as Ck = A−1
k · Bk,
2.3 Systems without almost periodic solutions 68
i.e., B(j+1)rn−1 · · · Bjrn as Bk and A(j+1)rn−1 · · · Ajrn as Ak. If (2.32) is not valid for any
ε > 0, then it suﬃces to use Lemma 2.27.
Let us consider the general system
xj+1 = C(j+1)rn−1 · · · Cjrn · xj, j ∈ Z. (2.56)
Clearly, if {xk}k∈Z is a solution of xk+1 = Ck xk, k ∈ Z, then {xjrn}j∈Z is a solution
of (2.56). Indeed, it is valid that
xk+rn =
k
i=k+rn−1
Ci · xk, k ∈ Z,
i.e.,
x(j+1)rn = C(j+1)rn−1 · · · Cjrn · xjrn, j ∈ Z.
One can easily show that {xk}k∈Z cannot be almost periodic if {xjrn}j∈Z is not almost
periodic. Non-trivial solutions of {A(j+1)rn−1 · · · Ajrn · ˜Aj}j∈Z are not almost periodic.
Thus, the system {Bk} (deﬁned in (2.55)) does not have an almost periodic solution other
than the trivial one. Note that {Bk} ∈ Oσ
ε ({Ak}) follows from (2.54).
The most important case is covered by the following corollary.
Corollary 2.33. Let X be weakly transformable and have a dense countable subset. Let
{Ak} ∈ AP (X) and ε > 0 be arbitrarily given. If for any vector u = o, u ∈ Fm
, there
exists M(u) ∈ X0 for which M(u) u = u, then there exists {Bk} ∈ Oσ
ε ({Ak}) which does
not possess almost periodic solutions.
Proof. Let X be a dense countable subset of X. Since the operations ⊕ and are continuous
with respect to , the multiplication of matrices and vectors on Mat (F, m) and
Fm
is continuous as well. Thus, if (M u, u) > 0 for some M ∈ X and u ∈ Fm
, then
(MX u, u) > (M u, u) /2 for some MX ∈ X from a neighbourhood of M. Now it suﬃces
to use Theorem 2.32.
Example 2.34. All concrete transformable and weakly transformable matrix groups, mentioned
in Examples 2.2–2.10 and 2.15–2.17 (except O(2)⊕Ok(2)), satisfy the conditions of
Corollary 2.33. ♦
Applying the process used in the proof of Theorem 2.28 and considering Theorem 2.32
(by simple modiﬁcations), one can analogously prove the following theorem.
Theorem 2.35. Let X be weakly transformable and let {Ak} ∈ AP (X), ε > 0, and
u ∈ Fm
be arbitrarily given. If there exists a matrix M ∈ X0 such that M u = u, then there
exists {Bk} ∈ Oσ
ε ({Ak}) for which the solution of
xk+1 = Bk · xk, k ∈ Z, x0 = u (2.57)
is not almost periodic.
2.3 Systems without almost periodic solutions 69
Remark 2.36. Let us illustrate the signiﬁcance of Theorem 2.35. Let {Ak} ∈ AP (X)
and ε > 0 be arbitrary. We consider the group X of complex matrices (with the usual
metric) in the form
I O
O U
,
where I is the m1 × m1 identity matrix and U is a m2 × m2 unitary matrix. This group
is transformable, but it does not satisfy the condition of Theorem 2.32. Clearly, a system
{Bk} ∈ AP (X) without almost periodic solutions does not exist. Of course, using Theorem
2.35 for arbitrarily given non-zero u ∈ Fm
satisfying M u = u for some M ∈ X, we
obtain {Bk} ∈ Oσ
ε ({Ak}) with such a property that the solution of (2.57) is not almost
periodic. In fact, using the above process (see also Corollary 2.33), one can construct a
system {Bk} ∈ Oσ
ε ({Ak}) for which the solution of (2.57) is not almost periodic for all
u ∈ Fm
such that M u = u for some M ∈ X; i.e., it is possible to obtain the same {Bk}
for all considered u.
At the end, we say that it is possible to obtain various generalizations and modiﬁcations
of results presented in this section. For simplicity, we consider only sufﬁciently general and,
at the same time, important cases. Especially, in Chapter 1, the constructions are used for
a ring with a pseudometric. For almost periodic sequences deﬁned for k ∈ N0, it suﬃces to
replace Corollary 1.5 by Remark 1.2 (see also [100]) and Corollary 1.28 by Theorem 1.23.
Note that the basic theory of almost periodic sequences on N0 is established, e.g., in [57].
Chapter 3
Values of almost periodic and limit
periodic sequences
The goal of this chapter is to ﬁnd limit periodic and almost periodic sequences whose
ranges consist of arbitrarily given sets. To ﬁnd them, we apply a construction from Chapter
1. In addition, using a diﬀerent construction, we obtain another result concerning values
of limit periodic sequences. The obtained results are also used in the study of complex
almost periodic (weakly) transformable diﬀerence systems (introduced in Chapter 2).
3.1 Preliminaries
As in the ﬁrst chapter, let X = ∅ be an arbitrary set and let d : X × X → [0, ∞) be a
pseudometric on X. For given ε > 0 and x ∈ X, we deﬁne the ε-neighbourhood of x in X
as the set {y ∈ X; d(x, y) < ε}. The ε-neighbourhood of x is denoted by Oε(x).
3.2 Sequences with given values
In this section, we construct limit periodic (and almost periodic) sequences whose ranges
consist of arbitrarily given sets satisfying only necessary conditions. We are motivated by
paper [73], where a similar problem is investigated for real valued sequences. In that
paper, using an explicit construction, it is shown that, for any bounded countable set of
real numbers, there exists an almost periodic sequence whose range is this set and which
attains each value in this set periodically. We extend this result to limit periodic sequences
attaining values in X.
Concerning almost periodic sequences with indices k ∈ N0 = N∪{0} (or asymptotically
almost periodic sequences), we refer to [100], where it is proved that, for any precompact
sequence {xk}k∈N0 in a metric space X, there exists a permutation P of the set of nonnegative
integers such that the sequence {xP(k)}k∈N0 is almost periodic. Let us point out
that the deﬁnition of the almost periodicity (in fact, asymptotic almost periodicity) in
70
3.2 Sequences with given values 71
[100] is based on the Bochner concept; i.e., a bounded sequence {xk}k∈N0 in X is called
almost periodic if the set of sequences {xk+p}k∈N0 , p ∈ N, is precompact in the space of
all bounded sequences in X. We repeat that, for sequences with values in complete metric
spaces, the Bochner deﬁnition is equivalent with the Bohr one which we prefer. Moreover,
we know that these deﬁnitions remain also equivalent in an arbitrary pseudometric space
if one replaces the convergence in the Bochner deﬁnition by the Cauchy property (see
Theorem 1.3 and Corollary 1.5). However, it can be shown that the result of [100] for the
almost periodicity on N0 cannot be true for the almost periodicity on Z or R (see also
Remark 1.2).
In Banach spaces, another important necessary and suﬃcient condition for a function
to be almost periodic is that it has the so-called approximation property; i.e., a function is
almost periodic if and only if there exists a sequence of trigonometric polynomials which
converges uniformly to the function on the whole real line in the norm topology (see, e.g.,
[46, Theorems 6.8, 6.15]). There exist generalizations of this result (see [38, 177]). For
example, it is proved in [13] that an almost periodic function with fuzzy real numbers as
values can be uniformly approximated by a sequence of generalized trigonometric polynomials.
We add that fuzzy real numbers form a complete metric space. One shows that
the approximation theorem is generally invalid if one does not require the completeness
of the space of values. Thus, we cannot use this idea in our constructions for general
pseudometric spaces.
We prove that, for a countable subset of X, there exists a limit periodic sequence whose
range is exactly this set. Since the range of any limit periodic sequence is totally bounded
(see Theorems 1.14 and 1.22), this condition on the set is necessary. Now we prove that
this condition is suﬃcient as well.
Theorem 3.1. For any countable and totally bounded set X ⊆ X, there exists a limit
periodic sequence {ψk}k∈Z satisfying
{ψk; k ∈ Z} = X (3.1)
with the property that, for any l ∈ Z, there exists q(l) ∈ N such that
ψl = ψl+jq(l), j ∈ Z. (3.2)
Proof. Let us put X = {ϕk; k ∈ N}. Without loss of generality, we can assume that the
set {ϕk; k ∈ N} is inﬁnite because, for only ﬁnitely many diﬀerent ϕk, we can deﬁne {ψk}
as periodic. Since {ϕk; k ∈ N} is totally bounded, for any ε > 0, it can be embedded into
a ﬁnite number of spheres of radius ε. Let us denote by xi
1, . . . , xi
m(i) the centres of the
spheres of radius 2−i
which cover the set for all i ∈ N. Evidently, we can also assume that
xi
1, . . . , xi
m(i) ∈ {ϕk; k ∈ N}, i ∈ N, (3.3)
and that
xi
1 = ϕi, i ∈ N. (3.4)
We will construct {ψk} applying Corollary 1.28. We choose arbitrary n(1) ∈ N for
which 22n(1)
> m(1). We put
ψ0 := x1
1, ψ1 := x1
2, . . . , ψm(1)−1 := x1
m(1),
3.2 Sequences with given values 72
ψk := x1
1, k ∈ {−22n(1)−1
− · · · − 23
− 2, . . . , −1} ∪ {m(1), . . . , 2 + 22
+ · · · + 22n(1)
− 1},
and
εk := L, k ∈ {1, . . . , 2n(1) + 1}, (3.5)
where
L := max
i,j∈{1,...,m(1)}
d x1
i , x1
j + 1.
In the second step, we choose n(2) > n(1) + m(2) (n(2) ∈ N). We deﬁne
ψk := ψk+22n(1)+1 , k ∈ {−22n(1)+1
− · · · − 23
− 2, · · · , −22n(1)−1
− · · · − 23
− 2 − 1},
ψk := ψk−22n(1)+2 , k ∈ {2 + 22
+ · · · + 22n(1)
, . . . , 2 + 22
+ · · · + 22n(1)+2
− 1},
...
ψk := ψk+22n(2)−1 , k ∈ {−22n(2)−1
− · · · − 23
− 2, · · · , −22n(2)−3
− · · · − 23
− 2 − 1},
and we put
εk := 0, k ∈ {2n(1) + 2, . . . , 2n(2)}, ε2n(2)+1 := 2−1
. (3.6)
Since n(2) > n(1) + m(2), from the above deﬁnition of ψk, it follows that, for each
j ∈ {1, . . . , m(1)}, there exist at least 2m(2) + 2 integers
l ∈ {−22n(2)−1
− · · · − 23
− 2, . . . , 22n(2)−2
+ · · · + 22
+ 2 − 1}
such that ψl = x1
j . Thus, we can deﬁne
ψk ∈ Oε2n(2)+1
(ψk−22n(2) ) , k ∈ {2 + 22
+ · · · + 22n(2)−2
, . . . , 2 + 22
+ · · · + 22n(2)
− 1},
with the property that
{ψk; k ∈ {2 + 22
+ · · · + 22n(2)−2
, . . . , 2 + 22
+ · · · + 22n(2)−2
+ 22n(2)
− 1}}
= {x1
1, . . . , x1
m(1), x2
1, . . . , x2
m(2)}.
In addition, we can put
ψ22n(2) := ψ0 = x1
1 (3.7)
and we can assume that
ψk = x1
1 for some k ∈ {2 + · · · + 22n(2)−2
, . . . , 2 + · · · + 22n(2)
− 1} {22n(2)
}.
In the third step, we choose n(3) > n(2)+m(3) (n(3) ∈ N) and we proceed analogously.
We construct {ψk} for
k ∈ {−22n(2)+1
− · · · − 23
− 2, . . . , −22n(2)−1
− · · · − 23
− 2 − 1},
...
k ∈ {−22n(3)−1
− · · · − 23
− 2, . . . , −22n(3)−3
− · · · − 23
− 2 − 1}
3.2 Sequences with given values 73
as in the 2(n(2) + 1)-th, . . . , 2n(3)-th steps of the process (mentioned in Corollary 1.28)
for
εk := 0, k ∈ {2n(2) + 2, . . . , 2n(3)}. (3.8)
Especially, we have
ψk = x1
1, k ∈ J3
0 := {j 22n(2)
; j ∈ Z}∩{−22n(3)−1
−· · ·−2, . . . , 2+· · ·+22n(3)−2
−1}. (3.9)
As in the second step, for all j(1) ∈ {1, 2} and j(2) ∈ {1, . . . , m(j(1))}, there exist at
least 2m(3) + 2 integers
l ∈ {−22n(3)−1
− · · · − 23
− 2, . . . , 2 + 22
+ · · · + 22n(3)−2
− 1} {j 22n(2)
; j ∈ Z}
such that ψl = x
j(1)
j(2). It is seen that, to obtain
ψk ∈ Oε2n(3)+1
(ψk−22n(3) ) , k ∈ {2 + 22
+ · · · + 22n(3)−2
, . . . , 2 + 22
+ · · · + 22n(3)
− 1},
where
ε2n(3)+1 := 2−2
, (3.10)
satisfying
{ψk; k ∈ {2 + 22
+ · · · + 22n(3)−2
, . . . , 2 + 22
+ · · · + 22n(3)−2
+ 22n(3)
− 1}}
= {x1
1, . . . , x1
m(1), . . . , x3
1, . . . , x3
m(3)},
we need less than (or equal to) m(3) + 1 such integers l. Thus, we can deﬁne these ψk so
that
ψk = x1
1, k ∈ I3
0 := {j 22n(2)
; j ∈ Z} ∩ {2 + · · · + 22n(3)−2
, . . . , 2 + · · · + 22n(3)
− 1}, (3.11)
ψ22n(3)+1 = ψ1 = x1
2, ψ22n(3)−1 = ψ−1 = x1
1 (3.12)
and
ψk = ψ1 for some k ∈ {2 + · · · + 22n(3)−2
, . . . , 2 + · · · + 22n(3)
− 1} {22n(3)
+ 1},
ψk = ψ−1 for some k ∈ {2 + · · · + 22n(3)−2
, . . . , 2 + · · · + 22n(3)
− 1} {22n(3)
− 1}.
We proceed further in the same way. In the i-th step, we have n(i) > n(i − 1) + m(i)
(n(i) ∈ N) and
ψk := ψk+22n(i−1)+1 , k ∈ {−22n(i−1)+1
− · · · − 2, · · · , −22n(i−1)−1
− · · · − 2 − 1},
...
ψk := ψk+22n(i)−1 , k ∈ {−22n(i)−1
− · · · − 2, · · · , −22n(i)−3
− · · · − 2 − 1},
and we denote
εk := 0, k ∈ {2n(i − 1) + 2, . . . , 2n(i)}, ε2n(i)+1 := 2−i+1
. (3.13)
3.2 Sequences with given values 74
We also have
ψk = ψ0, k ∈ Ji
0 := {j 22n(2)
; j ∈ Z}∩{−22n(i)−1
−· · ·−2, . . . , 2+· · ·+22n(i)−2
−1}, (3.14)
ψk = ψ1, k ∈ Ji
1,
Ji
1 := {1 + j 22n(3)
; j ∈ Z} ∩ {−22n(i)−1
− · · · − 2, . . . , 2 + · · · + 22n(i)−2
− 1},
(3.15)
ψk = ψ−1, k ∈ Ji
−1,
Ji
−1 := {−1 + j 22n(3)
; j ∈ Z}
∩ {−22n(i)−1
− · · · − 23
− 2, . . . , 2 + 22
+ · · · + 22n(i)−2
− 1},
(3.16)
...
ψk = ψi−3, k ∈ Ji
i−3,
Ji
i−3 := {i − 3 + j 22n(i−1)
; j ∈ Z}
∩ {−22n(i)−1
− · · · − 23
− 2, . . . , 2 + 22
+ · · · + 22n(i)−2
− 1},
ψk = ψ−i+3, k ∈ Ji
−i+3,
Ji
−i+3 := {−i + 3 + j 22n(i−1)
; j ∈ Z}
∩ {−22n(i)−1
− · · · − 23
− 2, . . . , 2 + 22
+ · · · + 22n(i)−2
− 1},
if i − 3 < 22n(2)
. If 22n(2)
≤ i − 3 < 22n(2)+1
, we have
...
ψk = ψ−22n(2)+1, k ∈ Ji
−22n(2)+1,
Ji
−22n(2)+1 := {−22n(2)
+ 1 + j 22n(22n(2)+1)
; j ∈ Z}
∩ {−22n(i)−1
− · · · − 23
− 2, . . . , 2 + 22
+ · · · + 22n(i)−2
− 1},
ψk = ψ22n(2)+1, k ∈ Ji
22n(2)+1,
Ji
22n(2)+1 := {22n(2)
+ 1 + j 22n(22n(2)+2)
; j ∈ Z}
∩ {−22n(i)−1
− · · · − 23
− 2, . . . , 2 + 22
+ · · · + 22n(i)−2
− 1},
...
If 22n(2)+1
≤ i−3, then we omit the values ψj 22n(2) , ψ1+j 22n(3) , ψ−1+j 22n(3) , . . . For simplicity,
let i − 2 < 22n(2)
.
Considering the construction, for all j(1) ∈ {1, . . . , i − 1}, j(2) ∈ {1, . . . , m(j(1))},
there exist at least 2m(i) + 2 integers
l ∈ {−22n(i)−1
− · · · − 23
− 2, . . . , 2 + 22
+ · · · + 22n(i)−2
− 1} (Ji
0 ∪ · · · ∪ Ji
−i+3)
such that ψl = x
j(1)
j(2). Evidently (similarly as in the third step), we can obtain
ψk ∈ Oε2n(i)+1
(ψk−22n(i) ) , k ∈ {2 + · · · + 22n(i)−2
, . . . , 2 + · · · + 22n(i)
− 1},
3.2 Sequences with given values 75
for which
{ψk; k ∈ {2 + · · · + 22n(i)−2
, . . . ,2 + · · · + 22n(i)−2
+ 22n(i)
− 1}}
= {x1
1, . . . , x1
m(1), . . . , xi
1, . . . , xi
m(i)}
(3.17)
and, in addition, we have
ψk = ψ0, k ∈ Ii
0,
Ii
0 := {j 22n(2)
; j ∈ Z} ∩ {2 + · · · + 22n(i)−2
, . . . , 2 + · · · + 22n(i)
− 1},
(3.18)
ψk = ψ1, k ∈ Ii
1,
Ii
1 := {1 + j 22n(3)
; j ∈ Z} ∩ {2 + · · · + 22n(i)−2
, . . . , 2 + · · · + 22n(i)
− 1},
(3.19)
ψk = ψ−1, k ∈ Ii
−1,
Ii
−1 := {−1 + j 22n(3)
; j ∈ Z} ∩ {2 + · · · + 22n(i)−2
, . . . , 2 + · · · + 22n(i)
− 1},
(3.20)
...
ψk = ψi−3, k ∈ Ii
i−3,
Ii
i−3 := {i − 3 + j 22n(i−1)
; j ∈ Z} ∩ {2 + · · · + 22n(i)−2
, . . . , 2 + · · · + 22n(i)
− 1},
ψk = ψ−i+3, k ∈ Ii
−i+3,
Ii
−i+3 := {−i + 3 + j 22n(i−1)
; j ∈ Z} ∩ {2 + · · · + 22n(i)−2
, . . . , 2 + · · · + 22n(i)
− 1},
and
ψk = ψi−2, k = 22n(i)
+ i − 2, ψk = ψ−i+2, k = 22n(i)
− i + 2,
ψk = ψi−2 for some k ∈ {2 + · · · + 22n(i)−2
, . . . , 2 + · · · + 22n(i)
− 1} {22n(i)
+ i − 2},
ψk = ψ−i+2 for some k ∈ {2 + · · · + 22n(i)−2
, . . . , 2 + · · · + 22n(i)
− 1} {22n(i)
− i + 2}.
Using this construction, we get the sequence {ψk}k∈Z ⊆ X with the property that
(see (3.7), (3.9), (3.11), (3.14), (3.18))
ψk = ψ0, k ∈ {j 22n(2)
; j ∈ Z},
and that (see (3.12), (3.15), (3.16), (3.19), (3.20))
ψk = ψ1, k ∈ {1 + j 22n(3)
; j ∈ Z},
ψk = ψ−1, k ∈ {−1 + j 22n(3)
; j ∈ Z},
and so on; i.e., for any l ∈ Z, there exists i(l) ∈ N satisfying
ψk = ψl, k ∈ {l + j 22n(i(l))
; j ∈ Z}. (3.21)
Now it suﬃces to show that the sequence {ψk} is limit periodic. Indeed, (3.1) follows
from the process, (3.3), (3.4), and (3.17); (3.2) follows from (3.21) for q(l) = 22n(i(l))
.
Since we construct {ψk} using Corollary 1.28, {ψk} is limit periodic if (1.25) is satisﬁed.
Considering (3.5), (3.6), (3.8), (3.10), and (3.13), we have
∞
i=1
εi = L (2n(1) + 1) + 1 (3.22)
which completes the proof.
3.2 Sequences with given values 76
From Theorem 3.1, we immediately obtain the following result concerning almost periodic
sequences.
Corollary 3.2. For any countable and totally bounded set X ⊆ X, there exists an almost
periodic sequence {ψk}k∈Z satisfying (3.1) with the property that, for any l ∈ Z, there exists
q(l) ∈ N such that (3.2) is valid.
Remark 3.3. We can mention several other corollaries which follow from Theorem 3.1,
when the limit periodicity of {ψk} is replaced by a concrete type of almost periodicity in the
statement. For example, one can consider almost automorphic sequences which generalize
classical almost periodicity and which have totally bounded ranges as well (see, e.g., [79,
Deﬁnition 3.2 and Theorem 3.3, (v)]).
Before the formulation of the next theorem, we add that a sequence {xk}k∈N ⊆ X is
dense in itself if for any k ∈ N and ε > 0, there exist inﬁnitely many x ∈ {xk; k ∈ N} with
the property that d(xk, x) < ε.
Theorem 3.4. Let a sequence {xk}k∈N ⊆ X be totally bounded and dense in itself. There
exists an injective limit periodic sequence {fk}k∈Z satisfying
{fk; k ∈ Z} = {xk; k ∈ N} . (3.23)
Proof. Without loss of generality, we can assume that the sequence {xk}k∈N is injective.
We put X := {xk; k ∈ N}. We know that, for any n ∈ N, there exists an odd integer
m = m(n) ≥ n such that
min
i∈{1,...,m}
d (xi, xl) <
1
2n
, l ∈ N. (3.24)
In the ﬁrst step, let us consider the periodic sequence {f1
k }k∈Z given by values
f1
0 = x1, f1
1 = x2, f1
−1 = x3, . . . , f1
m(1) = x2m(1), f1
−m(1) = x2m(1)+1
and period 2m(1) + 1. In the second step, we deﬁne a sequence {f2
k }k∈Z ⊂ X with period
[2m(1) + 1]m(2) so that
f2
−m(1) = f1
−m(1), . . . , f2
0 = f1
0 , . . . , f2
m(1) = f1
m(1),
x1, . . . , xm(2) ⊂ f2
l ; l ∈ {1, . . . , [2m(1) + 1] m(2)} , (3.25)
d f2
l , f1
l <
1
2
, l ∈ {1, . . . , [2m(1) + 1] m(2)}, (3.26)
f2
i = f2
j , i = j, i, j ∈ {1, . . . , [2m(1) + 1] m(2)}. (3.27)
We can ﬁnd such a sequence {f2
k }k∈Z. Conditions (3.25), (3.26) follow from (3.24) and
(3.27) can be satisﬁed because, in any neighbourhood of each considered value xi, there
are inﬁnitely many values from X.
3.2 Sequences with given values 77
We proceed further in the same way. In the n-th step, we deﬁne a sequence {fn
k }k∈Z ⊂ X
with period [2m(1) + 1] m(2) · · · m(n) arbitrarily so that
fn
−[2m(1)+1]m(2)···m(n−1)+1
2
= fn−1
−[2m(1)+1]m(2)···m(n−1)+1
2
, . . . , fn
0 = fn−1
0 , . . .
. . . , fn
[2m(1)+1]m(2)···m(n−1)−1
2
= fn−1
[2m(1)+1]m(2)···m(n−1)−1
2
and
x1, . . . , xm(n) ⊂ {fn
l ; l ∈ {1, . . . , [2m(1) + 1] m(2) · · · m(n)}} , (3.28)
d fn
l , fn−1
l <
1
2n−1
, l ∈ {1, . . . , [2m(1) + 1] m(2) · · · m(n)}, (3.29)
fn
i = fn
j , i = j, i, j ∈ {1, . . . , [2m(1) + 1] m(2) · · · m(n)}. (3.30)
We put
fk := lim
n→∞
fn
k = f
| k |+j
k , k ∈ Z, j ∈ N.
We have (see (3.29))
d (fk, fn
k ) ≤ d fn
k , fn+1
k + d fn+1
k , fn+2
k + · · · <
1
2n−1
, k ∈ Z, n ∈ N.
Thus, the sequence {fk}k∈Z is limit periodic. Since
fl = fn
l , l ∈
− [2m(1) + 1] m(2) · · · m(n) + 1
2
, . . . , 0,
. . . ,
[2m(1) + 1] m(2) · · · m(n) − 1
2
, n ∈ N {1},
from (3.30), we obtain
fi = fj, i = j, i, j ∈
− [2m(1) + 1] m(2) · · · m(n) + 1
2
, . . . , 0,
. . . ,
[2m(1) + 1] m(2) · · · m(n) − 1
2
, n ∈ N {1},
i.e., we have (m(n) → ∞ for n → ∞)
fi = fj, i = j, i, j ∈ Z.
It remains to prove that {fk} satisﬁes (3.23). Of course, this fact can be easily shown
considering (3.28).
Again, we obtain a new result in the almost periodic case.
Corollary 3.5. Let a sequence {xk}k∈N ⊆ X be totally bounded and dense in itself. There
exists an injective almost periodic sequence {fk}k∈Z satisfying (3.23).
3.3 Applications related to almost periodic diﬀerence systems 78
Remark 3.6. For any sequence {xk}k∈N ⊆ X which is not dense in itself, Theorem 3.4
cannot be valid. Indeed, if there exist l ∈ N and ε > 0 such that the intersection of the
ε-neighbourhood of xl and the set {xk; k ∈ N} contains only a ﬁnite number of elements of
{xk; k ∈ N}, then any almost periodic sequence {fk}k∈Z satisfying (3.23) attains a value
from this neighbourhood inﬁnitely many times (consider directly Deﬁnition 1.1).
At the end of this section, we assume that X is a uniform space with a non-empty
family U of entourages. We generalize the deﬁnition of limit periodicity from Chapter 1 as
follows.
Deﬁnition 3.7. A sequence {fk}k∈Z ⊆ X is called limit periodic if it is the uniform limit
of periodic sequences {fn
k }k∈Z ⊆ X, n ∈ N, i.e., for any U ∈ U, there exists n0 ∈ N such
that (fk, fn
k ) ∈ U for all k ∈ Z, n ≥ n0, n ∈ N.
We can also formulate the corresponding generalizations of almost periodicity and
asymptotic almost periodicity. Especially (see, e.g., [10]), a continuous multivalued map
f : R → X is said to be almost periodic if for any entourage U ∈ U, there exists a positive
number p ∈ R such that every real interval of length p contains a number s for which
f(x + s) is in the U-neighbourhood of f(x) and intersects the U-neighbourhood of all
y ∈ f(x), x ∈ R. For the fundamental properties of this type of almost periodic functions,
we refer to [10, 11, 109, 158] (and also [50, 51, 139]).
If there exists a countable fundamental system of entourages of X, then the uniform
structure of X can be deﬁned by a pseudometric (see, e.g., [107, Theorem 13 on p. 186]
and references cited therein). Hence, we have:
Theorem 3.8. The statement of Theorem 3.1 remains true if X is a uniform space with
a countable fundamental system of entourages.
Considering Theorem 3.4, we also obtain the following result.
Theorem 3.9. Let X have a countable fundamental system of entourages and let a totally
bounded sequence {xk}k∈N ⊆ X be such that, for any k ∈ N and any entourage U ∈ U,
there exist inﬁnitely many x ∈ {xk; k ∈ N} with the property that (xk, x) ∈ U. Then, there
exists an injective limit periodic sequence {fk}k∈Z satisfying (3.23).
3.3 Applications related to almost periodic diﬀerence
systems
Let m ∈ N be arbitrarily given. We consider m-dimensional homogeneous linear difference
systems of the form
yk+1 = Ak · yk, k ∈ Z,
where {Ak} is an almost periodic sequence of matrices from a given group X of m × m
matrices with complex elements. Of course, in C, we consider the usual metric d given by
the absolute value of diﬀerence. This metric induces the metric on the set Cm
of all m × 1
complex vectors and on the set Cm×m
of all m × m complex matrices as the sum of m and
m2
non-negative numbers, respectively. For the sake of simplicity and convenience, these
metrics are denoted by d as well. Let I ∈ Cm×m
be the identity matrix.
3.3 Applications related to almost periodic diﬀerence systems 79
Lemma 3.10. Let X ⊂ Cm×m
be a bounded group. If {fk}k∈Z ⊆ X is almost periodic,
then the sequence {f−1
k }k∈Z of the inverse matrices is almost periodic as well.
Proof. For an arbitrary matrix norm (especially, for the 1-norm which corresponds to d)
denoted by || · ||, we have (2.3) for any matrices A, E such that A is non-singular and
A−1
E < 1. For the bounded group X, (2.3) implies that the map C → C−1
has
the Lipschitz property on X. Hence, the almost periodicity of {f−1
k }k∈Z is guaranteed by
Theorem 1.6.
Theorem 3.11. Let X ⊂ Cm×m
be a bounded group. There exists an almost periodic
sequence {Ck}k∈Z ⊆ X satisfying {Ck; k ∈ Z} = X such that all solutions of the system
yk+1 = Ck · yk, k ∈ Z, (3.31)
are almost periodic.
Proof. It is seen that every bounded group of complex matrices has a dense countable
subgroup. Thus, we can assume that X = {xk; k ∈ N}. We apply Corollary 3.2. There
exists an almost periodic sequence {fk}k∈Z satisfying (3.23). From Corollary 1.10 and
Lemma 3.10, it follows that the sequence
C2k := fk, C2k+1 := f−1
k , k ∈ Z, (3.32)
is almost periodic as well. Note that f−1
k denotes the inverse matrix of fk and that f−1
k = fj
for some j = j(k) ∈ Z. The principal fundamental matrix {Φk}k∈Z of the system (3.31)
given by (3.32) attains values
Φ(0) = I, Φ(1) = f0, Φ(2) = I, . . . Φ(2n) = I, Φ(2n + 1) = fn, . . .
Φ(−1) = f−1, Φ(−2) = I, . . . Φ(−2n) = I, Φ(−2n − 1) = f−n−1, . . .
The almost periodicity of {Φk} is equivalent to the almost periodicity of {fk} (see directly
Deﬁnition 1.1).
To obtain counterparts of Theorem 3.11 (the below given Theorems 3.12 and 3.13), we
use Corollary 2.33 and Theorem 2.35.
Theorem 3.12. Let X ⊂ Cm×m
be a weakly transformable group. If there exists a matrix
M(u) ∈ X0 for any non-zero vector u ∈ Cm
such that M(u) u = u, then there exists an
almost periodic sequence {Dk}k∈Z ⊆ X satisfying {Dk; k ∈ Z} = X for which the system
yk+1 = Dk · yk, k ∈ Z, (3.33)
does not have any non-trivial almost periodic solution.
Proof. Evidently, in the complex case, any weakly transformable group has a dense countable
weakly transformable subgroup. Hence, as in the proof of Theorem 3.11, we can
assume that X = {xk; k ∈ N} and we can apply Corollary 3.2. We obtain an almost
periodic sequence {fk}k∈Z with the property that {fk; k ∈ Z} = {xk; k ∈ N} .
3.3 Applications related to almost periodic diﬀerence systems 80
Let {Bk}k∈Z ⊆ {xk; k ∈ N} be an arbitrary almost periodic sequence mentioned in the
statement of Corollary 2.33. We put
D3k := fk, D3k+1 := f−1
k , D3k+2 := Bk, k ∈ Z. (3.34)
Corollary 1.10 and Lemma 3.10 give the almost periodicity of {Dk}k∈Z. For the principal
fundamental matrix {Ψk}k∈Z of the system (3.33) determined by (3.34), we have
Ψ(0) = I, Ψ(1) = f0, Ψ(2) = I, Ψ(3) = B0,
...
Ψ(3n) = Bn−1 · · · B1 · B0,
Ψ(3n + 1) = fn · Bn−1 · · · B1 · B0,
Ψ(3n + 2) = Bn−1 · · · B1 · B0,
...
and
Ψ(−1) = B−1
−1, Ψ(−2) = f−1 · B−1
−1, Ψ(−3) = B−1
−1,
...
Ψ(−3n + 2) = (B−1 · B−2 · · · B−n)−1
,
Ψ(−3n + 1) = f−n · (B−1 · B−2 · · · B−n)−1
,
Ψ(−3n) = (B−1 · B−2 · · · B−n)−1
,
...
From Corollary 2.33, we know that the sequence {Ψ3k u}k∈Z is not almost periodic for any
non-zero vector u ∈ Cm
. Thus, the sequence {Ψk u}k∈Z cannot be almost periodic.
Analogously, using Theorem 2.35, one can prove:
Theorem 3.13. Let X ⊂ Cm×m
be a weakly transformable group and u ∈ Cm
be an
arbitrary non-zero vector. If there exists a matrix M(u) ∈ X0 such that M(u) u = u, then
there exists an almost periodic sequence {Dk}k∈Z ⊆ X satisfying {Dk; k ∈ Z} = X for
which the solution of
xk+1 = Dk · xk, k ∈ Z, x0 = u
is not almost periodic.
From the proofs of Theorems 3.11 and 3.12, we also get the following theorems which
ﬁnish this chapter. Note that we partially study almost periodic systems in Chapter 4 as
well.
3.3 Applications related to almost periodic diﬀerence systems 81
Theorem 3.14. For any countable bounded group X ⊂ Cm×m
, there exists an almost
periodic sequence {Ck}k∈Z satisfying {Ck; k ∈ Z} = X such that all solutions of the system
yk+1 = Ck · yk, k ∈ Z,
are almost periodic.
Theorem 3.15. Let X ⊂ Cm×m
be a countable weakly transformable group and let there
exist a matrix M(u) ∈ X0 for any non-zero vector u ∈ Cm
such that M(u) u = u. There
exists an almost periodic sequence {Dk}k∈Z satisfying {Dk; k ∈ Z} = X for which the
system
yk+1 = Dk · yk, k ∈ Z,
does not have any non-trivial almost periodic solution.
Theorem 3.16. Let X ⊂ Cm×m
be a countable weakly transformable group and let u ∈ Cm
be an arbitrary non-zero vector. If there exists a matrix M(u) ∈ X0 such that M(u) u = u,
then there exists an almost periodic sequence {Dk}k∈Z satisfying {Dk; k ∈ Z} = X for
which the solution of
xk+1 = Dk · xk, k ∈ Z, x0 = u
is not almost periodic.
Chapter 4
Solutions of limit periodic diﬀerence
systems
Now we consider limit periodic homogeneous linear difference systems of the form
xk+1 = Ak · xk, k ∈ Z. (4.1)
Limit periodic systems (4.1) form the smallest class of systems which generalize pure
periodic systems in the form (4.1) and which can have at least one non-almost periodic
solution (for complex matrices Ak from a bounded group, cf. Corollary 2.22). The basic
motivation comes from Chapter 2, where we study non-almost periodic solutions of almost
periodic systems. In this chapter, we improve the main results of Chapter 2 in a certain
sense, because we analyse non-almost periodic solutions in the limit periodic case. In
addition, we obtain results about non-asymptotically almost periodic solutions. Since the
methods used in this chapter are substantially diﬀerent from the process from Chapter 1
applied in Chapter 2, we also obtain new results for almost periodic systems.
The necessity of generalizations of periodic mathematical models is implied by various
oscillatory phenomena in natural sciences. The models induce the research of limit periodic,
almost periodic, and asymptotically almost periodic sequences in connection with diﬀerence
equations. There are many signiﬁcant books dealing with (asymptotically) almost periodic
solutions of diﬀerence and diﬀerential equations, e.g., [32, 39, 46, 72, 183]. We also refer
to references given in these books.
In this chapter, we use methods based on constructions of limit periodic sequences. A similar
method was ﬁrstly applied in [78], where non-almost periodic solutions of homogeneous
linear diﬀerence equations are found as classes of constructible sequences. A method
of constructions of minimal cocycles, which are obtained as solutions of homogeneous linear
diﬀerential systems, is described in [141] (see also [140]). Special constructions of homogeneous
linear diﬀerential systems with almost periodic coeﬃcients are used in [124, 125] as
well.
This chapter is organized as follows. We begin with the used notations. In Section 4.2,
we introduce properties (denoted by P and P∗
) which allow us to improve the results of
Chapter 2 for bounded groups of coeﬃcient matrices in Section 4.3. In Section 4.4, the
coeﬃcient matrices of the considered systems belong to a given commutative group. We
82
4.1 Preliminaries 83
ﬁnd a condition on the group under which the systems, whose fundamental matrices are not
almost periodic, form an everywhere dense subset in the space of all considered systems.
The treated problems are discussed for the elements of the coeﬃcient matrices from an
arbitrary inﬁnite ﬁeld with an absolute value. Nevertheless, the presented results are new
even for the ﬁeld of complex numbers.
4.1 Preliminaries
Let (F, ⊕, ) be an inﬁnite ﬁeld with a zero e0 and a unit e1. Let | · | : F → R be an
absolute value on F; i.e., let
(a) | f | ≥ 0 and | f | = 0 if and only if f = e0,
(b) | f g | = | f | · | g |,
(c) | f ⊕ g | ≤ | f | + | g |
for all f, g ∈ F. As F, we understand each one of the ﬁelds C, R, Q with the usual absolute
value. For arbitrary p ∈ N, we put pN := {pj; j ∈ N}. We remark that i ∈ C stands for
the imaginary unit.
Let m ∈ N be arbitrarily given (as the dimension of systems under consideration).
Symbol Mat(F, m) denotes the set of all m × m matrices with elements from F and Fm
the set of all m × 1 vectors with entries from F. Next, ·, + stand for the multiplication
and the addition on spaces Mat(F, m), Fm
. As usual, we denote the identity matrix
I ∈ Mat(F, m), the zero matrix O ∈ Mat(F, m), and the zero vector o ∈ Fm
.
The absolute value on F gives the norm on Fm
and Mat(F, m) as the sum of m and
m2
non-negative numbers which are the absolute values of the elements, respectively. For
simplicity, both of the norms are denoted by · . Especially (consider (a), (b), (c)), we
have
(A) M ≥ 0 and M = 0 if and only if M = O,
(B) u ≥ 0 and u = 0 if and only if u = o,
(C) M + N ≤ M + N ,
(D) u + v ≤ u + v ,
(E) M · N ≤ M · N ,
(F) M · u ≤ M · u
for all M, N ∈ Mat(F, m), u, v ∈ Fm
. Henceforth, we will use properties (A), (B), (C),
(D), (E), (F) without emphasizing.
The absolute value on F and the norms on Fm
, Mat(F, m) induce the metrics. For the
sake of convenience, we denote each one of these metrics by . The ε-neighbourhood of α
is denoted by Oε (α) in all above mentioned spaces with metric . In addition, we assume
that the valued ﬁeld F (with | · |) is separable. Hence, all considered spaces are separable.
We remark that metric space (F, ) does not need to be complete.
4.2 General homogeneous linear diﬀerence systems 84
4.2 General homogeneous linear diﬀerence systems
Let X ⊂ Mat(F, m) be a group. We study the homogeneous linear diﬀerence systems
xk+1 = Ak · xk, k ∈ Z, where {Ak} ⊆ X. (4.2)
Let P (X), LP (X), and AP (X) denote the set of all systems (4.2) for which the sequence
of matrices Ak is periodic, limit periodic, and almost periodic, respectively. Note that we
identify the sequence {Ak} with the system (4.2) which is determined by {Ak}. In AP (X),
we deﬁne the metric
σ ({Ak}, {Bk}) := sup
k∈Z
(Ak, Bk) , {Ak}, {Bk} ∈ AP (X).
Symbol Oσ
ε ({Ak}) stands for the ε-neighbourhood of {Ak} in AP (X).
To study limit periodic systems of the form (4.2), we introduce properties of X denoted
by P and P∗
as follows.
Deﬁnition 4.1. We say that X has property P if for every a > 0 and δ > 0, there exist
ζ(a) > 0 and l = l(a, δ) ∈ N such that, for any vector u ∈ Fm
satisfying u > a, one can
ﬁnd matrices M1, . . . , Ml ∈ X for which
M1 ∈ Oδ (I), Mi ∈ Oδ (Mi+1), i ∈ {1, . . . , l − 1}, Ml · u − u > ζ(a).
Deﬁnition 4.2. We say that X has property P∗
if for any a > 0 and δ > 0, there exist
M(a) ∈ X, ζ(a) > 0, and l = l(a, δ) ∈ N such that, for any N ∈ X, one can ﬁnd matrices
M1, . . . , Ml ∈ X satisfying
M1 ∈ Oδ (N), Mi ∈ Oδ (Mi+1), i ∈ {1, . . . , l − 1}, Ml = M(a)
and
M(a) · u − u > ζ(a), u ∈ Fm
, u > a.
We formulate Deﬁnitions 4.1 and 4.2 in the above form for general a > 0, because we
apply symbols ζ(a), l(a, δ) later (i.a., in the proofs of the main results of this chapter).
Of course, we obtain the identical deﬁnitions if we consider only one number a < 1 as
arbitrarily given. Indeed, we have
fl
= | f |l
, f u = | f | · u , f ∈ F, u ∈ Fm
, l ∈ Z.
Thus, we can simplify them into the following forms.
Deﬁnition 4.3. The group X has property P if there exists ζ > 0 and if for all δ > 0,
there exists l ∈ N such that, for any vector u ∈ Fm
satisfying u ≥ 1, one can ﬁnd
matrices M1, . . . , Ml ∈ X with the property that
M1 ∈ Oδ (I), Mi ∈ Oδ (Mi+1), i ∈ {1, . . . , l − 1}, Ml · u − u > ζ.
4.2 General homogeneous linear diﬀerence systems 85
Deﬁnition 4.4. The group X has property P∗
if there exist M ∈ X and ζ > 0 such that,
for every δ > 0, there exists l ∈ N with the property that, for any N ∈ X, one can ﬁnd
matrices M1, . . . , Ml ∈ X satisfying
M1 ∈ Oδ (N), Mi ∈ Oδ (Mi+1), i ∈ {1, . . . , l − 1}, Ml = M
and
M · u − u > ζ, u ∈ Fm
, u ≥ 1.
Remark 4.5. Especially, X has property P if there exist ζ > 0 and continuous functions
fi : [0, 1] → Mat(F, m) for i ∈ {1, . . . , n} with the properties that
fi(r) ∈ X, r ∈ Q ∩ [0, 1], fi(0) = I, i ∈ {1, . . . , n},
and
max
i∈{1,...,n}
fi(1) · u − u > ζ, u ∈ Fm
, u ≥ 1.
Remark 4.6. Let F = C and let X be weakly transformable. If there exists a matrix
M ∈ X0 such that Mu = u for all non-zero vectors u ∈ Cm
, then X has property P. It
follows directly from Deﬁnitions 2.14 and 4.3 and, considering the compactness of the set
C := {u ∈ Cm
; u = 1}, from the inequality
inf
u∈C
M · u − u > 0.
Note that the number l ∈ N considered in Deﬁnition 4.3 has to exist for any δ > 0, because
the set X is totally bounded.
Analogously, if for every non-zero vector u ∈ Cm
, there exists a matrix M(u) ∈ X0
satisfying M(u)u = u, then we can assume that
inf
u∈C
M(u) · u − u > 0.
Indeed, there exists a ﬁnite number of matrices M1, . . . , Mj ∈ X0 satisfying
max
i∈{1,...,j}
Mi · u − u > 0, u ∈ Cm
, u = 1. (4.3)
Hence, in this case, X has property P as well.
Example 4.7. Based on Remark 4.6, we can mention many examples of weakly transformable
groups X ⊂ Mat(F, m) with property P. Since each one of concrete transformable
and weakly transformable groups mentioned in Examples 2.2–2.10, 2.15, and 2.17 contains
matrices M1, . . . , Mj from a transformable subgroup such that (4.3) is satisﬁed, all these
groups have property P (it also follows from Remark 4.5). ♦
Remark 4.8. It is seen that X has property P if it has property P∗
. One can trivially
show that the converse implication is not true. It suﬃces to consider the real case for m = 3
and the group X = SO(3) which consists of all orthogonal matrices with determinant 1.
This group has property P (see Example 4.7). For any matrix M ∈ SO(3), there exists
a vector u ∈ R3
with the properties that Mu = u and u = 1. This fact implies that
X = SO(3) cannot have property P∗
.
4.2 General homogeneous linear diﬀerence systems 86
To formulate our results in a simple and consistent form, we also introduce the following
deﬁnition concerning non-asymptotically almost periodic solutions of systems (4.2).
Deﬁnition 4.9. We say that a system {Ak}k∈Z ∈ AP (X) does not have any non-zero
asymptotically almost periodic solution in the strong sense if there exists a sequence
{ln}n∈N ⊆ N satisfying limn→∞ ln = ∞ with the property that, for any non-zero solution
{xk}k∈Z, there exist ϑ > 0 and Q ∈ N such that the inequality
xk+li
− xk+lj
> ϑ
is valid for all i > j ≥ Q and for a set of k which is relatively dense in N and which depends
on i and j.
Remark 4.10. Evidently, the systems considered in Deﬁnition 4.9 form a special class
of systems whose non-zero solutions are not asymptotically almost periodic. For example,
the scalar system xk+1 = eik
xk (for k ∈ Z) does not have non-zero asymptotically
almost periodic solutions, but it is not true that this system does not have any non-zero
asymptotically almost periodic solution in the strong sense.
In Section 4.4, we intend to improve results of Section 4.3. To show how the results of
Section 4.4 improve theorems from Section 4.3, we need to reformulate Deﬁnition 4.3 for
bounded groups applying the following two lemmas (which we will need later as well).
Lemma 4.11. Let p ∈ N be given. The multiplication of p matrices is continuous in the
Lipschitz sense on any bounded subset of Mat(F, m).
Proof. Let K > 0 be given. Since the addition and the multiplication have the Lipschitz
property on the set of f ∈ F satisfying | f | < K, the statement of the lemma is true.
Lemma 4.12. Let a bounded group X ⊆ Mat(F, m) be given. There exists L > 1 such
that
M · N−1
, N−1
· M ∈ OaL(I) if M, N ∈ X, M ∈ Oa(N). (4.4)
Proof. We know that the inequality
M < K, M ∈ X, i.e., M−1
< K, M ∈ X, (4.5)
holds for some K > 0. The map f → −f, the multiplication, and the addition have the
Lipschitz property on the set of all f ∈ F satisfying | f | < K. In addition, for any M ∈ X,
we have (see (4.5))
det M < m!Km
, det M =
1
det M−1
>
1
m!Km
.
Hence, the map
M →
1
det M
, M ∈ X,
has the Lipschitz property as well. Let a matrix M ∈ X be given. If we use the expression
m−1
i,j =
Mj,i
det M
, i, j ∈ {1, . . . , m},
4.3 Systems without asymptotically almost periodic solutions 87
where m−1
i,j are elements of M−1
∈ X and Mj,i are the algebraic complements of the
elements mj,i of M, then it is seen that the map M → M−1
is continuous in the Lipschitz
sense on X.
Evidently, Lemma 4.11 and the Lipschitz continuity of M → M−1
on X imply the
existence of L > 1 for which (4.4) is valid.
Using Lemmas 4.11 and 4.12 for bounded X and for M2 replaced by M2 · M1, . . . , Ml
by Ml · · · M2 · M1, we can rewrite Deﬁnition 4.3 as follows.
Deﬁnition 4.13. A bounded group X ⊂ Mat(F, m) has property P if there exists ζ > 0
and if for all δ > 0, there exists l ∈ N such that, for any vector u ∈ Fm
satisfying u ≥ 1,
one can ﬁnd matrices M1, . . . , Ml ∈ X with the property that
Mi ∈ Oδ (I), i ∈ {1, . . . , l}, Ml · · · M1 · u − u > ζ. (4.6)
We introduce the following direct generalization of Deﬁnition 4.13.
Deﬁnition 4.14. Let a non-zero vector u ∈ Fm
be given. We say that X has property P
with respect to u if there exists ζ > 0 such that, for all δ > 0, one can ﬁnd matrices
M1, . . . , Ml ∈ X satisfying (4.6).
Remark 4.15. Since a group with property P has property P with respect to any non-zero
vector u (consider || f u || = | f | · || u ||, f ∈ F, u ∈ Fm
), we can refer to a lot
of examples recalled in Example 4.7. Furthermore, we point out that any group, which
contains a subgroup having property P with respect to a vector u, has property P with
respect to u as well.
4.3 Systems without asymptotically almost periodic
solutions
In this section, we consider that the given group X is bounded. We can directly prove
the main result of this section which reads as follows.
Theorem 4.16. Let X have property P. For any {Ak} ∈ LP(X) and ε > 0, there exists
a system {Bk} ∈ Oσ
ε ({Ak}) ∩ LP(X) which does not have any non-zero asymptotically
almost periodic solution.
Proof. Let {un}n∈N ⊂ Fm
be a sequence of non-zero vectors such that {un; n ∈ N} = Fm
.
We know that there exists K > 1 satisfying M < K for all M ∈ X. In addition, for
any un, there exists Ln > 0 satisfying
Mun > Ln, M ∈ X. (4.7)
Indeed, limj→∞ Mjun = 0 for a sequence of Mj ∈ X implies the following contradiction
un = lim
j→∞
M−1
j Mjun ≤ lim sup
j→∞
M−1
j · Mjun ≤ K lim
j→∞
Mjun = 0.
4.3 Systems without asymptotically almost periodic solutions 88
Since {Ak} ∈ LP(X), there exist sequences {Bn
k }k∈Z ⊆ X for n ∈ N with the property
that
Ak − Bn
k <
ε
2n+1K
, k ∈ Z, n ∈ N, (4.8)
where {Bn
k } has period pn ∈ N. We can assume that pn ≥ 2, n ∈ N.
For M, M1, . . . , Ml ∈ X, it is seen that
(M1 · · · Ml)−1
· M · M1 · · · Ml − I
≤ (M1 · · · Ml)−1
· M − I · M1 · · · Ml ≤ K2
M − I .
(4.9)
Let matrices Ak, . . . , Ak+i−1+l, Ek, . . . , Ek+i−1 ∈ X be arbitrary. The product
Ek · Ek+1 · · · Ek+i−1 · Ak · Ak+1 · · · Ak+i−1 · · · Ak+i−1+l
can be expressed in the form
Ak · Fk · · · Ak+i−1 · Fk+i−1 · · · Ak+i−1+l · Fk+i−1+l, (4.10)
where
Fk = (Ak)−1
· Ek · Ak,
...
Fk+i−1 = (Ak · Ak+1 · · · Ak+i−1)−1
· Ek+i−1 · Ak · Ak+1 · · · Ak+i−1,
and
Fk+i = I, . . . Fk+i−1+l = I.
Analogously, one can express
Ak · Ak+1 · · · Ak+i−1 · · · Ak+i−1+l · Ek · Ek+1 · · · Ek+i−1
in the above given form (4.10) for
Fk = Ak+1 · · · Ak+i−1+l · Ek · (Ak+1 · · · Ak+i−1+l)−1
,
...
Fk+i−1 = Ak+i · · · Ak+i−1+l · Ek+i−1 · (Ak+i · · · Ak+i−1+l)−1
,
and
Fk+i = I, . . . Fk+i−1+l = I.
In the both cases, using (4.9), we have
Fk, . . . , Fk+i−1+l ∈ OK2a(I) if Ek, . . . , Ek+i−1 ∈ Oa(I). (4.11)
Considering Lemma 4.12, let L > 1 be such that
M · N−1
, N−1
· M ∈ OaL(I) if M, N ∈ X, M ∈ Oa(N). (4.12)
4.3 Systems without asymptotically almost periodic solutions 89
For all ui, there exists ξ(Li) = ξ(ui) > 0 such that
M · v − u >
ζ(Li)
2
if M · u − u > ζ(Li), v − u ≤ ξ(Li), (4.13)
where M ∈ X is arbitrary. Indeed, we have
M · u − u ≤ M · u − M · v + M · v − u ≤ K v − u + M · v − u .
Assume that ξ(Li) < ζ(Li)/2. In fact, we can put ξ(Li) := ζ(Li)/(2K).
Let us consider ζ(L1) and matrices D1,1
1 , . . . , D1,1
l(1,1) ∈ X satisfying
D1,1
1 ∈ O ε
2K3L
(I), D1,1
i ∈ O ε
2K3L
D1,1
i+1 , i ∈ {1, . . . , l(1, 1) − 1},
D1,1
l(1,1) · u1 − u1 > ζ(L1).
Expressing
D1,1
l(1,1) = D1,1
l(1,1) · D1,1
l(1,1)−1
−1
· · · D1,1
2 · D1,1
1
−1
· D1,1
1 ,
where (see (4.12))
D1,1
l(1,1) · D1,1
l(1,1)−1
−1
, . . . , D1,1
2 · D1,1
1
−1
, D1,1
1 ∈ O ε
2K3
(I),
we know (see also (4.11) and (4.13)) that there exist matrices C1
0 , C1
1 , . . . , C1
2p1l(1,1)−1 ∈ X
such that
Ap1l(1,1)−2 · C1
p1l(1,1)−2 · · · A1 · C1
1 · A0 · C1
0 · u1 − u1 > ξ(L1)
and that
C1
0 , . . . , C1
p1l(1,1)−2 ∈ O ε
2K
(I), C1
p1l(1,1)−1 = · · · = C1
2p1l(1,1)−1 = I.
Indeed, one can put
C1
0 = · · · = C1
p1l(1,1)−2 = C1
p1l(1,1)−1 = · · · = C1
2p1l(1,1)−1 = I
if
Ap1l(1,1)−2 · · · A1 · A0 · u1 − u1 > ξ(L1).
We deﬁne the periodic sequence {C1
k}k∈Z ⊂ X with period 2p1l(1, 1) by the matrices
C1
0 , . . . , C1
p1l(1,1)−2, C1
p1l(1,1)−1, . . . , C1
2p1l(1,1)−1.
In the second step, we consider ζ(L1), ζ(L2) and the numbers (see Deﬁnition 4.1)
l(2, 1) := l L1,
ε
22K3L
, l(2, 2) := l L2,
ε
22K3L
.
We recall that, for any v1, v2 ∈ Fm
satisfying v1 > L1, v2 > L2, there exist matrices
D2,1
1 , . . . , D2,1
l(2,1) ∈ X, D2,2
1 , . . . , D2,2
l(2,2) ∈ X with the property that
D2,1
1 ∈ O ε
22K3L
(I), D2,1
i ∈ O ε
22K3L
D2,1
i+1 , i ∈ {1, . . . , l(2, 1) − 1},
4.3 Systems without asymptotically almost periodic solutions 90
D2,2
1 ∈ O ε
22K3L
(I), D2,2
i ∈ O ε
22K3L
D2,2
i+1 , i ∈ {1, . . . , l(2, 2) − 1},
and
D2,1
l(2,1) · v1 − v1 > ζ(L1), D2,2
l(2,2) · v2 − v2 > ζ(L2).
Assume that l(2, 2) ≥ l(2, 1) ≥ l(1, 1) > 1. Denote
h1 := (0!)2
p1l(1, 1), h2 := (1!)2
p1p2l(1, 1)l(2, 2).
Analogously as in the ﬁrst step (consider also (4.7)), we can show that there exist matrices
C2
0 , C2
1 , . . . , C2
222h2−1 ∈ X such that
Ah2+h2−2 · C1
h2+h2−2 · C2
h2+h2−2 · · · A0 · C1
0 · C2
0 · u1
− Ah2−1 · C1
h2−1 · C2
h2−1 · · · A0 · C1
0 · C2
0 · u1 > ξ(L1),
A2h2+h2−h1−1 · C1
2h2+h2−h1−1 · C2
2h2+h2−h1−1 · · · A0 · C1
0 · C2
0 · u1
− A2h2−1 · C1
2h2−1 · C2
2h2−1 · · · A0 · C1
0 · C2
0 · u1 > ξ(L1),
A3h2+h2−2 · C1
3h2+h2−2 · C2
3h2+h2−2 · · · A0 · C1
0 · C2
0 · u2
− A3h2−1 · C1
3h2−1 · C2
3h2−1 · · · A0 · C1
0 · C2
0 · u2 > ξ(L2),
A4h2+h2−h1−1 · C1
4h2+h2−h1−1 · C2
4h2+h2−h1−1 · · · A0 · C1
0 · C2
0 · u2
− A4h2−1 · C1
4h2−1 · C2
4h2−1 · · · A0 · C1
0 · C2
0 · u2 > ξ(L2)
and, at the same time, such that
C2
0 , C2
1 , . . . , C2
222h2−1 ∈ O ε
22K
(I),
C2
j2h1+0 = C2
j2h1+1 = · · · = C2
j2h1+h1−1 = I, j ∈ {0, 1, . . . , 22
p2l(2, 2) − 1},
C2
j22h1+h1
= C2
j22h1+h1+1 = · · · = C2
j22h1+2h1−1 = I, j ∈ {0, 1, . . . , 2p2l(2, 2) − 1},
i.e., only matrices
C2
j22h1+3h1
, C2
j22h1+3h1+1, . . . , C2
j22h1+4h1−1 ∈ O ε
22K
(I), j ∈ {0, 1, . . . , 2p2l(2, 2) − 1},
do not need to be I. Especially, we have
C2
j = I if C1
j = I for some j ∈ {0, 1, . . . , 22
2h2 − 1}.
We consider the periodic sequence {C2
k}k∈Z ⊂ X given by period 22
2h2 and the matrices
C2
0 , C2
1 , . . . , C2
222h2−1.
We continue in the same manner. Before the n-th step, we have
Ci
j2n−1h1+h1
= Ci
j2n−1h1+h1+1 = · · · = Ci
j2n−1h1+2h1−1 = I, j ∈ Z, i ∈ {1, . . . , n − 1},
4.3 Systems without asymptotically almost periodic solutions 91
for the resulting periodic sequences {Ci
k}k∈Z ⊂ X with periods 2i2
hi, i ∈ {3, . . . , n − 1},
where
hi := ((i − 1)!)2
p1p2 · · · pil(1, 1)l(2, 2) · · · l(i, i).
In the n-th step, we consider ζ(L1), . . . , ζ(Ln−1), ζ(Ln) and the integers
l(n, 1) := l L1,
ε
2nK3L
, . . . , l(n, n − 1) := l Ln−1,
ε
2nK3L
, l(n, n) := l Ln,
ε
2nK3L
.
Let
l(n, n) ≥ l(n, n − 1) ≥ · · · ≥ l(n, 1) ≥ · · · ≥ l(2, 2) ≥ l(2, 1) ≥ l(1, 1) > 1.
We put
hn := ((n − 1)!)2
p1p2 · · · pnl(1, 1)l(2, 2) · · · l(n, n).
For all v1, . . . , vn ∈ Fm
satisfying v1 > L1, . . . , vn > Ln, there exist matrices
Dn,1
1 , . . . , Dn,1
l(n,1), . . . , Dn,n
1 , . . . , Dn,n
l(n,n) ∈ X
with the property that
Dn,1
1 ∈ O ε
2nK3L
(I), Dn,1
i ∈ O ε
2nK3L
Dn,1
i+1 , i ∈ {1, . . . , l(n, 1) − 1},
...
Dn,n
1 ∈ O ε
2nK3L
(I), Dn,n
i ∈ O ε
2nK3L
Dn,n
i+1 , i ∈ {1, . . . , l(n, n) − 1},
and
Dn,1
l(n,1) · v1 − v1 > ζ(L1),
...
Dn,n
l(n,n) · vn − vn > ζ(Ln).
Thus, considering
hn − 1 > · · · > hn − hn−1 = hn−1 (n − 1)2
pnl(n, n) − 1 > hn−1l(n, n),
we know that there exist matrices Cn
0 , Cn
1 , . . . , Cn
2n2hn−1 ∈ X such that
Ahn+hn−2 · C1
hn+hn−2 · · · Cn
hn+hn−2 · · · A0 · C1
0 · · · Cn
0 · u1
− Ahn−1 · C1
hn−1 · · · Cn
hn−1 · · · A0 · C1
0 · · · Cn
0 · u1 > ξ(L1),
...
Anhn+hn−hn−1−1 · C1
nhn+hn−hn−1−1 · · · Cn
nhn+hn−hn−1−1 · · · A0 · C1
0 · · · Cn
0 · u1
− Anhn−1 · C1
nhn−1 · · · Cn
nhn−1 · · · A0 · C1
0 · · · Cn
0 · u1 > ξ(L1),
...
4.3 Systems without asymptotically almost periodic solutions 92
A[(i−1)n+1]hn+hn−2 · C1
[(i−1)n+1]hn+hn−2 · · · Cn
[(i−1)n+1]hn+hn−2 · · · A0 · C1
0 · · · Cn
0 · ui
− A[(i−1)n+1]hn−1 · C1
[(i−1)n+1]hn−1 · · · Cn
[(i−1)n+1]hn−1 · · · A0 · C1
0 · · · Cn
0 · ui > ξ(Li),
...
Ainhn+hn−hn−1−1 · C1
inhn+hn−hn−1−1 · · · Cn
inhn+hn−hn−1−1 · · · A0 · C1
0 · · · Cn
0 · ui
− Ainhn−1 · C1
inhn−1 · · · Cn
inhn−1 · · · A0 · C1
0 · · · Cn
0 · ui > ξ(Li),
...
A[(n−1)n+1]hn+hn−2 · C1
[(n−1)n+1]hn+hn−2 · · · Cn
[(n−1)n+1]hn+hn−2 · · · A0 · C1
0 · · · Cn
0 · un
− A[(n−1)n+1]hn−1 · C1
[(n−1)n+1]hn−1 · · · Cn
[(n−1)n+1]hn−1 · · · A0 · C1
0 · · · Cn
0 · un > ξ(Ln),
...
An2hn+hn−hn−1−1 · C1
n2hn+hn−hn−1−1 · · · Cn
n2hn+hn−hn−1−1 · · · A0 · C1
0 · · · Cn
0 · un
− An2hn−1 · C1
n2hn−1 · · · Cn
n2hn−1 · · · A0 · C1
0 · · · Cn
0 · un > ξ(Ln)
and such that
Cn
0 , Cn
1 , . . . , Cn
2n2hn−1 ∈ O ε
2nK
(I),
where
Cn
j = I, j ∈ {0, 1, . . . , 2(n − 1)2
hn−1 − 1}. (4.14)
In addition, we can assume that
Cn
j = I if Ci
j = I for some j ∈ {0, 1, . . . , 2n2
hn − 1}, i ∈ {1, . . . , n − 1}. (4.15)
It follows from the inequalities
hn − 1 > · · · > hn − hn−1 > 22
hn−1l(n, n) > 24
hn−2l(n, n) > · · ·
· · · > 22n−4
h2l(n, n) ≥ 22n−2
h1l(n, n) ≥ 22n
l(n, n)
and from the fact that it suﬃces to choose only the matrices
Cn
j2nh1+(2n−1+1)h1
, Cn
j2nh1+(2n−1+1)h1+1, . . . , Cn
j2nh1+(2n−1+1)h1+h1−1
as diﬀerent from I. We consider the periodic sequence {Cn
k }k∈Z ⊂ X given by the above
values Cn
0 , Cn
1 , . . . , Cn
2n2hn−1 and period 2n2
hn.
We deﬁne
Bk := Ak · C1
k · C2
k · · · Cn
k · · · , k ∈ Z. (4.16)
For all k ∈ Z, there exists i ∈ N such that Bk = Ak · Ci
k (consider (4.15) together with
(4.14)). Especially, the deﬁnition of the sequence of Bk is correct and Bk ∈ X, k ∈ Z. We
have to prove that {Bk}k∈Z is limit periodic and that
sup
k∈Z
Bk − Ak < ε. (4.17)
Since
Cn
k ∈ O ε
2nK
(I), k ∈ Z, n ∈ N, (4.18)
4.3 Systems without asymptotically almost periodic solutions 93
we have
Bk − Ak = Ak · Ci
k − Ak ≤ Ak · Ci
k − I < K
ε
2iK
≤
ε
2
for some i ∈ N. Thus, (4.17) is satisﬁed. Similarly, if we express
Bk − Bn
k · C1
k · · · Cn
k
≤ Bk − Bn
k · C1
k · · · Cn
k · · · + Bn
k · C1
k · · · Cn
k · · · − Bn
k · C1
k · · · Cn
k
≤ Ak − Bn
k · C1
k · · · Cn
k · · · + Bn
k · C1
k · · · Cn
k · Cn+1
k · · · Cn+j
k · · · − I ,
then we obtain (see (4.8), (4.18))
Bk − Bn
k · C1
k · · · Cn
k <
ε
2n+1K
· K + K
ε
2n+1K
=
ε
2n
, k ∈ Z, n ∈ N.
It means that {Bk} is the uniform limit of the sequence of Bn
k · C1
k · · · Cn
k as n → ∞.
Any sequence {Bn
k · C1
k · · · Cn
k }k∈Z has period 2(n!)2
p1 · · · pnl(1, 1) · · · l(n, n). Hence, {Bk}
is limit periodic.
It remains to show that the system {Bk} ∈ Oσ
ε ({Ak}) does not possess any non-zero
asymptotically almost periodic solution. On contrary, suppose that the solution {xk}k∈Z
of the Cauchy problem
xk+1 = Bk · xk, k ∈ Z, x0 = ul
is asymptotically almost periodic. Applying Theorem 1.15 for l1 := 1, ln+1 := hn, n ∈ N,
we obtain
xk+hn1
− xk+hn2
< ξ(Ll), k ∈ N, (4.19)
for inﬁnitely many n1, n2 ∈ N. Let integers n1 > n2 ≥ l be arbitrarily given. It holds
x[(l−1)n1+n2+1]hn1 +hn1 −hn2
− x[(l−1)n1+n2+1]hn1
= B[(l−1)n1+n2+1]hn1 +hn1 −hn2 −1 · · · B1 · B0 · ul − B[(l−1)n1+n2+1]hn1 −1 · · · B1 · B0 · ul .
From (consider (4.14) and (4.16))
B[(l−1)n1+n2+1]hn1 +hn1 −hn2 −1 = A[(l−1)n1+n2+1]hn1 +hn1 −hn2 −1×
× C1
[(l−1)n1+n2+1]hn1 +hn1 −hn2 −1 · · · Cn1
[(l−1)n1+n2+1]hn1 +hn1 −hn2 −1,
...
B[(l−1)n1+n2+1]hn1 −1 = A[(l−1)n1+n2+1]hn1 −1 · C1
[(l−1)n1+n2+1]hn1 −1 · · · Cn1
[(l−1)n1+n2+1]hn1 −1,
...
B1 = A1 · C1
1 · · · Cn1
1 , B0 = A0 · C1
0 · · · Cn1
0 ,
we obtain
x[(l−1)n1+n2+1]hn1 +hn1 −hn2
− x[(l−1)n1+n2+1]hn1
> ξ(Ll).
4.3 Systems without asymptotically almost periodic solutions 94
Especially, it is valid that
sup
k∈N
xk+hn1
− xk+hn2
> ξ(Ll), n1 > n2 ≥ l, n1, n2 ∈ N. (4.20)
This contradiction (see (4.19)) proves that the solution of
xk+1 = Bk · xk, k ∈ Z, x0 = un
is not asymptotically almost periodic for any un, n ∈ N.
Let us consider an arbitrary non-zero vector u ∈ Fm
. We know that there exists a
sequence of ui(n) for which limn→∞ ui(n) = u. For the solution {zk}k∈Z of
zk+1 = Bk · zk, k ∈ Z, z0 = ui(n),
we have (see (4.20))
sup
k∈N
zk+hn1
− zk+hn2
> ξ(Li(n)), n1 > n2 ≥ i(n), n1, n2 ∈ N.
The value ξ(Li(n)) can be chosen as ξ(Li(n)) := ζ(Li(n))/(2K) and Li(n) can be chosen as a
constant value L∗
for suﬃciently large n, because there exist ϑ > 0 and L∗
> 0 satisfying
M · v > L∗
, v ∈ Oϑ(u), M ∈ X. (4.21)
Note that (4.21) follows directly from (4.7) and from the Lipschitz continuity of multiplication.
Without loss of generality, we assume that ui(n) ∈ Oϑ(u), n ∈ N. In this case, it is
valid
sup
k∈N
zk+hn1
− zk+hn2
>
ζ (L∗
)
2K
, n1 > n2 ≥ i(n), n, n1, n2 ∈ N.
Therefore, for the solution {yk}k∈Z of the following problem
yk+1 = Bk · yk, k ∈ Z, y0 = u, (4.22)
it holds
sup
k∈N
yk+hn1
− yk+hn2
>
ζ (L∗
)
4K
, n1 > n2 ≥ i(n), n1, n2 ∈ N, (4.23)
where n ∈ N is suﬃciently large. Indeed, it follows from
zk − yk = Bk−1 · · · B0 · ui(n) − Bk−1 · · · B0 · u ≤ K ui(n) − u , k ∈ N.
Finally, (4.23) implies that the solution {yk} of (4.22) cannot be asymptotically almost
periodic (consider again Theorem 1.15).
Remark 4.17. Since P(X) is a dense subset of LP(X) (consider directly Deﬁnition 1.17),
the statement of Theorem 4.16 does not change if one replaces the system {Ak} ∈ LP(X)
by {Ak} ∈ P(X).
Example 4.18. Theorem 4.16 can be applied for all groups of matrices recalled in Example
4.7. In addition, considering Remark 4.6, other matrix groups with property P can
be constructed using the direct sums, because the direct sum of a weakly transformable
group and a ﬁnite group and the direct sum of two weakly transformable groups are weakly
transformable as well (see Examples 2.16 and 2.17). ♦
4.3 Systems without asymptotically almost periodic solutions 95
Remark 4.19. Let us consider that the statement of Theorem 4.16 is true for a bounded
matrix group X ⊂ Mat(F, m). Especially, it is valid when Ak := I, k ∈ Z. Since the
multiplication of matrices is continuous in the Lipschitz sense on X (see Lemma 4.11), for
any non-zero vector u ∈ Fm
and δ > 0, there exist matrices M1, . . . , Ml (where Mi is the
product of i matrices from a neighbourhood of I) with the property that
M1 ∈ Oδ (I), Mi ∈ Oδ (Mi+1), i ∈ {1, . . . , l − 1}, Ml · u − u > 0.
Similarly as in Remark 4.6, one can show that X has property P. Thus, the condition of
Theorem 4.16 (that X has property P) is necessary if F = F.
Theorem 4.20. Let X have property P∗
. For any {Ak} ∈ LP(X) and ε > 0, there exists
a system {Bk} ∈ Oσ
ε ({Ak}) ∩ LP(X) which does not have any non-zero asymptotically
almost periodic solution in the strong sense.
Proof. We can proceed as in the proof of Theorem 4.16 and construct the limit periodic
system {Bk}k∈Z for a decreasing sequence {Li}i∈N of positive numbers with the property
that limi→∞ Li = 0. The value Li > 0 is given by a non-zero vector ui ∈ Fm
in the sense
that Mui > Li, M ∈ X. Let the sequence {ui}i∈N be such that, for any non-zero
vector u ∈ Fm
, one can ﬁnd j ∈ N for which ui ≤ u , i ≥ j.
Especially, from the construction, we obtain
Bhn+hn−2 · · · B1 · B0 · u1 − Bhn−1 · · · B1 · B0 · u1 > ξ(L1),
...
Bnhn+hn−hn−1−1 · · · B1 · B0 · u1 − Bnhn−1 · · · B1 · B0 · u1 > ξ(L1),
...
B[(i−1)n+1]hn+hn−2 · · · B1 · B0 · ui − B[(i−1)n+1]hn−1 · · · B1 · B0 · ui > ξ(Li),
...
Binhn+hn−hn−1−1 · · · B1 · B0 · ui − Binhn−1 · · · B1 · B0 · ui > ξ(Li),
...
B[(n−1)n+1]hn+hn−2 · · · B1 · B0 · un − B[(n−1)n+1]hn−1 · · · B1 · B0 · un > ξ(Ln),
...
Bn2hn+hn−hn−1−1 · · · B1 · B0 · un − Bn2hn−1 · · · B1 · B0 · un > ξ(Ln),
...
Let a non-zero vector u ∈ Fm
be given. We consider the solution {xk}k∈Z of the Cauchy
problem
xk+1 = Bk · xk, k ∈ Z, x0 = u.
4.3 Systems without asymptotically almost periodic solutions 96
From the proof of Theorem 4.16, we know that
inf
k∈Z
xk > Lj for some j ∈ N (4.24)
and that (see (4.23))
sup
k∈N
xk+hn1
− xk+hn2
> ϑ
for all suﬃciently large integers n1 > n2 and for some ϑ > 0. Hence, for all n1 > n2 ≥ n0,
n1, n2 ∈ N, where n0 ∈ N is suﬃciently large, there exists an integer l ≥ hn2 with the
property that
xl+hn1 −hn2 +1 − xl+1 = Bl+hn1 −hn2
· · · Bl · · · B0 · u − Bl · · · B0 · u
= Bl+hn1 −hn2
· · · Bl+1 − I Bl · · · B0 · u > ϑ.
Considering (4.24), the construction, and the property P∗
of X, we can assume that
Bl+hn1 −hn2
· · · Bl+1 − I xk > ϑ, k ∈ N. (4.25)
Since the multiplication and addition are continuous in the Lipschitz sense on bounded
subsets of F and {Bk} is limit periodic, Theorem 1.3 and Corollary 1.11 (together with
Theorem 1.22) imply that the sequence {Bk+hn1 −hn2
· · · Bk+1}k∈Z is almost periodic for all
given integers n1 > n2 ≥ n0. Thus, for any η > 0, we get the existence of a sequence
{qn}n∈N ⊆ N which is relatively dense in N and which has the property that
Bl+hn1 −hn2
· · · Bl+1 − Bqn+hn1 −hn2
· · · Bqn+1 < η, n ∈ N. (4.26)
Combining (4.25) and (4.26) for a suﬃciently small number η, we have
xk+hn1
− xk+hn2
= Bk+hn1 −1 · · · Bk+hn2
− I xk+hn2
>
ϑ
2
for all k = qn − hn2 + 1, n ∈ N.
Remark 4.21. Based on the proofs of Theorems 4.16 and 4.20, it is possible to prove that
the group X is transformable if it has property P∗
.
Example 4.22. One can easily show that the transformable groups of matrices recalled in
Example 4.7 have actually property P∗
(except SO(m) if m is odd); i.e., for these groups,
one can apply Theorem 4.20 which improves Theorem 4.16. ♦
The process used in the proof of Theorem 4.16 can be applied for almost periodic
systems as well.
Theorem 4.23. Let X have property P. For any {Ak} ∈ AP(X) and ε > 0, there exists a
system {Bk} ∈ Oσ
ε ({Ak}) which does not have any non-zero asymptotically almost periodic
solution.
Proof. The statement of this theorem can be proved using the construction from the proof
of Theorem 4.16, where it suﬃces to put pn := 2 and Bn
k := Ak for all n ∈ N, k ∈ Z. Indeed,
the almost periodicity of the sequence {Ak · C1
k · · · Cn
k }k∈Z follows from Theorem 1.3 and
Corollary 1.11 (the multiplication and addition have the Lipschitz property on any bounded
subset of F) and the almost periodicity of {Bk}k∈Z comes from Theorem 1.7.
4.3 Systems without asymptotically almost periodic solutions 97
Corollary 4.24. Let X have property P. For any {Ak} ∈ AP(X) and ε > 0, there exists
a system {Bk} ∈ Oσ
ε ({Ak}) which does not have any non-zero almost periodic solution.
Proof. See Theorems 1.22 and 4.23.
In the complex case, Theorem 4.23 is a generalization of Corollary 2.33 (see also Corollary
4.24). It follows from Remark 4.6. Of course, in the general case, the results presented
in Chapter 2 do not follow from ones presented here.
Remark 4.25. In fact, considering the proofs of the above theorems, we see that it is
not necessary to introduce the map | · | on the whole ﬁeld. It suﬃces to deﬁne it on a
neighbourhood of e0 and for all elements of matrices from the matrix group X in such a
way that condition (a) is replaced by | f | ≥ 0, | e0 | = 0, | e1 | = 1 and condition (b) by
| f g | ≤ | f | · | g |, | f | = | − f | (consider also Deﬁnitions 4.1 and 4.2). Let X have
property P with respect to a given non-zero vector u and let there exist a neighbourhood
of u with the property that, for all vectors v from the neighbourhood and all M ∈ X, there
exist the norms M v . In this case, for any limit periodic (or almost periodic) sequence
{Ak}k∈Z ⊆ X, there exists a limit periodic (or almost periodic) system given by {Bk}k∈Z
in an arbitrarily small neighbourhood of {Ak} for which the solution of
xk+1 = Bk · xk, k ∈ Z, x0 = u
is not asymptotically almost periodic.
Example 4.26. Now we show that there exists a group X with property P which is
not weakly transformable. Especially, we show that Corollary 4.24 does not follow from
Corollary 2.33. For simplicity, we use Remark 4.25. Let F be the ﬁeld of all meromorphic
functions deﬁned on an open connected set which contains the set
D = {z ∈ C; Re z ∈ [−π, π], Im z = 0}.
Note that, using the analytic continuation to eliminate removable singularities, meromorphic
functions can be added, subtracted, multiplied, and the quotient can be formed. For
the set F0 ⊂ F of all bounded functions on D, we put
| f | := sup
z∈D
| f(z) | , f ∈ F0.
Let m = 1, i.e., let us consider the scalar case. Especially, f = | f |, f ∈ F0.
We put
X := {c [sin(nz) ± i cos(nz)]; c ∈ C, | c | = 1, n ∈ Z}.
The set X forms a group. Indeed, the identity element
1 ≡ i [sin(0z) − i cos(0z)] ∈ X,
associativity is obvious, the inverse elements
(c [sin(nz) ± i cos(nz)])−1
= c−1
[sin(nz) i cos(nz)] , c ∈ C, | c | = 1, n ∈ Z,
4.3 Systems without asymptotically almost periodic solutions 98
belong to X, and the closure of multiplication follows from the formulas
[sin(nz) + i cos(nz)] · [sin(lz) ± i cos(lz)] = cos[(n ± l)z] ± i sin[(n ± l)z],
[sin(nz) − i cos(nz)] · [sin(lz) − i cos(lz)] = − cos[(n + l)z] − i sin[(n + l)z].
Evidently, this group is bounded.
Firstly, we show that X is not weakly transformable. The subgroup
X0 := {c [sin 0 ± i cos 0]; c ∈ C, | c | = 1} = {f ≡ c ∈ C; | c | = 1}
is transformable. It can be directly veriﬁed that
c − d [sin(nz) ± i cos(nz)] > 1, c, d ∈ C, | c | = | d | = 1, n ∈ Z {0}.
Thus, there does not exist a transformable subgroup of X which contains X0 and at least
one other element. To prove that X is not weakly transformable, it suﬃces to consider
that the distance of any two diﬀerent components
X0 = {f ≡ c ∈ C; | c | = 1},
X+
n := {c [sin(nz) + i cos(nz)]; c ∈ C, | c | = 1}, n ∈ Z {0},
X−
n := {c [sin(nz) − i cos(nz)]; c ∈ C, | c | = 1}, n ∈ Z {0},
is greater that 1.
Secondly, we show that X has property P. It suﬃces to put ζ(a) := 2a for all a > 0.
For δ = π/l, l ∈ N, we can choose matrices
M1 = eiδ
, M2 = ei2δ
, . . . , Ml−1 = ei(l−1)δ
, Ml = eilδ
= (−1)
for which we have
M1 − I < δ, M2 − M1 < δ, . . . , Ml−1 − Ml < δ.
Evidently, it is also valid that
Ml · u − u = −u − u = 2 u > ζ(a) = 2a, u > a, u ∈ F0.
♦
Analogously as Theorem 4.23 (see the proof of Theorem 4.20), one can obtain the
following result.
Theorem 4.27. Let X have property P∗
. For any {Ak} ∈ AP(X) and ε > 0, there
exists a system {Bk} ∈ Oσ
ε ({Ak}) which does not have any non-zero asymptotically almost
periodic solution in the strong sense.
Remark 4.28. We recall that Theorems 4.23 and 4.27 do not follow from Theorems 4.16
and 4.20. See Theorem 1.21.
4.4 Systems with non-almost periodic solutions 99
4.4 Systems with non-almost periodic solutions
Henceforth, we assume that X is commutative. To prove the announced result (the
below given Theorem 4.31), we use Lemmas 4.29 and 4.30.
Lemma 4.29. Let {Ak} ∈ LP(X) and ε > 0 be arbitrarily given. Let {δn}n∈N ⊂ R be a
decreasing sequence satisfying
lim
n→∞
δn = 0 (4.27)
and let {Bn
k }k∈Z ⊂ X be periodic sequences for n ∈ N such that
Bn
k ∈ Oδn
(I), k ∈ Z, n ∈ N, (4.28)
Bj
k = I or Bi
k = I, k ∈ Z, i = j, i, j ∈ N. (4.29)
If one puts
Bk := Ak · B1
k · B2
k · · · Bn
k · · · , k ∈ Z,
then {Bk} ∈ LP(X). In addition, if
δ1 <
ε
sup
l∈Z
Al
, (4.30)
then {Bk} ∈ Oσ
ε ({Ak}).
Proof. Condition (4.29) means that, for any k ∈ Z, there exists i ∈ N such that
Bk = Ak · Bi
k. (4.31)
Especially, the deﬁnition of {Bk}k∈Z is correct and Bk ∈ X, k ∈ Z.
We show that {Bk} is limit periodic. Since {Ak} is limit periodic and Ak ∈ X, k ∈ Z,
there exist periodic sequences {Cn
k }k∈Z ⊂ X for n ∈ N with the property that
Ak − Cn
k <
1
n
, k ∈ Z, n ∈ N. (4.32)
Let {Bn
k } and {Cn
k } have period pn ∈ N and qn ∈ N for n ∈ N, respectively. The sequence
{Cn
k · B1
k · B2
k · · · Bn
k }k∈Z ⊂ X has period qn · p1 · p2 · · · pn; i.e., it is periodic for all n ∈ N.
It is valid that
Bk − Cn
k · B1
k · B2
k · · · Bn
k
≤ Bk − Cn
k · B1
k · B2
k · · · Bn
k · · · + Cn
k · B1
k · B2
k · · · Bn
k · · · − Cn
k · B1
k · B2
k · · · Bn
k
≤ Ak − Cn
k · B1
k · B2
k · · · Bn
k · · · + Cn
k · B1
k · B2
k · · · Bn
k · Bn+1
k · · · Bn+j
k · · · − I
and that
Cn
k · B1
k · B2
k · · · Bn
k ≤ Cn
k · B1
k · B2
k · · · Bn
k
≤ ( Ak + Cn
k − Ak ) · B1
k · B2
k · · · Bn
k .
4.4 Systems with non-almost periodic solutions 100
Hence (see (4.28), (4.29), (4.32)), we have
Bk − Cn
k · B1
k · B2
k · · · Bn
k <
1
n
(m + δ1) + sup
l∈Z
Al +
1
n
(m + δ1) δn+1
for all k ∈ Z, n ∈ N. Considering (4.27), we get that {Bk} is the uniform limit of the
sequence of periodic sequences {Cn
k · B1
k · B2
k · · · Bn
k }. Especially, {Bk} ∈ LP(X).
Let (4.30) be true. We have to prove that {Bk} ∈ Oσ
ε ({Ak}), i.e.,
sup
k∈Z
Ak − Bk < ε. (4.33)
Since
Bn
k ∈ Oδ1
(I), k ∈ Z, n ∈ N,
considering (4.31), we have
Ak − Bk ≤ Ak · I − Bi
k ≤ δ1 sup
l∈Z
Al
for some i ∈ N and for all k ∈ Z. Thus (see (4.30)), we obtain (4.33).
Lemma 4.30. If for any δ > 0 and K > 0, there exist matrices M1, . . . , Ml ∈ X such that
Mi ∈ Oδ (I), i ∈ {1, . . . , l}, Ml · · · M1 > K, (4.34)
then, for any {Ak} ∈ LP(X) and ε > 0, there exists a system {Bk} ∈ Oσ
ε ({Ak}) ∩ LP(X)
whose fundamental matrix is not almost periodic.
Proof. We can assume that all solutions of {Ak} are almost periodic. Especially (consider
Corollary 1.11), for any ϑ > 0, there exist inﬁnitely many positive integers p with the
property that
Ap−1 · · · A1 · A0 − I < ϑ. (4.35)
Let {δn}n∈N ⊂ R be a decreasing sequence satisfying (4.27) and (4.30). For δn and Kn := n,
n ∈ N, we consider matrices
M1
1 , M1
2 , . . . , M1
l1
∈ X,
M2
1 , M2
2 , . . . , M2
l2
∈ X,
...
Mj
1 , Mj
2 , . . . , Mj
lj
∈ X,
...
such that
Mj
i ∈ Oδj
(I) , i ∈ {1, 2, . . . , lj}, j ∈ N, (4.36)
and
Mj
lj
· · · Mj
2 · Mj
1 > Kj = j, j ∈ N. (4.37)
Let a sequence of positive numbers ϑn for n ∈ N be given.
4.4 Systems with non-almost periodic solutions 101
Let us consider p1
1, p1
2 ∈ N such that p1
2 − p1
1 > 2l1 and that (see (4.35))
Ap1
2−1 · · · A1 · A0 − I < ϑ1. (4.38)
In addition, let p1
1 and p1
2 be even (consider Corollary 1.4). We deﬁne the periodic sequence
{B1
k}k∈Z with period p1
2 by values
B1
0 := I, B1
1 := I, . . . , B1
p1
1−2 := I,
B1
p1
1−1 := I, B1
p1
1
:= I, B1
p1
1+1 := M1
1 , B1
p1
1+2 := I, B1
p1
1+3 := M1
2 , B1
p1
1+4 := I,
...
B1
p1
1+2l1−3 := M1
l1−1, B1
p1
1+2l1−2 := I, B1
p1
1+2l1−1 := M1
l1
,
B1
p1
1+2l1
:= I, B1
p1
1+2l1+1 := I, B1
p1
1+2l1+2 := I,
...
B1
p1
2−1 := I.
We put
˜B1
k := Ak · B1
k, k ∈ Z.
We have
˜B1
p1
2−1 · · · ˜B1
1 · ˜B1
0 = M1
l1
· · · M1
2 · M1
1 · Ap1
2−1 · · · A1 · A0 .
Again, we can assume that, for any ϑ > 0, there exist inﬁnitely many positive integers p
with the property that
˜B1
p−1 · · · ˜B1
1 · ˜B1
0 − I < ϑ. (4.39)
Otherwise, we obtain the system {Bk} ≡ { ˜B1
k} with a non-almost periodic solution. Indeed,
it suﬃces to consider Lemma 4.29 for Bn+1
k = I, k ∈ Z, n ∈ N.
Analogously, let us consider p2
1, p2
2 ∈ N satisfying p2
2 − 4l2 > p2
1 > p1
2 and (see (4.39))
˜B1
p2
2−1 · · · ˜B1
1 · ˜B1
0 − I < ϑ2. (4.40)
Let p2
1, p2
2 ∈ 4N (see Corollary 1.4). We deﬁne the periodic sequence {B2
k}k∈Z with period
p2
2 by values
B2
0 := I, B2
1 := I, . . . , B2
p2
1−1 := I,
B2
p2
1
:= I, B2
p2
1+1 := I, B2
p2
1+2 := M2
1 , B2
p2
1+3 := I,
B2
p2
1+4 := I, B2
p2
1+5 := I, B2
p2
1+6 := M2
2 , B2
p2
1+7 := I,
...
B2
p2
1+4l2−4 := I, B2
p2
1+4l2−3 := I, B2
p2
1+4l2−2 := M2
l2
, B2
p2
1+4l2−1 := I,
4.4 Systems with non-almost periodic solutions 102
B2
p2
1+4l2
:= I, B2
p2
1+4l2+1 := I, B2
p2
1+4l2+2 := I, B2
p2
1+4l2+3 := I,
...
B2
p2
2−1 := I.
For
˜B2
k := Ak · B1
k · B2
k, k ∈ Z,
it holds
˜B2
p2
2−1 · · · ˜B2
1 · ˜B2
0 = M2
l2
· · · M2
2 · M2
1 · ˜B1
p2
2−1 · · · ˜B1
1 · ˜B1
0 .
Especially, for all k ∈ Z, there exists i ∈ {1, 2} such that ˜B2
k := Ak · Bi
k.
We continue in the same manner. Let us assume that all obtained systems { ˜Bj
k}k∈Z
have only almost periodic solutions. Thus, for every ϑ > 0 and j ∈ N, one can ﬁnd
inﬁnitely many p ∈ N such that
˜Bj
p−1 · · · ˜Bj
1 · ˜Bj
0 − I < ϑ.
In the n-th step, we consider pn
1 , pn
2 ∈ 2n
N such that pn
2 − 2n
ln > pn
1 > pn−1
2 and
˜Bn−1
pn
2 −1 · · · ˜Bn−1
1 · ˜Bn−1
0 − I < ϑn. (4.41)
We deﬁne the periodic sequence {Bn
k }k∈Z with period pn
2 by values
Bn
0 := I, Bn
1 := I, . . . , Bn
pn
1 −1 := I,
Bn
pn
1
:= I,Bn
pn
1 +1 := I, . . . , Bn
pn
1 +2n−1−1 := I,
Bn
pn
1 +2n−1 := Mn
1 , Bn
pn
1 +2n−1+1 := I, . . . , Bn
pn
1 +2n−1 := I,
Bn
pn
1 +2n := I,Bn
pn
1 +2n+1 := I, . . . , Bn
pn
1 +2n+2n−1−1 := I,
Bn
pn
1 +2n+2n−1 := Mn
2 , Bn
pn
1 +2n+2n−1+1 := I, . . . , Bn
pn
1 +2·2n−1 := I,
...
Bn
pn
1 +(ln−1)2n := I,Bn
pn
1 +(ln−1)2n+1 := I, . . . , Bn
pn
1 +(ln−1)2n+2n−1−1 := I,
Bn
pn
1 +(ln−1)2n+2n−1 := Mn
ln
, Bn
pn
1 +(ln−1)2n+2n−1+1 := I, . . . , Bn
pn
1 +ln2n−1 := I,
Bn
pn
1 +ln2n := I,Bn
pn
1 +ln2n+1 := I, . . . , Bn
pn
1 +ln2n+2n−1−1 := I,
Bn
pn
1 +ln2n+2n−1 := I, Bn
pn
1 +ln2n+2n−1+1 := I, . . . , Bn
pn
1 +(ln+1)2n−1 := I,
...
Bn
pn
2 −1 := I.
4.4 Systems with non-almost periodic solutions 103
If we put
˜Bn
k := Ak · B1
k · B2
k · · · Bn
k , k ∈ Z,
then
˜Bn
pn
2 −1 · · · ˜Bn
1 · ˜Bn
0 = Mn
ln
· · · Mn
2 · Mn
1 · ˜Bn−1
pn
2 −1 · · · ˜Bn−1
1 · ˜Bn−1
0 . (4.42)
Finally, we put
Bk := Ak · B1
k · B2
k · · · Bn
k · · · , k ∈ Z.
From the construction, we obtain that, for any k ∈ Z, there exists i ∈ N such that
Bk = Ak · Bi
k. It means that (4.29) is satisﬁed. Since (4.28) follows from the construction
and from (4.36), we can use Lemma 4.29 which guarantees that {Bk} ∈ Oσ
ε ({Ak})∩LP(X).
It remains to prove that the fundamental matrix of {Bk} is not almost periodic. On
contrary, let us suppose its almost periodicity. Then, the fundamental matrix is bounded
(see Theorem 1.14); i.e., there exists K0 > 0 with the property that
Bk · · · B1 · B0 < K0, k ∈ N. (4.43)
Let us choose n ∈ N for which n ≥ K0 + 1. We repeat that the multiplication of matrices
is continuous (see also Lemma 4.11). Hence, for given matrix
Mn
1 · Mn
2 · · · Mn
ln
= Mn
ln
· · · Mn
2 · Mn
1 ∈ X,
there exists θn > 0 such that
Mn
ln
· · · Mn
2 · Mn
1 − 1 < Mn
ln
· · · Mn
2 · Mn
1 · C , C ∈ Oθn
(I). (4.44)
We can assume that ϑn = θn in (4.41) (see also (4.38), (4.40)). We construct sequences
{Bj
k} in such a way that
Bj
0 = I, Bj
1 = I, . . . , Bj
pn
2 −1 = I, j > n, j, n ∈ N.
Indeed, pj+1
1 > pj
2 > pj
1, j ∈ N. Thus, (4.37), (4.41), (4.42), and (4.44) imply
Bpn
2 −1 · · · B1 · B0 = ˜Bn
pn
2 −1 · · · ˜Bn
1 · ˜Bn
0
= Mn
ln
· · · Mn
2 · Mn
1 · ˜Bn−1
pn
2 −1 · · · ˜Bn−1
1 · ˜Bn−1
0
> Mn
ln
· · · Mn
2 · Mn
1 − 1 > n − 1 ≥ K0.
(4.45)
This contradiction (cf. (4.43) and (4.45)) completes the proof.
Theorem 4.31. Let X have property P with respect to a vector u. For any {Ak} ∈ LP(X)
and ε > 0, there exists a system {Bk} ∈ Oσ
ε ({Ak}) ∩ LP(X) whose fundamental matrix is
not almost periodic.
Proof. Let us consider the solution {x0
k}k∈Z of the Cauchy problem
xk+1 = Ak · xk, k ∈ Z, x0 = u.
4.4 Systems with non-almost periodic solutions 104
If {x0
k} is not almost periodic, then the statement of the theorem is true for Bk := Ak,
k ∈ Z. Hence, we assume that {x0
k} is almost periodic.
We put
δn :=
1
n + 1
·
ε
sup
l∈Z
Al
, n ∈ N. (4.46)
We know that there exist ζ > 0 and matrices
M1
1 , M1
2 , . . . , M1
l1
∈ X,
M2
1 , M2
2 , . . . , M2
l2
∈ X,
...
Mj
1 , Mj
2 , . . . , Mj
lj
∈ X,
...
such that
Mj
i ∈ Oδj
(I) , i ∈ {1, . . . , lj}, (4.47)
Mj
lj
· · · Mj
2 · Mj
1 · u − u > ζ (4.48)
for all j ∈ N. Of course, we can consider lj such that
lj ≥ · · · ≥ l2 ≥ l1 ≥ 2. (4.49)
Denote
Kj := Mj
lj
· · · Mj
2 · Mj
1 , j ∈ N. (4.50)
For
ϑj :=
ζ
2 (Kj + 2)
, j ∈ N, (4.51)
we have
M · v − w >
ζ
2
if M · u − u > ζ, M ∈ OKj+1 (O) ∩ X, v, w ∈ Oϑj
(u) . (4.52)
Indeed, for considered u, v, w ∈ Fm
and M ∈ X, it holds (see (4.51))
M · u − u ≤ M · u − M · v + M · v − w + w − u
< (Kj + 1) u − v + w − u + M · v − w <
ζ
2
+ M · v − w .
The almost periodicity of {x0
k} (see Corollary 1.4) implies that there exists an even
positive integer j(1,0) such that
x0
0 − x0
j(1,0)
= u − x0
j(1,0)
<
ϑ1
2
. (4.53)
4.4 Systems with non-almost periodic solutions 105
Let us deﬁne a periodic sequence {B1
k} with period j(1, 0) + r1, where r1 := 2l1. If
x0
j(1,0)
− x0
j(1,0)+r1
≥
ϑ1
2
, (4.54)
then we put B1
k := I, k ∈ Z; and if
x0
j(1,0)
− x0
j(1,0)+r1
<
ϑ1
2
, (4.55)
then
B1
0 := I, B1
1 := I, . . . , B1
j(1,0)−1 := I,
B1
j(1,0)
:= I, B1
j(1,0)+1 := M1
1 , B1
j(1,0)+2 := I, B1
j(1,0)+3 := M1
2 ,
...
B1
j(1,0)+2l1−4 := I, B1
j(1,0)+2l1−3 := M1
l1−1, B1
j(1,0)+2l1−2 := I, B1
j(1,0)+2l1−1 := M1
l1
.
For ˜B1
k := Ak · B1
k, k ∈ Z, we consider the solution {x1
k}k∈Z of the initial problem
xk+1 = ˜B1
k · xk, k ∈ Z, x0 = u.
Lemma 4.29 gives that { ˜B1
k} ∈ Oσ
ε ({Ak}) ∩ LP(X). In the case, when {x1
k} is not almost
periodic, we can put Bk := ˜B1
k, k ∈ Z. Thus, we have to consider the almost periodicity
of {x1
k}. Especially (see Corollary 1.4), there exist inﬁnitely many numbers j ∈ 4N with
the property that
x1
0 − x1
j = u − x1
j <
ϑ2
2
. (4.56)
Let us consider an integer j(1,1) ∈ 4N satisfying (4.56) and the inequality
j(1,1) ≥ j(1,0) + r1. (4.57)
For r2 := 8l1l2, we deﬁne a sequence {B
(1,2)
k }k∈Z with period j(1,1) + r2. We put B
(1,2)
k := I
for all k ∈ Z if
x1
j(1,1)
− x1
j(1,1)+r2
≥
ϑ2
2
. (4.58)
In the second case, when (4.58) is not valid, we deﬁne
B
(1,2)
0 := I, B
(1,2)
1 := I, . . . , B
(1,2)
j(1,1)−1 := I,
B
(1,2)
j(1,1)
:= I, B
(1,2)
j(1,1)+1 := I, B
(1,2)
j(1,1)+2 := M2
1 , B
(1,2)
j(1,1)+3 := I,
B
(1,2)
j(1,1)+4 := I, B
(1,2)
j(1,1)+5 := I, B
(1,2)
j(1,1)+6 := M2
2 , B
(1,2)
j(1,1)+7 := I,
...
B
(1,2)
j(1,1)+4l2−4 := I, B
(1,2)
j(1,1)+4l2−3 := I, B
(1,2)
j(1,1)+4l2−2 := M2
l2
, B
(1,2)
j(1,1)+4l2−1 := I,
B
(1,2)
j(1,1)+4l2
:= I, B
(1,2)
j(1,1)+4l2+1 := I, B
(1,2)
j(1,1)+4l2+2 := I, B
(1,2)
j(1,1)+4l2+3 := I,
4.4 Systems with non-almost periodic solutions 106
...
B
(1,2)
j(1,1)+r2−1 := I.
For ˜B
(1,2)
k := Ak · B1
k · B
(1,2)
k , k ∈ Z, we consider the solution {x
(1,2)
k }k∈Z of the initial
problem
xk+1 = ˜B
(1,2)
k · xk, k ∈ Z, x0 = u.
Again, we can assume that {x
(1,2)
k }k∈Z is almost periodic. Let an integer j(2,1) ∈ 8N have
the properties that
x
(1,2)
0 − x
(1,2)
j(2,1)
= u − x
(1,2)
j(2,1)
<
ϑ2
2
(4.59)
and that
j(2,1) ≥ j(1,1) + r2. (4.60)
We deﬁne a periodic sequence {B
(2,2)
k }k∈Z with period j(2,1)(r2 − r1). If
x
(1,2)
j(2,1)
− x
(1,2)
j(2,1)+r2−r1
≥
ϑ2
2
, (4.61)
we put B
(2,2)
k := I for all k ∈ Z; and, in the other case, we deﬁne
B
(2,2)
0 := I, B
(2,2)
1 := I, . . . , B
(2,2)
j(2,1)−1 := I,
B
(2,2)
j(2,1)
:= I, B
(2,2)
j(2,1)+1 := I, B
(2,2)
j(2,1)+2 := I, B
(2,2)
j(2,1)+3 := I,
B
(2,2)
j(2,1)+4 := M2
1 , B
(2,2)
j(2,1)+5 := I, B
(2,2)
j(2,1)+6 := I, B
(2,2)
j(2,1)+7 := I,
B
(2,2)
j(2,1)+8 := I, B
(2,2)
j(2,1)+9 := I, B
(2,2)
j(2,1)+10 := I, B
(2,2)
j(2,1)+11 := I,
B
(2,2)
j(2,1)+12 := M2
2 , B
(2,2)
j(2,1)+13 := I, B
(2,2)
j(2,1)+14 := I, B
(2,2)
j(2,1)+15 := I,
...
B
(2,2)
j(2,1)+8l2−8 := I, B
(2,2)
j(2,1)+8l2−7 := I, B
(2,2)
j(2,1)+8l2−6 := I, B
(2,2)
j(2,1)+8l2−5 := I,
B
(2,2)
j(2,1)+8l2−4 := M2
l2
, B
(2,2)
j(2,1)+8l2−3 := I, B
(2,2)
j(2,1)+8l2−2 := M2
l2
, B
(2,2)
j(2,1)+8l2−1 := I,
B
(2,2)
j(2,1)+8l2
:= I, B
(2,2)
j(2,1)+8l2+1 := I, B
(2,2)
j(2,1)+8l2+2 := I, B
(2,2)
j(2,1)+8l2+3 := I,
B
(2,2)
j(2,1)+8l2+4 := I, B
(2,2)
j(2,1)+8l2+5 := I, B
(2,2)
j(2,1)+8l2+6 := I, B
(2,2)
j(2,1)+8l2+7 := I,
...
B
(2,2)
j(2,1)+r2−r1−1 := I, . . . , B
(2,2)
j(2,1)(r2−r1)−1 := I.
Finally, in the second step, we consider the periodic sequence of
B2
k := B
(1,2)
k · B
(2,2)
k , k ∈ Z. (4.62)
4.4 Systems with non-almost periodic solutions 107
Note that its period is [j(1,1) + r2][j(2,1)(r2 − r1)]. Consequently, we consider
˜B2
k := Ak · B1
k · B2
k, k ∈ Z, (4.63)
and the solution {x2
k}k∈Z of
xk+1 = ˜B2
k · xk, k ∈ Z, x0 = u.
In the case, when {x2
k} is not almost periodic, we can put Bk := ˜B2
k for k ∈ Z and use
Lemma 4.29 for Bj+2
k = I, k ∈ Z, j ∈ N (see also (4.63)). Thus, we have to assume that
{x2
k} is almost periodic.
We continue in the same manner. Before the n-th step, we deﬁne
˜Bn−1
k := Ak · B1
k · B2
k · · · Bn−1
k , k ∈ Z.
Let { ˜Bn−1
k }k∈Z have period qn−1, e.g., let
qn−1 :=[j(1,0) + r1][j(1,1) + r2][j(2,1)(r2 − r1)] × · · ·
· · · × [j(1,n−2) + rn−1][j(2,n−2)(rn−1 − r1)] · · · [j(n−1,n−2)(rn−1 − rn−2)].
Consider the solution {xn−1
k } of
xk+1 = ˜Bn−1
k · xk, k ∈ Z, x0 = u.
Again, we consider that the sequence {xn−1
k } is almost periodic. Otherwise, we can put
Bk := ˜Bn−1
k , k ∈ Z. Especially, for all p ∈ N, there exist inﬁnitely many numbers j ∈ pN
with the property that
xn−1
0 − xn−1
j = u − xn−1
j <
ϑn
2
. (4.64)
Denote
pn := 21+ n−1
i=1 i
, n ≥ 2, n ∈ N, (4.65)
rn := 2pnl1l2 · · · ln, n ≥ 2, n ∈ N. (4.66)
Let us consider an integer j(1,n−1) ∈ pnN satisfying (4.64) and
j(1,n−1) ≥ qn−1. (4.67)
We deﬁne {B
(1,n)
k }k∈Z with period j(1,n−1) + rn. If
xn−1
j(1,n−1)
− xn−1
j(1,n−1)+rn
≥
ϑn
2
, (4.68)
we put B
(1,n)
k := I, k ∈ Z. In the other case, we put
B
(1,n)
0 := I, B
(1,n)
1 := I, . . . , B
(1,n)
j(1,n−1)−1 := I,
B
(1,n)
j(1,n−1)
:= I, B
(1,n)
j(1,n−1)+1 := I, . . . , B
(1,n)
j(1,n−1)+pn
2
−1
:= I,
4.4 Systems with non-almost periodic solutions 108
B
(1,n)
j(1,n−1)+pn
2
:= Mn
1 , B
(1,n)
j(1,n−1)+pn
2
+1
:= I, . . . , B
(1,n)
j(1,n−1)+pn−1 := I,
B
(1,n)
j(1,n−1)+pn
:= I, B
(1,n)
j(1,n−1)+pn+1 := I, . . . , B
(1,n)
j(1,n−1)+pn+pn
2
−1
:= I,
B
(1,n)
j(1,n−1)+pn+pn
2
:= Mn
2 , B
(1,n)
j(1,n−1)+pn+pn
2
+1
:= I, . . . , B
(1,n)
j(1,n−1)+2pn−1 := I,
...
B
(1,n)
j(1,n−1)+(ln−1)pn
:= I, B
(1,n)
j(1,n−1)+(ln−1)pn+1 := I, . . . , B
(1,n)
j(1,n−1)+(ln−1)pn+pn
2
−1
:= I,
B
(1,n)
j(1,n−1)+(ln−1)pn+pn
2
:= Mn
ln
, B
(1,n)
j(1,n−1)+(ln−1)pn+pn
2
+1
:= I, . . . , B
(1,n)
j(1,n−1)+lnpn−1 := I,
B
(1,n)
j(1,n−1)+lnpn
:= I, B
(1,n)
j(1,n−1)+lnpn+1 := I, . . . , B
(1,n)
j(1,n−1)+lnpn+pn
2
−1
:= I,
B
(1,n)
j(1,n−1)+lnpn+pn
2
:= I, B
(1,n)
j(1,n−1)+lnpn+pn
2
+1
:= I, . . . , B
(1,n)
j(1,n−1)+(ln+1)pn−1 := I,
...
B
(1,n)
j(1,n−1)+rn−1 := I.
For
˜B
(1,n)
k := ˜Bn−1
k · B
(1,n)
k = Ak · B1
k · B2
k · · · Bn−1
k · B
(1,n)
k , k ∈ Z,
we consider the solution {x
(1,n)
k } of
xk+1 = ˜B
(1,n)
k · xk, k ∈ Z, x0 = u.
Again, we assume that {x
(1,n)
k } is almost periodic. Let a number j(2,n−1) ∈ 2pnN have the
properties that
x
(1,n)
0 − x
(1,n)
j(2,n−1)
= u − x
(1,n)
j(2,n−1)
<
ϑn
2
(4.69)
and
j(2,n−1) ≥ j(1,n−1) + rn. (4.70)
We deﬁne the following periodic sequence {B
(2,n)
k }k∈Z with period j(2,n−1)(rn − r1). If
x
(1,n)
j(2,n−1)
− x
(1,n)
j(2,n−1)+rn−r1
≥
ϑn
2
, (4.71)
then B
(2,n)
k := I, k ∈ Z. In the other case, we put
B
(2,n)
0 := I, B
(2,n)
1 := I, . . . , B
(2,n)
j(2,n−1)−1 := I,
B
(2,n)
j(2,n−1)
:= I, B
(2,n)
j(2,n−1)+1 := I, . . . , B
(2,n)
j(2,n−1)+pn−1 := I,
B
(2,n)
j(2,n−1)+pn
:= Mn
1 , B
(2,n)
j(2,n−1)+pn+1 := I, . . . , B
(2,n)
j(2,n−1)+2pn−1 := I,
B
(2,n)
j(2,n−1)+2pn
:= I, B
(2,n)
j(2,n−1)+2pn+1 := I, . . . , B
(2,n)
j(2,n−1)+2pn+pn−1 := I,
B
(2,n)
j(2,n−1)+2pn+pn
:= Mn
2 , B
(2,n)
j(2,n−1)+2pn+pn+1 := I, . . . , B
(2,n)
j(2,n−1)+4pn−1 := I,
4.4 Systems with non-almost periodic solutions 109
...
B
(2,n)
j(2,n−1)+(ln−1)2pn
:= I, B
(2,n)
j(2,n−1)+(ln−1)2pn+1 := I, . . . , B
(2,n)
j(2,n−1)+(ln−1)2pn+pn−1 := I,
B
(2,n)
j(2,n−1)+(ln−1)2pn+pn
:= Mn
ln
, B
(2,n)
j(2,n−1)+(ln−1)2pn+pn+1 := I, . . . , B
(2,n)
j(2,n−1)+2lnpn−1 := I,
B
(2,n)
j(2,n−1)+2lnpn
:= I, B
(2,n)
j(2,n−1)+2lnpn+1 := I, . . . , B
(2,n)
j(2,n−1)+2lnpn+pn−1 := I,
B
(2,n)
j(2,n−1)+2lnpn+pn
:= I, B
(2,n)
j(2,n−1)+2lnpn+pn+1 := I, . . . , B
(2,n)
j(2,n−1)+2(ln+1)pn−1 := I,
...
B
(2,n)
j(2,n−1)+rn−r1−1 := I, . . . , B
(2,n)
j(2,n−1)(rn−r1)−1 := I.
We continue in the n-th step. We deﬁne
˜B
(n−1,n)
k := ˜Bn−1
k · B
(1,n)
k · B
(2,n)
k · · · B
(n−1,n)
k , k ∈ Z.
We consider the solution {x
(n−1,n)
k }k∈Z of
xk+1 = ˜B
(n−1,n)
k · xk, k ∈ Z, x0 = u.
Again, we have to assume that {x
(n−1,n)
k } is almost periodic. Let j(n,n−1) ∈ 2n−1
pnN satisfy
x
(n−1,n)
0 − x
(n−1,n)
j(n,n−1)
= u − x
(n−1,n)
j(n,n−1)
<
ϑn
2
(4.72)
and
j(n,n−1) ≥ j(n−1,n−1)(rn − rn−2). (4.73)
We deﬁne a periodic sequence {B
(n,n)
k }k∈Z with period j(n,n−1)(rn − rn−1). If
x
(n−1,n)
j(n,n−1)
− x
(n−1,n)
j(n,n−1)+rn−rn−1
≥
ϑn
2
, (4.74)
we put B
(n,n)
k := I, k ∈ Z. If inequality (4.74) is not valid, we put
B
(n,n)
0 := I, B
(n,n)
1 := I, . . . , B
(n,n)
j(n,n−1)−1 := I,
B
(n,n)
j(n,n−1)
:= I, B
(n,n)
j(n,n−1)+1 := I, . . . , B
(n,n)
j(n,n−1)+2n−2pn−1 := I,
B
(n,n)
j(n,n−1)+2n−2pn
:= Mn
1 , B
(n,n)
j(n,n−1)+2n−2pn+1 := I, . . . , B
(n,n)
j(n,n−1)+2n−1pn−1 := I,
B
(n,n)
j(n,n−1)+2n−1pn
:= I, B
(n,n)
j(n,n−1)+2n−1pn+1 := I, . . . , B
(n,n)
j(n,n−1)+2n−1pn+2n−2pn−1 := I,
B
(n,n)
j(n,n−1)+2n−1pn+2n−2pn
:= Mn
2 , B
(n,n)
j(n,n−1)+2n−1pn+2n−2pn+1 := I, . . . , B
(n,n)
j(n,n−1)+2npn−1 := I,
...
4.4 Systems with non-almost periodic solutions 110
B
(n,n)
j(n,n−1)+(ln−1)2n−1pn
:= I, B
(n,n)
j(n,n−1)+(ln−1)2n−1pn+1 := I,
. . . , B
(n,n)
j(n,n−1)+(ln−1)2n−1pn+2n−2pn−1 := I,
B
(n,n)
j(n,n−1)+(ln−1)2n−1pn+2n−2pn
:= Mn
ln
, B
(n,n)
j(n,n−1)+(ln−1)2n−1pn+2n−2pn+1 := I,
. . . , B
(n,n)
j(n,n−1)+ln2n−1pn−1 := I,
B
(n,n)
j(n,n−1)+ln2n−1pn
:= I, B
(n,n)
j(n,n−1)+ln2n−1pn+1 := I,
. . . , B
(n,n)
j(n,n−1)+ln2n−1pn+2n−2pn−1 := I,
B
(n,n)
j(n,n−1)+ln2n−1pn+2n−2pn
:= I, B
(n,n)
j(n,n−1)+ln2n−1pn+2n−2pn+1 := I,
. . . , B
(n,n)
j(n,n−1)+(ln+1)2n−1pn−1 := I,
...
B
(n,n)
j(n,n−1)+rn−rn−1−1 := I, . . . , B
(n,n)
j(n,n−1)(rn−rn−1)−1 := I,
where (see (4.49))
rn − rn−1 = rn−1 2n−1
ln − 1 ≥ pn−12n
2n−1
ln − 1 = 2pn 2n−1
ln − 1 > ln2n−1
pn.
Finally, in the n-th step, we deﬁne
Bn
k := B
(1,n)
k · B
(2,n)
k · · · B
(n,n)
k , k ∈ Z, (4.75)
and
˜Bn
k := Ak · B1
k · B2
k · · · Bn
k , k ∈ Z.
Then, we consider the solution {xn
k }k∈Z of
xk+1 = ˜Bn
k · xk, k ∈ Z, x0 = u.
Applying Lemma 4.29 for Bn+j
k = I, k ∈ Z, j ∈ N, it suﬃces to consider the case, when
{xn
k } is almost periodic, and to continue in the construction.
All sequence {Bn
k }k∈Z is periodic as the product of n periodic sequences. Let qn be a
period of {Bn
k }, n ∈ N. In the construction, we can obtain matrices diﬀerent from I only
for
B1
2l+1, B
(1,2)
4l+2, B
(2,2)
8l+4, . . . , B
(1,n)
lpn+pn
2
, B
(2,n)
2lpn+pn
, . . . , B
(n,n)
2n−1lpn+2n−2pn
, . . . , (4.76)
where l ∈ Z. Considering (4.65) and (4.75) (see also (4.62)), the structure of the indices of
matrices in (4.76) gives (4.29). It is seen that (4.27) and (4.30) follow from (4.46) and from
the construction. Analogously, (4.28) follows from (4.47). Thus, applying Lemma 4.29 for
the sequence of
Bk := Ak · B1
k · B2
k · · · Bn
k · · · , k ∈ Z,
4.4 Systems with non-almost periodic solutions 111
we have that {Bk} ∈ Oσ
ε ({Ak}) ∩ LP(X).
To complete the proof, it suﬃces to show that the solution {xk}k∈Z of the problem
xk+1 = Bk · xk, k ∈ Z, x0 = u
is not almost periodic. On contrary, let us suppose that {xk} is almost periodic. We use
Theorem 1.3 for h1 = 0, hn+1 = rn, n ∈ N (see (4.66)). We know that, for any ξ > 0, there
exist inﬁnitely many i, j ∈ N satisfying
xk+li
− xk+lj
< ξ, k ∈ Z. (4.77)
From the construction (consider (4.57), (4.60), . . . , (4.67), (4.70), . . . , (4.73)), we obtain
Bn+j
k = I, k ∈ {0, 1, . . . , qn − 1}, n, j ∈ N. (4.78)
Hence, we get
xj(1,0) − xj(1,0)+r1 = Bj(1,0)−1 · · · B1 · B0 · u − Bj(1,0)+r1−1 · · · B1 · B0 · u
= B1
j(1,0)−1 · · · B1
1 · B1
0 · Aj(1,0)−1 · · · A1 · A0 · u
− B1
j(1,0)+r1−1 · · · B1
1 · B1
0 · Aj(1,0)+r1−1 · · · A1 · A0 · u
= B1
j(1,0)−1 · · · B1
1 · B1
0 · x0
j(1,0) − B1
j(1,0)+r1−1 · · · B1
1 · B1
0 · x0
j(1,0)+r1
,
i.e.,
xj(1,0) − xj(1,0)+r1
= B1
j(1,0)−1 · · · B1
1 · B1
0 · x0
j(1,0) − B1
j(1,0)+r1−1 · · · B1
1 · B1
0 · x0
j(1,0)+r1
.
(4.79)
If (4.54) is valid, then we can rewrite (4.79) into
xj(1,0) − xj(1,0)+r1 = I · · · I · I · x0
j(1,0) − I · · · I · I · x0
j(1,0)+r1
≥
ϑ1
2
.
If (4.55) is true, then we have
xj(1,0) − xj(1,0)+r1 = I · · · I · I · x0
j(1,0) − M1
l1
· · · M1
2 · M1
1 · x0
j(1,0)+r1
>
ζ
2
≥
ϑ1
2
,
which follows from (4.48), (4.51), (4.52), and from (see (4.53), (4.55))
u − x0
j(1,0)+r1
≤ u − x0
j(1,0) + x0
j(1,0) − x0
j(1,0)+r1
<
ϑ1
2
+
ϑ1
2
= ϑ1.
In the both cases, we get
xj(1,0) − xj(1,0)+r1 ≥
ϑ1
2
. (4.80)
4.4 Systems with non-almost periodic solutions 112
Considering (4.78) and the construction, we can express
xj(1,1) − xj(1,1)+r2 = Bj(1,1)−1 · · · B1 · B0 · u − Bj(1,1)+r2−1 · · · B1 · B0 · u
= B
(1,2)
j(1,1)−1 · · · B
(1,2)
1 · B
(1,2)
0 · ˜B1
j(1,1)−1 · · · ˜B1
1 · ˜B1
0 · u
− B
(1,2)
j(1,1)+r2−1 · · · B
(1,2)
1 · B
(1,2)
0 · ˜B1
j(1,1)+r2−1 · · · ˜B1
1 · ˜B1
0 · u
= B
(1,2)
j(1,1)−1 · · · B
(1,2)
1 · B
(1,2)
0 · x1
j(1,1)
− B
(1,2)
j(1,1)+r2−1 · · · B
(1,2)
1 · B
(1,2)
0 · x1
j(1,1)+r2
,
i.e.,
xj(1,1) − xj(1,1)+r2
= B
(1,2)
j(1,1)−1 · · · B
(1,2)
1 · B
(1,2)
0 · x1
j(1,1) − B
(1,2)
j(1,1)+r2−1 · · · B
(1,2)
1 · B
(1,2)
0 · x1
j(1,1)+r2
.
(4.81)
If (4.58) is valid, then (4.81) takes the form
xj(1,1) − xj(1,1)+r2 = I · · · I · I · x1
j(1,1) − I · · · I · I · x1
j(1,1)+r2
≥
ϑ2
2
. (4.82)
If (4.58) is not valid, then we have
xj(1,1) − xj(1,1)+r2
= I · · · I · I · x1
j(1,1) − M2
l2
· · · M2
2 · M2
1 · x1
j(1,1)+r2
>
ζ
2
≥
ϑ2
2
.
(4.83)
Indeed, it suﬃces to consider (4.48), (4.51), (4.52), and the inequality (see also (4.56))
u − x1
j(1,1)+r2
≤ u − x1
j(1,1) + x1
j(1,1) − x1
j(1,1)+r2
<
ϑ2
2
+
ϑ2
2
= ϑ2.
Again, one can express
xj(2,1) − xj(2,1)+r2−r1 = Bj(2,1)−1 · · · B1 · B0 · u − Bj(2,1)+r2−r1−1 · · · B1 · B0 · u
= B
(2,2)
j(2,1)−1 · · · B
(2,2)
1 · B
(2,2)
0 · ˜B
(1,2)
j(2,1)−1 · · · ˜B
(1,2)
1 · ˜B
(1,2)
0 · u
− B
(2,2)
j(2,1)+r2−r1−1 · · · B
(2,2)
0 · ˜B
(1,2)
j(2,1)+r2−r1−1 · · · ˜B
(1,2)
0 · u
= B
(2,2)
j(2,1)−1 · · · B
(2,2)
1 · B
(2,2)
0 · x
(1,2)
j(2,1)
− B
(2,2)
j(2,1)+r2−r1−1 · · · B
(2,2)
1 · B
(2,2)
0 · x
(1,2)
j(2,1)+r2−r1
,
4.4 Systems with non-almost periodic solutions 113
i.e.,
xj(2,1) − xj(2,1)+r2−r1 = B
(2,2)
j(2,1)−1 · · · B
(2,2)
1 · B
(2,2)
0 · x
(1,2)
j(2,1)
− B
(2,2)
j(2,1)+r2−r1−1 · · · B
(2,2)
1 · B
(2,2)
0 · x
(1,2)
j(2,1)+r2−r1
.
(4.84)
If (4.61) is valid, then (4.84) gives
xj(2,1) − xj(2,1)+r2−r1 = I · · · I · I · x
(1,2)
j(2,1) − I · · · I · I · x
(1,2)
j(2,1)+r2−r1
≥
ϑ2
2
. (4.85)
If (4.61) is not valid, then (4.84) gives
xj(2,1) − xj(2,1)+r2−r1
= I · · · I · I · x
(1,2)
j(2,1) − M2
l2
· · · M2
2 · M2
1 · x
(1,2)
j(2,1)+r2−r1
>
ζ
2
≥
ϑ2
2
,
(4.86)
where (4.48), (4.51), (4.52), (4.59), and (4.61) are used.
Finally (see (4.82), (4.83), (4.85), and (4.86)), from the second step of the construction,
we have
xj(1,1) − xj(1,1)+r2 ≥
ϑ2
2
, xj(2,1) − xj(2,1)+r2−r1 ≥
ϑ2
2
. (4.87)
Analogously as (4.80) and (4.87) (consider again (4.48), (4.51), (4.52) and the construction
with (4.64), (4.68), (4.69), (4.71), . . . , (4.72), (4.74)), one can obtain
xj(1,n−1)
− xj(1,n−1)+rn ≥
ϑn
2
,
xj(2,n−1)
− xj(2,n−1)+rn−r1 ≥
ϑn
2
,
...
xj(n,n−1)
− xj(n,n−1)+rn−rn−1 ≥
ϑn
2
for all n ∈ N.
Considering Lemma 4.30, we can assume that (see (4.50) and (4.51))
sup
j∈N
Kj < ∞, i.e., ϑ := inf
j∈N
ϑj > 0. (4.88)
Thus, for all n ∈ N, we obtain
xj(1,n−1)
− xj(1,n−1)+rn ≥
ϑ
2
,
xj(2,n−1)
− xj(2,n−1)+rn−r1 ≥
ϑ
2
,
...
4.4 Systems with non-almost periodic solutions 114
xj(n,n−1)
− xj(n,n−1)+rn−rn−1 ≥
ϑ
2
.
Especially, for all i = j, i, j ∈ N, there exists l ∈ Z such that
xl+li
− xl+lj
≥
ϑ
2
.
This contradiction (consider (4.77) for 2ξ ≤ ϑ) proves that {xk} is not almost periodic.
Remark 4.32. It is easy to see that the statement of Theorem 4.31 does not change if
one replaces system {Ak} ∈ LP(X) by a periodic one. Indeed, it follows directly from
Deﬁnition 1.17.
Remark 4.33. To illustrate Theorem 4.31, let us consider an arbitrary periodic system
{Mk} in the complex case (i.e., for F = C with the usual absolute value). It means that
we have a system
xk+1 = Mk · xk, k ∈ Z, where Mk = Mk+p, k ∈ Z,
for a positive integer p and arbitrarily given non-singular complex matrices M0, . . . , Mp−1.
We know that a solution of {Mk} is almost periodic if and only if it is bounded (see
Corollary 2.22). The fundamental matrix Φ(k, 0) of {Mk} satisfying Φ(0, 0) = I is given
by
Φ(lp + i, 0) = Mi−1 · · · M1 · M0 · (Mp−1 · · · M1 · M0)l
, l ∈ N ∪ {0}, i ∈ {1, . . . , p}.
Thus, to describe the structure of almost periodic solutions, it suﬃces to consider the
multiples (Mp−1 · · · M1 · M0)l
and, in fact, the constant system
xk+1 = Mp−1 · · · M1 · M0 · xk, k ∈ Z.
For any constant system given by a non-singular complex matrix M, one can easily ﬁnd
a commutative matrix group X containing M and having property P with respect to a
vector (e.g., one can consider the group generated by matrices cM for all complex numbers
c = sin l + i cos l, l ∈ Z). Applying Theorem 4.31, we know that, in any neighbourhood of
the considered system, there exists a limit periodic system whose coeﬃcient matrices are
from the group and whose fundamental matrix is not almost periodic. In addition, such a
limit periodic system can be found for any commutative group X which contains M and
which has property P with respect to at least one vector.
Remark 4.34. We repeat that the basic motivation comes from the previous section, where
non-asymptotically almost periodic solutions of limit periodic systems are considered. Of
course, systems with coeﬃcient matrices from bounded groups are analysed in Section 4.3.
For general groups, it is not possible to prove the main results of Section 4.3. It suﬃces
to consider the constant system given by matrix I/2 in the complex case. Any solution
{xk}k∈Z of this system has the property that
|| xl+1 || =
|| xl ||
2
, l ∈ N.
4.4 Systems with non-almost periodic solutions 115
Thus, there exists a neighbourhood of the system such that, for any solution {yk}k∈Z of an
almost periodic system from the neighbourhood, we obtain limk→∞ || yk || = 0, which gives
the asymptotic almost periodicity of {yk} (see Remark 1.16).
At the same time, in Section 4.3, there is required that the studied matrix group
has property P. Since the group X has property P only with respect to one vector in
the statement of Theorem 4.31, we can apply this theorem for groups of matrices in the
following form 




X 0 · · · 0
0 1 · · · 0
...
...
...
...
0 0 · · · 1





,
where X is taken from a commutative matrix group having property P with respect to a
concrete vector. In this sense, Theorem 4.31 generalizes Theorem 4.16 as well.
The construction from the proof of Theorem 4.31 can be applied for the Cauchy (initial)
problem. Especially, we immediately obtain the following result.
Theorem 4.35. Let a non-zero vector u ∈ Fm
be given. Let X have the property that there
exist ζ > 0 and K > 0 such that, for all δ > 0, one can ﬁnd matrices M1, . . . , Ml ∈ X
satisfying
Mi ∈ Oδ (I), i ∈ {1, . . . , l}, Ml · · · M1 · u − u > ζ, Ml · · · M1 < K. (4.89)
For any {Ak} ∈ LP(X) and ε > 0, there exists a system {Bk} ∈ Oσ
ε ({Ak}) ∩ LP(X) for
which the solution of
xk+1 = Bk · xk, k ∈ Z, x0 = u
is not almost periodic.
Proof. The theorem follows from the proof of Theorem 4.31, where (4.88) is satisﬁed (i.e.,
the case, which is covered by Lemma 4.30, does not happen).
Remark 4.36. We point out that, in a certain sense, Theorem 4.35 has been improved
in [41].
Similarly to Theorem 4.23 which is the almost periodic version of Theorem 4.16, we
formulate the below given Theorem 4.39 as the almost periodic version of Theorem 4.31.
We need the next two lemmas.
Lemma 4.37. Let {Ak} ∈ AP(X) and ε > 0 be arbitrarily given. Let {δn}n∈N ⊂ R be a
decreasing sequence satisfying (4.27) and let {Bn
k }k∈Z ⊂ X be periodic sequences for n ∈ N
such that (4.28) and (4.29) are valid. Then, {Bk} ∈ AP(X) if
Bk := Ak · B1
k · B2
k · · · Bn
k · · · , k ∈ Z.
In addition, {Bk} ∈ Oσ
ε ({Ak}) if (4.30) is fulﬁlled.
4.4 Systems with non-almost periodic solutions 116
Proof. The lemma can be proved analogously as Lemma 4.29. In the proof of Lemma 4.29,
it suﬃces to put Cn
k = Ak for all k ∈ Z, n ∈ N, to use Theorem 1.7, and to consider the
almost periodicity of {Cn
k · B1
k · B2
k · · · Bn
k }k∈Z ⊂ X which follows from Theorems 1.3, 1.14,
and 1.22 and from Lemma 4.11.
Using the same way which is applied in the proof of Lemma 4.30, we can prove its
almost periodic counterpart. Indeed, we do not use the limit periodicity of {Ak} in the
proof (consider also Lemma 4.37).
Lemma 4.38. If for any δ > 0 and K > 0, there exist matrices M1, . . . , Ml ∈ X such
that (4.34) is valid, then, for any {Ak} ∈ AP(X) and ε > 0, there exists a system {Bk} ∈
Oσ
ε ({Ak}) whose fundamental matrix is not almost periodic.
Theorem 4.39. Let X have property P with respect to a vector. For any {Ak} ∈ AP(X)
and ε > 0, there exists a system {Bk} ∈ Oσ
ε ({Ak}) whose fundamental matrix is not almost
periodic.
Proof. The theorem can be proved using the same construction as Theorem 4.31. It suﬃces
to replace Lemma 4.29 by Lemma 4.37 and Lemma 4.30 by Lemma 4.38.
Analogously, we get the following result as well.
Theorem 4.40. Let a non-zero vector u ∈ Fm
be given. Let X have the property that there
exist ζ > 0 and K > 0 such that, for all δ > 0, one can ﬁnd matrices M1, . . . , Ml ∈ X satisfying
(4.89). For any {Ak} ∈ AP(X) and ε > 0, there exists a system {Bk} ∈ Oσ
ε ({Ak})
for which the solution of
xk+1 = Bk · xk, k ∈ Z, x0 = u
is not almost periodic.
Remark 4.41. We add that Theorems 4.39 and 4.40 do not follow from Theorems 4.31
and 4.35. Consider Theorem 1.21.
At the end, we remark that all main results presented in this chapter remain true if
one replaces k ∈ Z by k ∈ N; and we repeat that the main results of Chapter 2 are not
covered by results about almost periodic systems presented in this chapter.
Chapter 5
Almost periodic and limit periodic
functions in pseudometric spaces
This chapter is analogous to Chapter 1, where almost periodic and limit periodic sequences
are considered. Here we consider almost periodic and limit periodic functions.
Our aim is to mention basic properties of considered functions and to show a way one can
generate functions with several prescribed properties. Since our process can be used for
generalizations of classical (complex valued) almost periodic (and limit periodic) functions,
we introduce the almost and limit periodicity in pseudometric spaces and we present our
method for functions with values in a pseudometric space X as in Chapter 1.
We point out that we obtain the most important case if X is a Banach space, and
that the theory of almost periodic functions of the real variable with values in a Banach
space, given by S. Bochner in [22], is in its essential lines similar to the theory of classical
almost periodic functions which is due to H. Bohr in [27, 28]. We introduce almost periodic
and limit periodic functions in pseudometric spaces using a trivial extension of the Bohr
concept, where the modulus is replaced by the distance. In the classical case, we refer to
the monographs [18, 72, 128]; for functions with values in Banach spaces, to [7, 46, 117];
for other extensions, to [9, 12, 19, 20, 31, 77, 84, 99, 181]; for modiﬁcations, to [46, 85] and
references cited therein; for applications, to [32, 47, 160, 164].
Necessary and suﬃcient conditions for a continuous function with values in a Banach
space to be almost periodic may be no longer valid for continuous functions in general
metric spaces. For the approximation condition, it is seen that the completeness of the
space of values is necessary and H. Tornehave (in [177]) also required the local connection by
arcs of the space of values. In the Bochner condition, it suﬃces to replace the convergence
by the Cauchy condition. Since we need the Bochner concept as well, we recall that the
Bochner condition means that any sequence of translates of a given continuous function
has a subsequence which converges uniformly on the domain of the function. The fact,
that this condition is equivalent with the Bohr deﬁnition of almost periodicity in Banach
spaces, was proved by S. Bochner in [22].
We begin with the used notation in Section 5.1. The above mentioned Bohr deﬁnition
and Bochner condition are formulated in Section 5.2 (with some basic properties of considered
functions). In Section 5.2, processes from [46] are generalized. Analogously, the
117
5.1 Preliminaries 118
theory of almost periodic functions of real variable with fuzzy real numbers as values is
developed in [13] (see also [157]). In Section 5.3, we mention the way one can construct
almost periodic functions with prescribed properties in a pseudometric space. We present
it in Theorems 5.20, 5.22, and 5.24 below. Note that it is possible to obtain many modiﬁcations
and generalizations of our method. A special construction of almost periodic
functions with given properties is published (and applied) in [101] as well.
5.1 Preliminaries
Let X be an arbitrary pseudometric space with a pseudometric . Symbol Oε(x) denotes
the ε-neighbourhood of x in X for arbitrary ε > 0, x ∈ X. The set of all non-negative real
numbers is denoted by R+
0 .
5.2 Generalizations of pure periodicity
As in the ﬁrst chapter, we deﬁne the notion of almost and limit periodicity in pseudometric
spaces.
5.2.1 Almost periodic functions
At ﬁrst, we introduce the almost periodicity in X. Observe that we are not able to
distinguish between x ∈ X and y ∈ X if (x, y) = 0.
Deﬁnition 5.1. A continuous function ψ : R → X is almost periodic if for any ε > 0,
there exists a number p (ε) > 0 with the property that any interval of length p (ε) of the
real line contains at least one point s such that
(ψ(t + s), ψ(t)) < ε, t ∈ R.
The number s is called an ε-translation number and the set of all ε-translation numbers
of ψ is denoted by T(ψ, ε).
Remark 5.2. It is possible to introduce almost periodic functions deﬁned on various sets.
For almost periodic functions deﬁned on the torus (on the annuloid), see [136, 154]; on
a tube, see [68]; on a circle, see [30].
If X is a Banach space, then a continuous function ψ is almost periodic if and only
if any set of translates of ψ has a subsequence, uniformly convergent on R in the sense
of the norm. See, e.g., [46, Theorem 6.6]. Evidently, this result cannot be longer valid if
the space of values is not complete. Nevertheless, we prove the below given Theorem 5.5,
where the convergence is replaced by the Cauchy condition. Before proving this result, we
mention two simple lemmas. Their proofs are easily obtained by modifying the proofs of
[46, Theorem 6.2] and [46, Theorem 6.5].
5.2 Generalizations of pure periodicity 119
Lemma 5.3. An almost periodic function with values in X is uniformly continuous on the
real line.
Proof. Let an almost periodic function ψ : R → X be given and let p = p (ε/3), where
ε > 0 is arbitrary, be from Deﬁnition 5.1. Since ψ is uniformly continuous on the interval
I := [−1, 1 + p], there exists δ = δ(ε) ∈ (0, 1) such that
(ψ(t1), ψ(t2)) <
ε
3
, t1, t2 ∈ I, | t1 − t2 | < δ.
Let t1, t2 ∈ R satisfying | t1 − t2 | < δ be arbitrary and s = s(t1, δ) ∈ [−t1, −t1 + p] be an
(ε/3)-translation number of ψ. Evidently, t1 + s ∈ I, t2 + s ∈ I. Finally, we have
(ψ(t1), ψ(t2)) ≤ (ψ(t1), ψ(t1 + s)) + (ψ(t1 + s), ψ(t2 + s))
+ (ψ(t2 + s), ψ(t2)) <
ε
3
+
ε
3
+
ε
3
= ε,
which terminates the proof.
Lemma 5.4. The set of all values of an almost periodic function ψ : R → X is totally
bounded in X.
Proof. Let p = p (ε/2) be from Deﬁnition 5.1 for arbitrarily given ε > 0. Obviously, the
set of all values of ψ on [0, p] is a subset of a ﬁnite number of neighbourhoods of radius
ε/2. Let us denote by x1, x2, . . . , xq the centres of these neighbourhoods which cover
the set {ψ(t); t ∈ [0, p]}. For an arbitrary t ∈ R, we take an (ε/2)-translation number
s = s(t) ∈ [−t, −t + p] of ψ. Thus, t + s ∈ [0, p]. Let x(t) ∈ {x1, x2, . . . , xq} be the centre
of the neighbourhood of radius ε/2 which contains ψ(t + s). We obtain
(x(t), ψ(t)) ≤ (x(t), ψ(t + s)) + (ψ(t + s), ψ(t)) <
ε
2
+
ε
2
= ε.
It shows that, for any ε > 0, the set of all values of ψ is covered by a ﬁnite number of
neighbourhoods of radius ε.
Theorem 5.5. Let ψ : R → X be a continuous function. Then, ψ is almost periodic if and
only if, from any sequence of the form {ψ(t + sn)}n∈N, where sn are real numbers, one can
extract a subsequence {ψ (t + rn)}n∈N satisfying the Cauchy uniform convergence condition
on R, i.e., for any ε > 0, there exists l(ε) ∈ N with the property that
(ψ(t + ri), ψ(t + rj)) < ε, t ∈ R,
for all i, j > l(ε), i, j ∈ N.
Proof. We prove the suﬃciency of the condition using a simple extension of the argument
used in the proof of [46, Theorem 1.10]. Suppose, on contrary, that ψ is not almost periodic.
Then, there exists a number ε > 0 such that, for any p ∈ N, one can ﬁnd an interval of
length p which does not contain any ε-translation number of ψ. Consider an arbitrary
number l1 ∈ N and an interval (a1, b1) ⊆ R of the length greater than 2(l1 + 1) which
contains no ε-translation number of ψ. We choose l2 ∈ Z such that l2 − l1 ∈ (a1, b1). Thus,
l2 − l1 is not an ε-translation number of ψ. Next, there exists an interval (a2, b2) ⊆ R of
5.2 Generalizations of pure periodicity 120
the length greater than 2(l1 +l2 +1) such that there exists no ε-translation number of ψ in
(a2, b2). We can also ﬁnd l3 ∈ Z for which l3 − l1, l3 − l2 ∈ (a2, b2) and, hence, l3 − l1, l3 − l2
cannot be ε-translation numbers of ψ.
Proceeding in a similar way, we get a sequence {ln}n∈N satisfying that none of the numbers
ln1 − ln2 , where n1 = n2 (n1, n2 ∈ N), is an ε-translation number of ψ. Therefore, we
obtain
(ψ(t + ln1 − ln2 ), ψ(t)) ≥ ε
for all n1 = n2 (n1, n2 ∈ N) and at least one t ∈ R. This contradiction proves that ψ is
almost periodic.
To prove the converse implication, we assume that ψ is an almost periodic function.
We apply the well-known method of the diagonal extraction and modify the proof of [46,
Theorem 6.6].
Let {tn; n ∈ N} be a dense subset of R and {sn}n∈N ⊂ R be an arbitrarily given
sequence. From the sequence {ψ(t1 + sn)}n∈N, using Lemma 5.4, we choose a subsequence
{ψ(t1 + r1
n)}n∈N such that, for any ε > 0, there exists l1(ε) ∈ N with the property that
ψ t1 + r1
i ), ψ(t1 + r1
j < ε, i, j > l1(ε), i, j ∈ N.
Such a subsequence exists, because inﬁnitely many values ψ(t1 +sn) is in a neighbourhood
of radius 2−i
for all i ∈ N (consider the method of the diagonal extraction). Analogously,
from the sequence {ψ(t2 + r1
n)}n∈N, we get {ψ(t2 + r2
n)}n∈N such that, for any ε > 0, there
exists l2(ε) ∈ N for which
ψ(t2 + r2
i ), ψ(t2 + r2
j ) < ε, i, j > l2(ε), i, j ∈ N.
We proceed further in the same way. We obtain {rk
n} ⊆ · · · ⊆ {r1
n}, k ∈ N.
Let ε > 0 be arbitrarily given, p = p (ε/5) be from Deﬁnition 5.1, δ = δ (ε/5) correspond
to ε/5 from the deﬁnition of the uniform continuity of ψ (see Lemma 5.3) and let a ﬁnite
set {t1, . . . , tj} ⊂ {tn; n ∈ N} satisfy
min
i∈{1,...,j}
| ti − t | < δ, t ∈ [0, p].
Obviously, there exists l ∈ N such that, for all integers n1, n2 > l, it holds
ψ(ti + rn1
n1
), ψ(ti + rn2
n2
) <
ε
5
, i ∈ {1, . . . , j}.
Let t ∈ R be given, s = s(t) ∈ [−t, −t + p] be an (ε/5)-translation number of ψ, and
ti = ti(s) ∈ {t1, . . . , tj} be such that | t + s − ti | < δ. Finally, we have
ψ(t + rn1
n1
), ψ(t + rn2
n2
) ≤ ψ(t + rn1
n1
), ψ(t + rn1
n1
+ s)
+ ψ(t + rn1
n1
+ s), ψ(ti + rn1
n1
) + ψ(ti + rn1
n1
), ψ(ti + rn2
n2
)
+ ψ(ti + rn2
n2
), ψ(t + rn2
n2
+ s) + ψ(t + rn2
n2
+ s), ψ(t + rn2
n2
) .
Thus, we obtain
ψ(t + rn1
n1
), ψ(t + rn2
n2
) <
ε
5
+
ε
5
+
ε
5
+
ε
5
+
ε
5
= ε (5.1)
for all t ∈ R, n1, n2 > l, n1, n2 ∈ N. Evidently, (5.1) completes the proof of the theorem if
we put rn := rn
n, n ∈ N.
5.2 Generalizations of pure periodicity 121
Remark 5.6. Using a combination of the methods used in the proofs of Theorems 1.3
and 5.5, it is possible to prove that, for a continuous function f : G → X with G an abelian
topological group and X a complete metric space, the deﬁnitions of almost periodicity in
the sense of Bohr and Bochner are equivalent. For other generalizations, see [14]; for
other equivalent deﬁnitions (e.g., the von Neumann and the Maak deﬁnition) of almost
periodicity, see [117, 128]. We add that, for the ﬁrst time, almost periodic functions on
groups with values in Banach spaces were studied by S. Bochner and J. von Neumann
in [24, 25].
In the recent years, many researchers study the concept of almost periodicity on time
scales and analyse solutions of almost periodic (linear) dynamic equations. We refer at
least to papers [86, 119, 120, 121, 127, 180, 185].
Analogously as for complex valued almost periodic functions or almost periodic sequences
in Chapter 1, one can prove many properties of almost periodic functions with
values in pseudometric spaces.
Theorem 5.7. Let X1, X2 be pseudometric spaces and Φ : X1 → X2 be a uniformly continuous
map. If ψ : R → X1 is almost periodic, then Φ ◦ ψ is almost periodic as well.
Proof. We can proceed similarly as in the proof of Theorem 1.6. If δ(ε) > 0 is the number
corresponding to arbitrary ε > 0 from the deﬁnition of the uniform continuity of Φ, then
it is valid
T (ψ, δ(ε)) ⊆ T (Φ ◦ ψ, ε)
which proves the theorem.
Theorem 5.8. The limit of a uniformly convergent sequence of almost periodic functions
is almost periodic.
Proof. It is possible to prove the theorem using the process from the proof of [46, Theorem
6.4].
Directly from Theorem 5.5, we obtain the following corollaries.
Corollary 5.9. Let X be a Banach space. The sum of two almost periodic functions with
values in X is an almost periodic function.
Corollary 5.10. If X1, . . . , Xn are pseudometric spaces and ψ1, . . . , ψn are arbitrary almost
periodic functions with values in X1, . . . , Xn, respectively, then the function ψ, with values
in X1 × · · · × Xn given by ψ := (ψ1, . . . , ψn), is almost periodic.
We add that one can use Corollary 5.10 to obtain simple modiﬁcations of the below
presented method of constructions of almost periodic functions. Moreover, from Corollary
5.10, we get:
Corollary 5.11. The set
T(ψ1, ε) ∩ T(ψ2, ε) ∩ · · · ∩ T(ψn, ε)
is relatively dense in R for arbitrary almost periodic functions ψ1, ψ2, . . . , ψn and any ε > 0.
5.2 Generalizations of pure periodicity 122
Remark 5.12. For the ﬁrst time, Corollary 5.11 was proved for almost periodic functions
with values in an arbitrary metric space in [153].
To conclude this subsection, we establish theorems which show how almost periodic
functions can be characterized by almost periodic sequences.
Theorem 5.13. A uniformly continuous function ψ : R → X is almost periodic if and
only if there exists a sequence of positive numbers rn, n ∈ N, satisfying rn → 0 as n → ∞,
such that the sequence {ψ(rnk)}k∈Z is almost periodic for all n ∈ N.
Proof. One can prove the theorem using a corresponding extension of the proof of [46,
Theorem 1.29].
Theorem 5.14. Let X be a Banach space. A necessary and suﬃcient condition for a sequence
{ϕk}k∈Z ⊆ X to be almost periodic is the existence of an almost periodic function
ψ : R → X for which ψ(k) = ϕk, k ∈ Z.
Proof. The suﬃciency of the condition follows directly from Theorems 1.3 and 5.5. Conversely,
assume that an almost periodic sequence {ϕk}k∈Z is given. We deﬁne
ψ(t) := ϕk + (t − k)(ϕk+1 − ϕk), k ≤ t < k + 1, k ∈ Z. (5.2)
Evidently, ψ : R → X is continuous and ψ(k) = ϕk, k ∈ Z. The almost periodicity of ψ
follows from
T {ϕk} ,
ε
3
⊆ T(ψ, ε)
which can be proved using (5.2).
Remark 5.15. Many known theorems show how almost periodic functions can be characterized
by almost periodic sequences. Such theorems are used to study almost periodic
solutions of diﬀerential equations as well. General examples of diﬀerential equations, for
which a solution x(t) deﬁned for t ∈ R is almost periodic if and only if {x(k)}k∈Z is an
almost periodic sequence, are mentioned in [4, 133, 144].
5.2.2 Limit periodic functions
Now we brieﬂy recall the concept of limit periodicity for continuous functions with
ranges in X.
Deﬁnition 5.16. A function f : R → X is called limit periodic if f(x) = limn→∞ fn(x)
uniformly for x ∈ R, where all fn : R → X are periodic continuous functions.
Remark 5.17. As in the discrete case (cf. Remark 1.18), the periods of functions fn in
Deﬁnition 5.16 do not need to be the same for considered n.
Theorem 5.18. Any limit periodic function is almost periodic.
Proof. It suﬃces to consider Theorem 5.8 and the above deﬁnitions.
Theorem 5.19. There exist almost periodic functions f : R → C (with respect to the usual
metric) which are not limit periodic.
Proof. The theorem follows from the characterization of limit periodic functions using the
Fourier expansion, which can be found in [18] (see also [47, p. 129]).
5.3 Constructions of almost periodic functions 123
5.3 Constructions of almost periodic functions
Now we present the way one can generate almost periodic functions with given pro-
perties.
Theorem 5.20. For arbitrary a > 0, any continuous function ψ : R → X such that
ψ(t) ∈ Oa (ψ(t − 1)) , t ∈ (1, 2],
ψ(t) ∈ Oa (ψ(t + 2)) , t ∈ (−2, 0],
ψ(t) ∈ Oa/2 (ψ(t − 4)) , t ∈ (2, 6],
ψ(t) ∈ Oa/2 (ψ(t + 8)) , t ∈ (−10, −2],
ψ(t) ∈ Oa/4 ψ(t − 24
) , t ∈ (2 + 22
, 2 + 22
+ 24
],
ψ(t) ∈ Oa/4 ψ(t + 25
) , t ∈ (−25
− 23
− 2, −23
− 2],
...
ψ(t) ∈ Oa 2−n ψ(t − 22n
) , t ∈ (2 + 22
+ · · · + 22n−2
, 2 + 22
+ · · · + 22n−2
+ 22n
],
ψ(t) ∈ Oa 2−n ψ(t + 22n+1
) , t ∈ (−22n+1
− · · · − 23
− 2, −22n−1
− · · · − 23
− 2],
...
is almost periodic.
Proof. Let ε > 0 be arbitrary and k = k(ε) ∈ N be such that 2k
> 8a/ε. It suﬃces to
prove that l 22k
is an ε-translation number of ψ for any integer l.
First we deﬁne
ϕ(t) := ψ(t), t ∈ [−22k−1
− · · · − 23
− 2, 2 + 22
+ · · · + 22k−2
].
We see that
ψ(t) ∈ Oε/8 ϕ(t − 22k
) , t ∈ [2 + 22
+ · · · + 22k−2
, 2 + 22
+ · · · + 22k
],
ψ(t) ∈ Oε/8 ψ(t + 22k+1
) , t ∈ [−22k+1
− · · · − 23
− 2, −22k−1
− · · · − 23
− 2],
ψ(t) ∈ Oε/16 ψ(t − 22k+2
) , t ∈ [2 + 22
+ · · · + 22k
, 2 + 22
+ · · · + 22k+2
],
ψ(t) ∈ Oε/16 ψ(t + 22k+3
) , t ∈ [−22k+3
− · · · − 23
− 2, −22k+1
− · · · − 23
− 2],
...
In a pseudometric space X, this observation implies
ψ(t + 22k
) ∈ Oε/8 (ϕ(t)) , t ∈ [−22k−1
− · · · − 23
− 2, 2 + 22
+ · · · + 22k−2
],
ψ(t − 22k+1
) ∈ Oε/8 (ϕ(t)) , t ∈ [−22k−1
− · · · − 23
− 2, 2 + 22
+ · · · + 22k−2
],
5.3 Constructions of almost periodic functions 124
ψ(t − 22k
) ∈ Oε/8+ε/8 (ϕ(t)) , t ∈ [−22k−1
− · · · − 23
− 2, 2 + 22
+ · · · + 22k−2
],
ψ(t + 22k+1
) ∈ Oε/8+ε/16 (ϕ(t)) , t ∈ [−22k−1
− · · · − 23
− 2, 2 + 22
+ · · · + 22k−2
],
ψ(t + 3 22k
) ∈ Oε/8+ε/8+ε/16 (ϕ(t)) , t ∈ [−22k−1
− · · · − 23
− 2, 2 + 22
+ · · · + 22k−2
],
ψ(t + 22k+2
) ∈ Oε/16 (ϕ(t)) , t ∈ [−22k−1
− · · · − 23
− 2, 2 + 22
+ · · · + 22k−2
],
ψ(t + 22k
+ 22k+2
) ∈ Oε/8+ε/16 (ϕ(t)) , t ∈ [−22k−1
− · · · − 23
− 2, 2 + 22
+ · · · + 22k−2
],
...
Since
ε
8
+
ε
8
+
ε
16
+
ε
16
+
ε
32
+
ε
32
+ · · · =
ε
2
,
we have
ψ t + l 22k
∈ Oε/2 (ϕ(t)) , t ∈ [−22k−1
− · · · − 23
− 2, 2 + 22
+ · · · + 22k−2
], l ∈ Z. (5.3)
We express any t ∈ R as the sum of numbers p(t) and q(t) for which
p(t) ∈ [−22k−1
− · · · − 23
− 2, 2 + 22
+ · · · + 22k−2
]
and q(t) = j22k
for some j ∈ Z. Using (5.3), we obtain
ψ(t), ψ(t + l 22k
) ≤ (ψ(p(t) + q(t)), ϕ(p(t)))
+ ϕ(p(t)), ψ(p(t) + (j + l) 22k
) <
ε
2
+
ε
2
= ε
(5.4)
for any t ∈ R and l ∈ Z which terminates the proof.
Remark 5.21. Note that the range of a function ψ generating by Theorem 5.20 does not
need to be complete (for a general pseudometric space).
The process mentioned in the previous theorem is easily modiﬁable. We illustrate this
fact by the following two theorems.
Theorem 5.22. Let M > 0, x0 ∈ X, and j ∈ N be given. Let ϕ : [0, M] → X satisfy
ϕ (0) = ϕ (M) = x0. If {rn}n∈N ⊂ R+
0 has the property that
∞
n=1
rn < ∞, (5.5)
then an arbitrary continuous function ψ : R → X, ψ|[0,M] ≡ ϕ for which
ψ (t) = x0, t ∈ {iM, 2 ≤ i ≤ j + 1} ∪ {−i(j + 1)M, 1 ≤ i ≤ j}
∞
n=1
{((j + 1) + · · · + j(j + 1)2n−2
+ i(j + 1)2n
)M; 1 ≤ i ≤ j}
∞
n=1
{−((j + 1) + · · · + j(j + 1)2n−1
+ i(j + 1)2n+1
)M; 1 ≤ i ≤ j}
(5.6)
5.3 Constructions of almost periodic functions 125
and, at the same time, for which it is valid
ψ(t) ∈ Or1 (ψ(t − M)) , t ∈ (M, 2M),
...
ψ(t) ∈ Or1 (ψ(t − jM)) , t ∈ (jM, (j + 1)M) ,
ψ(t) ∈ Or2 (ψ(t + (j + 1)M)) , t ∈ (−(j + 1)M, 0) ,
...
ψ(t) ∈ Or2 (ψ(t + j(j + 1)M)) , t ∈ (−j(j + 1)M, −(j − 1)(j + 1)M) ,
ψ(t) ∈ Or3 ψ(t − (j + 1)2
M) , t ∈ (j + 1)M, ((j + 1) + (j + 1)2
)M ,
...
ψ(t) ∈ Or3 ψ(t − j(j + 1)2
M) ,
t ∈ ((j + 1) + (j − 1)(j + 1)2
)M, ((j + 1) + j(j + 1)2
)M ,
...
ψ(t) ∈ Or2n ψ(t + (j + 1)2n−1
M) ,
t ∈ − ((j + 1)2n−1
+ j(j + 1)2n−3
+ · · · + j(j + 1)3
+ j(j + 1))M,
− (j(j + 1)2n−3
+ · · · + j(j + 1)3
+ j(j + 1))M ,
...
ψ(t) ∈ Or2n ψ(t + j(j + 1)2n−1
M) ,
t ∈ − (j(j + 1)2n−1
+ j(j + 1)2n−3
+ · · · + j(j + 1)3
+ j(j + 1))M,
− ((j − 1)(j + 1)2n−1
+ j(j + 1)2n−3
+ · · · + j(j + 1)3
+ j(j + 1))M ,
ψ(t) ∈ Or2n+1 ψ(t − (j + 1)2n
M) ,
t ∈ ((j + 1) + j(j + 1)2
+ · · · + j(j + 1)2n−2
)M,
((j + 1) + j(j + 1)2
+ · · · + j(j + 1)2n−2
+ (j + 1)2n
)M ,
...
ψ(t) ∈ Or2n+1 ψ(t − j(j + 1)2n
M) ,
t ∈ ((j + 1) + j(j + 1)2
+ · · · + j(j + 1)2n−2
+ (j − 1)(j + 1)2n
)M,
((j + 1) + j(j + 1)2
+ · · · + j(j + 1)2n−2
+ j(j + 1)2n
)M ,
...
is almost periodic.
5.3 Constructions of almost periodic functions 126
Proof. We prove this theorem analogously as Theorem 5.20. Let ε be a positive number
and let an odd integer n(ε) ≥ 2 have the property that (see (5.5))
∞
n=n(ε)
rn <
ε
2
. (5.7)
We prove that l(j + 1)n(ε)−1
M is an ε-translation number of ψ for all l ∈ Z. Let l ∈ Z and
t ∈ R be arbitrary. If we put
s := l(j + 1)n(ε)−1
M, (5.8)
then it suﬃces to show that the inequality
(ψ(t), ψ(t + s)) < ε (5.9)
holds; i.e., this inequality proves the theorem.
We can write t as the sum of numbers t1 and t2, where
t1 ≥ − j(j + 1)n(ε)−2
+ · · · + j(j + 1)3
+ j(j + 1) M,
t1 ≤ j + 1 + j(j + 1)2
+ · · · + j(j + 1)n(ε)−3
M
(5.10)
and
t2 = i(j + 1)n(ε)−1
M for some i ∈ Z. (5.11)
Now we have (see (5.10) and the proof of Theorem 5.20)
(ψ(t), ψ(t + s)) ≤ (ψ(t1 + t2), ψ(t1)) + (ψ(t1), ψ(t1 + t2 + s))
<
n(ε)+p−1
n=n(ε)
rn +
n(ε)+q−1
n=n(ε)
rn.
(5.12)
Indeed, we can express (consider (5.8) and (5.11))
t2 = i1(j + 1)n(ε)−1
+ i2(j + 1)n(ε)
+ · · · + ip(j + 1)n(ε)+p−1
(m + 1),
t2 + s = l1(j + 1)n(ε)−1
+ l2(j + 1)n(ε)
+ · · · + lq(j + 1)n(ε)+q−1
(m + 1),
where i1, . . . , ip, l1, . . . , lq ⊆ {−j, . . . , 0, . . . , j} satisfy
i1 ≥ 0, i2 ≤ 0, · · · (−1)p
ip ≤ 0, l1 ≥ 0, l2 ≤ 0, · · · (−1)q
lq ≤ 0.
Evidently, (5.7) and (5.12) give (5.9).
For j = 1, we get the most important case of Theorem 5.22.
Corollary 5.23. Let M > 0 and x0 ∈ X be given and let ϕ : [0, M] → X be such that
ϕ(0) = ϕ(M) = x0.
If {εi}i∈N ⊂ R+
0 satisﬁes
∞
i=1
εi < ∞, (5.13)
5.3 Constructions of almost periodic functions 127
then any continuous function ψ : R → X, ψ|[0,M] ≡ ϕ for which
ψ (t) = x0, t ∈ {2M, −2M} ∪ {(2 + 22
+ · · · + 22(i−1)
+ 22i
)M; i ∈ N}
∪ {−(2 + 23
+ · · · + 22i−1
+ 22i+1
)M; i ∈ N}
(5.14)
and, at the same time, for which it is valid
ψ(t) ∈ Oε1 (ψ(t − M)) , t ∈ (M, 2M),
ψ(t) ∈ Oε2 (ψ(t + 2M)) , t ∈ (−2M, 0),
ψ(t) ∈ Oε3 ψ(t − 22
M) , t ∈ (2M, (2 + 22
)M),
ψ(t) ∈ Oε4 ψ(t + 23
M) , t ∈ (−(23
+ 2)M, −2M),
ψ(t) ∈ Oε5 ψ(t − 24
M) , t ∈ ((2 + 22
)M, (2 + 22
+ 24
)M),
...
ψ(t) ∈ Oε2i
ψ(t + 22i−1
M) , t ∈ (−(22i−1
+ · · · + 2)M, −(22i−3
+ · · · + 2)M),
ψ(t) ∈ Oε2i+1
ψ(t − 22i
M) , t ∈ ((2 + 22
+ · · · + 22i−2
)M, (2 + 22
+ · · · + 22i
)M),
...
is almost periodic.
Theorem 5.24. Let ϕ : (−r, r] → X, {rn}n∈N ⊂ R+
0 , and {jn}n∈N ⊆ N be arbitrary such
that ∞
n=1
rnjn < ∞ (5.15)
holds. Let a function ψ : R → X satisfy ψ|(−r,r] ≡ ϕ and
ψ(t) ∈ Or1 (ϕ(t − 2r)) , t ∈ (r, r + 2r] ,
...
ψ(t) ∈ Or1 (ϕ(t − 2r)) , t ∈ (r + (j1 − 1)2r, r + j12r] ,
ψ(t) ∈ Or1 (ϕ(t + 2r)) , t ∈ (−2r − r, −r] ,
...
ψ(t) ∈ Or1 (ϕ(t + 2r)) , t ∈ (−j12r − r, −(j1 − 1)2r − r] ,
...
ψ(t) ∈ Orn (ϕ(t − pn)) , t ∈ (p1 + · · · + pn−1, p1 + · · · + pn−1 + pn] ,
...
ψ(t) ∈ Orn (ϕ(t − pn)) , t ∈ (p1 + · · · + pn−1 + (jn − 1)pn, p1 + · · · + pn−1 + jnpn] ,
5.3 Constructions of almost periodic functions 128
ψ(t) ∈ Orn (ϕ(t + pn)) , t ∈ (−pn − pn−1 − · · · − p1, −pn−1 − · · · − p1] ,
...
ψ(t) ∈ Orn (ϕ(t + pn)) , t ∈ (−jnpn − pn−1 − · · · − p1, −(jn − 1)pn − pn−1 − · · · − p1] ,
...
where
p1 := r + j12r, p2 := 2(r + j12r),
p3 := (2j2 + 1)p2, . . . pn := (2jn−1 + 1)pn−1, . . .
If ψ is continuous on R, then it is almost periodic.
Proof. It is not diﬃcult to prove Theorem 5.24 analogously as Theorems 5.20 and 5.22.
For given ε > 0, let an integer n(ε) ≥ 2 satisfy
∞
n=n(ε)
rn jn <
ε
4
.
One can prove the inclusion
{l pn(ε); l ∈ Z} ⊆ T(ψ, ε) (5.16)
which guarantees the almost periodicity of ψ.
Remark 5.25. From the proofs of Theorems 5.20, 5.22, 5.24 (see (5.4), (5.8) and (5.9),
(5.16)), we get an important property of the set of all ε-translation numbers of the resulting
function ψ. For any ε > 0, there exists non-zero c ∈ R for which
{l c; l ∈ Z} ⊆ T(ψ, ε).
Hence, applying the method from the above theorems, one cannot construct almost periodic
functions without this property.
Chapter 6
Solutions of almost periodic
diﬀerential systems
In this chapter, we analyse (non-)almost periodic solutions of almost periodic homogeneous
linear differential systems. Sometimes this ﬁeld is called the Favard theory which
is based on the Favard contributions in [67] (see also [35, Theorem 1.2], [47, Chapter 5],
[72, Theorem 6.3] or [145, Theorem 1]; for homogeneous case, see [44, 66]). In this context,
suﬃcient conditions for the existence of almost periodic solutions are mentioned in
[42, 54, 97] (for generalizations, see [48, 49, 69, 90, 95, 98, 105, 116, 118, 135, 159, 161];
for other extensions and supplements of the Favard theory, e.g., see [2, 34, 36, 37, 52, 53,
55, 96, 122, 166, 167]). Certain suﬃcient conditions, under which homogeneous systems
that have non-trivial bounded solutions have also non-trivial almost periodic solutions, are
given in [146].
It is a corollary of the Favard (and the Floquet) theory that any bounded solution of an
almost periodic linear diﬀerential system is almost periodic if the matrix valued function,
which determines the system, is periodic (see [72, Corollary 6.5]; for a generalization in
the homogeneous case, see [89]). This result is no longer valid for systems with almost
periodic coeﬃcients. There exist systems for which all solutions are bounded, but none
of them is almost periodic (see [102, 103, 145, 155]). Homogeneous systems have the zero
solution which is almost periodic, but do not need to have other almost periodic solutions.
Note that the existence of a homogeneous system, which has bounded solutions (separated
from zero) and, at the same time, all systems from some neighbourhood of it do not have
non-trivial almost periodic solutions, is proved in [171].
In this chapter, we consider the set of all almost periodic skew-Hermitian diﬀerential
systems with the uniform topology of matrix functions on the real axis. In [170], there is
proved that the systems, whose all solutions are almost periodic, form a dense subset of
the set of all considered systems. We add that special cases of this result are proved in
[113, 114]. Using the method for constructing almost periodic functions from Section 5.3,
we prove that, in any neighbourhood of a skew-Hermitian system, there exists a system
which does not possess an almost periodic solution other than the trivial one (not only
with a fundamental matrix which is not almost periodic as in [172]). Then, we prove the
corresponding result in the real case, i.e., for the skew-symmetric diﬀerential systems.
129
6.1 Preliminaries 130
We use recurrent methods for constructing almost periodic functions. For non-almost
periodic solutions of homogeneous linear diﬀerential equations, we refer to [140] (and [141]),
where a method of constructions of minimal cocycles, which one gets as solutions of recurrent
homogeneous linear diﬀerential systems, is mentioned. Special constructions of
almost periodic homogeneous linear diﬀerential systems with given properties can be found
in [112, 123, 124, 125] as well. A method to construct fundamental matrices for almost
periodic homogeneous linear systems is introduced in [150].
6.1 Preliminaries
Let m ∈ N be arbitrarily given. In this chapter, we use the following notations: Im(ϕ)
for the range of a function ϕ, Mat(C, m) for the set of all m×m matrices with complex elements,
Mat (R, m) for the set of all m×m matrices with real elements, U(m) ⊂ Mat(C, m)
for the group of all unitary matrices of dimension m, SO(m) ⊂ Mat (R, m) for the
group of all orthogonal matrices with determinant 1, so(m) ⊂ Mat (R, m) for the set
of all skew-symmetric (i.e., antisymmetric) matrices, A∗
for the conjugate transpose of
A ∈ Mat(C, m), I for the identity matrix, O for the zero matrix, and symbol i for the
imaginary unit. We remark that the Lie algebra associated to the Lie group SO(m) consists
of the skew-symmetric m × m matrices (i.e., this Lie algebra is so(m) and it is sometimes
called the special orthogonal Lie algebra).
6.2 Skew-Hermitian systems without almost periodic
solutions
We consider systems of m homogeneous linear diﬀerential equations of the form
x (t) = A(t) · x(t), t ∈ R, (6.1)
where A is an almost periodic function with Im(A) ⊂ Mat(C, m) and with the property
that A(t) + A∗
(t) = O for any t ∈ R, i.e., A : R → Mat(C, m) is an almost periodic
function of skew-Hermitian matrices. Let S be the set of all systems (6.1). We identify the
function A with the system (6.1) which is determined by A. Especially, we write A ∈ S
and O ∈ S denotes the system (6.1) given by A(t) = O, t ∈ R.
In the vector space Cm
, we consider the absolute norm || · ||1 (one can also consider
the Euclidean norm or the maximum norm). Let || · || be the corresponding matrix norm.
Considering that any almost periodic function is bounded (see Lemma 5.4), the distance
between two systems A, B ∈ S is deﬁned by the norm of the matrix valued functions A, B,
uniformly on R; i.e., we introduce the metric
σ (A, B) := sup
t∈R
|| A(t) − B(t) || , A, B ∈ S. (6.2)
For ε > 0, symbol Oσ
ε (A) stands for the ε-neighbourhood of A in S.
6.2 Skew-Hermitian systems without almost periodic solutions 131
Now we recall the notion of the frequency module and its rational hull which can be
introduced for all almost periodic function with values in a Banach space. The frequency
module F of an almost periodic function A : R → Mat(C, m) is the Z-module of the real
numbers, generated by the numbers λ such that
lim
T→∞
1
T
T
0
e2πiλt
A(t) dt = O.
The rational hull of F is the set
{λ/l; λ ∈ F, l ∈ Z}.
For the frequency modules of almost periodic linear diﬀerential systems and their solutions,
we refer to [72, Chapters 4 and 6], [145].
In [170], there is proved that, in any neighbourhood of a system (6.1) with frequency
module F, there exists a system with a frequency module contained in the rational hull
of F possessing all almost periodic solutions with frequencies belonging to the rational hull
of F as well. From [174, Theorem 1] it follows that there exists a system (6.1) which cannot
be approximated by the so-called reducible systems with frequency module F (there exists
an open set of irreducible systems with a ﬁxed frequency module; see [173] in the real
case); i.e., a neighbourhood of a system (6.1) with frequency module F may not contain
a system with almost periodic solutions and frequency module F. In this case, see also
[62] and [175] for reducible constant systems and systems reducing to diagonal forms by
a Lyapunov transformation with frequency module F, respectively.
In addition, for all k ∈ N, it is proved in [172] that the systems with k-dimensional
frequency basis of A, having solutions which are not almost periodic, form a subset of the
second category of the space of all considered systems with k-dimensional frequency basis
of A. Thus, it is known (see also [170, Corollary 1]) that the systems with k-dimensional
frequency basis of A and with an almost periodic fundamental matrix form a dense set of
the ﬁrst category in the space of all systems (6.1) with k-dimensional frequency basis.
In this context, we formulate and prove the following result that the systems having no
non-trivial almost periodic solution form a dense subset of S.
Theorem 6.1. For any A ∈ S and ε > 0, there exists B ∈ Oσ
ε (A) which does not have an
almost periodic solution other than the trivial one.
Proof. Let A, C ∈ S and ε > 0 be arbitrary. Since the sum of skew-Hermitian matrices is
also skew-Hermitian and since the sum of two almost periodic functions is almost periodic
(see Corollary 5.9), we have that A + C ∈ S. Let XA(t), t ∈ R, and XC(t), t ∈ R, be the
principal (i.e., XA(0) = XC(0) = I) fundamental matrix of A ∈ S and C ∈ S, respectively.
If matrices C(t), XA(t) commute for all t ∈ R, then the matrix valued function XA(t) XC(t),
t ∈ R, is the principal fundamental matrix of A+C ∈ S. Indeed, from XA(t) = A(t) XA(t),
XC(t) = C(t) XC(t), t ∈ R, we obtain
(XA(t) · XC(t)) = A(t) · XA(t) · XC(t) + XA(t) · C(t) · XC(t)
= A(t) · XA(t) · XC(t) + C(t) · XA(t) · XC(t) = (A + C)(t) · XA(t) · XC(t), t ∈ R.
6.2 Skew-Hermitian systems without almost periodic solutions 132
This fact implies that it suﬃces to ﬁnd C ∈ Oσ
ε (O) for which all matrices C(t),
t ∈ R, have the form diag (ia, . . . , ia), a ∈ R, and for which the vector valued function
XA(t) XC(t) u, t ∈ R, is not almost periodic for any vector u ∈ Cm
, || u ||1 = 1.
We construct such an almost periodic function C using Theorem 5.20 for a = ε/4. First
of all, we put
C(t) ≡ O, t ∈ [0, 1].
Then, in the ﬁrst step of our construction, we deﬁne C on (1, 2] arbitrarily so that it is
constant on [1+1/4, 1+3/4] and || C(t) || < ε/4 for t from this interval, C(2) := C(1) = O,
and it is linear between values O, C(3/2) on [1, 1 + 1/4] and [1 + 3/4, 2].
In the second step, we deﬁne continuous C satisfying || C(t) − C(t + 2) || < ε/4 for
t ∈ [−2, 0) arbitrarily so that it is constant on
[−2 + 1/16, −2 + 1 − 1/16], [−2 + 1 + 1/4 + 1/16, −2 + 1 + 3/4 − 1/16];
at the same time, we put
C(−2) := C(0) = O, C(−1 + 1/4) := C(1 + 1/4) = C(3/2),
C(−1) := C(1) = O, C(−1/4) := C(2 − 1/4) = C(3/2),
and
C(t) ≡ C(3/2)/2, t ∈ [−1 + 1/16, −1 + 1/4 − 1/16] ∪ [−1/4 + 1/16, −1/16],
and we deﬁne C so that it is linear on
[−2, −2 + 1/16], [−1 − 1/16, −1], [−1, −1 + 1/16],
[−1 + 1/4 − 1/16, −1 + 1/4], [−1 + 1/4, −1 + 1/4 + 1/16],
[−1/4 − 1/16, −1/4], [−1/4, −1/4 + 1/16], [−1/16, 0].
Analogously, in the third step, we get C on (2, 6] for which we can choose constant
values on
[4 − 2 + 1/16 + 8−1
/16, 4 − 2 + 1 − 1/16 − 8−1
/16],
[4 − 2 + 1 + 1/4 + 1/16 + 8−1
/16, 4 − 2 + 1 + 3/4 − 1/16 − 8−1
/16],
[4 − 1 + 1/16 + 8−1
/16, 4 − 1 + 1/4 − 1/16 − 8−1
/16],
[4 − 1/4 + 1/16 + 8−1
/16, 4 − 1/16 − 8−1
/16],
[4 + 8−1
/16, 4 + 1 − 8−1
/16], [4 + 1 + 1/4 + 8−1
/16, 4 + 1 + 3/4 − 8−1
/16]
arbitrarily so that || C(t) − C(t − 4) || < ε/8, t ∈ (2, 6]; at the same time, we put
C(4 − 2 + 1/16) := C(−2 + 1/16) = C(−3/2),
C(4 − 2 + 1 − 1/16) := C(−1 − 1/16) = C(−3/2),
C(4 − 1) := C(−1) = O,
C(4 − 1 + 1/16) := C(−1 + 1/16) = C(3/2)/2,
6.2 Skew-Hermitian systems without almost periodic solutions 133
C(4 − 1 + 1/4 − 1/16) := C(−1 + 1/4 − 1/16) = C(3/2)/2,
C(4 − 1 + 1/4) := C(−1 + 1/4) = C(3/2),
C(4 − 2 + 1 + 1/4 + 1/16) := C(−2 + 1 + 1/4 + 1/16) = C(−1/2),
C(4 − 2 + 1 + 3/4 − 1/16) := C(−2 + 1 + 3/4 − 1/16) = C(−1/2),
C(4 − 1/4) := C(−1/4) = C(3/2),
C(4 − 1/4 + 1/16) := C(−1/4 + 1/16) = C(3/2)/2,
C(4 − 1/16) := C(−1/16) = C(3/2)/2,
C(4) := C(0), C(4 + 1) := C(1),
C(4 + 1 + 1/4) := C(1 + 1/4) = C(3/2),
C(4 + 1 + 3/4) := C(1 + 3/4) = C(3/2),
C(4 + 2) := C(2) = C(0) = O,
C(t) ≡ C(−3/2)/2, t ∈ [4 − 2 + 8−1
/16, 4 − 2 + 1/16 − 8−1
/16]
∪ [4 − 1 − 1/16 + 8−1
/16, 4 − 1 − 8−1
/16],
C(t) ≡ C(3/2)/4, t ∈ [4 − 1 + 8−1
/16, 4 − 1 + 1/16 − 8−1
/16]
∪ [4 − 1/16 + 8−1
/16, 4 − 8−1
/16],
C(t) ≡ 3 C(3/2)/4, t ∈ [4 − 1 + 1/4 − 1/16 + 8−1
/16, 4 − 1 + 1/4 − 8−1
/16]
∪ [4 − 1/4 + 8−1
/16, 4 − 1/4 + 1/16 − 8−1
/16],
C(t) ≡ (C(3/2) + C(−1/2))/2, t ∈ [4 − 1 + 1/4 + 8−1
/16, 4 − 1 + 1/4 + 1/16 − 8−1
/16]
∪ [4 − 1/4 − 1/16 + 8−1
/16, 4 − 1/4 − 8−1
/16],
C(t) ≡ (8 C(4 + 1) + 1 C(4 + 1 + 1/4))/9,
t ∈ [4 + 1 + 8−1
/16, 4 + 1 + 8−1
/16 · 3],
C(t) ≡ (7 C(4 + 1) + 2 C(4 + 1 + 1/4))/9,
t ∈ [4 + 1 + 8−1
/16 · 5, 4 + 1 + 8−1
/16 · 7],
...
C(t) ≡ (1 C(4 + 1) + 8 C(4 + 1 + 1/4))/9,
t ∈ [4 + 1 + 8−1
/16 · 29, 4 + 1 + 8−1
/16 · 31],
C(t) ≡ (8 C(4 + 1 + 3/4) + 1 C(4 + 2))/9,
t ∈ [4 + 1 + 3/4 + 8−1
/16, 4 + 1 + 3/4 + 8−1
/16 · 3],
C(t) ≡ (7 C(4 + 1 + 3/4) + 2 C(4 + 2))/9,
t ∈ [4 + 1 + 3/4 + 8−1
/16 · 5, 4 + 1 + 3/4 + 8−1
/16 · 7],
...
C(t) ≡ (1 C(4 + 1 + 3/4) + 8 C(4 + 2))/9,
t ∈ [4 + 1 + 3/4 + 8−1
/16 · 29, 4 + 1 + 3/4 + 8−1
/16 · 31],
6.2 Skew-Hermitian systems without almost periodic solutions 134
and we deﬁne continuous C so that it is linear on the rest of subintervals.
If we denote
a1
1 := 0, b1
1 := 0, c1
1 := 1,
a1
2 := 1, b1
2 := 1 + 1/4, c1
2 := 1 + 3/4, a1
3 := 2
and (compare with the situation after the second step)
a2
1 := −2, b2
1 := −2, c2
1 := −2,
a2
2 := −2, b2
2 := −2 + 1/16, c2
2 := −1 − 1/16,
a2
3 := −1, b2
3 := −1, c2
3 := −1,
a2
4 := −1, b2
4 := −1 + 1/16, c2
4 := −1 + 1/4 − 1/16,
a2
5 := −1 + 1/4, b2
5 := −1 + 1/4 + 1/16, c2
5 := −1 + 3/4 − 1/16,
a2
6 := −1 + 3/4, b2
6 := −1 + 3/4 + 1/16, c2
6 := −1/16,
we see that C does not need to be constant only on
[a1
j − 2, a1
j − 2 + 4−2
], [b1
2 − 2 − 4−2
, b1
2 − 2], [b1
j − 2, b1
j − 2 + 4−2
],
[c1
j − 2 − 4−2
, c1
j − 2], [c1
j − 2, c1
j − 2 + 4−2
], [a1
j+1 − 2 − 4−2
, a1
j+1 − 2]
for j ∈ {1, 2}, i.e., on
[a2
j , b2
j ], j ∈ {1, . . . , 6}, [c2
j , a2
j+1], j ∈ {1, . . . , 5}, [c2
6, 0],
and it has to be constant on each one of the intervals
[a1
2 − 2 + 4−2
, b1
2 − 2 − 4−2
], [c1
2 − 2 + 4−2
, a1
3 − 2 − 4−2
],
[b1
j − 2 + 4−2
, c1
j − 2 − 4−2
], j ∈ {1, 2},
i.e., on
[b2
j , c2
j ], j ∈ {1, . . . , 6}.
It is also seen that
a2
1 = d1
1, b2
1 = d1
2, c2
1 = d1
3, a2
2 = d1
4, · · · c2
6 = d1
18,
where d1
1, d1
2, . . . , d1
18 is the non-decreasing sequence of all numbers
a1
j − 2, b1
j − 2, c1
j − 2,
min{a1
j − 2 + 4−2
, b1
j − 2}, max{a1
j − 2, b1
j − 2 − 4−2
},
min{c1
j − 2, b1
j − 2 + 4−2
}, max{c1
j − 2 − 4−2
, b1
j − 2},
min{c1
j − 2 + 4−2
, a1
j+1 − 2}, max{c1
j − 2, a1
j+1 − 2 − 4−2
}
for j ∈ {1, 2}. We put a2
7 := 0.
6.2 Skew-Hermitian systems without almost periodic solutions 135
Let d2
1, d2
2, . . . , d2
168 be the non-decreasing sequence of all numbers
b1
1 + 4, b1
1 + 4, b1
1 + 4, b1
1 + 4, b1
1 + 4, b1
1 + 4,
b1
1 + 4, b1
1 + 4, b1
1 + 4, b1
1 + 4, b1
1 + 4, b1
1 + 4,
c1
1 + 4, min{c1
1 + 4, b1
1 + 4 + 8−1
/16}, max{c1
1 + 4 − 8−1
/16, b1
1 + 4},
c1
2 + 4, min{c1
2 + 4, b1
2 + 4 + 8−1
/16}, max{c1
2 + 4 − 8−1
/16, b1
2 + 4},
a1
1 + (4k + 1)(b1
1 − a1
1)/32 + 4, a1
1 + (4k + 3)(b1
1 − a1
1)/32 + 4,
a1
1 + (4k + 4)(b1
1 − a1
1)/32 + 4, k ∈ {0, 1, . . . , 7},
c1
1 + (4k + 1)(a1
2 − c1
1)/32 + 4, c1
1 + (4k + 3)(a1
2 − c1
1)/32 + 4,
c1
1 + (4k + 4)(a1
2 − c1
1)/32 + 4, k ∈ {0, 1, . . . , 7},
a1
2 + (4k + 1)(b1
2 − a1
2)/32 + 4, a1
2 + (4k + 3)(b1
2 − a1
2)/32 + 4,
a1
2 + (4k + 4)(b1
2 − a1
2)/32 + 4, k ∈ {0, 1, . . . , 7},
c1
2 + (4k + 1)(a1
3 − c1
2)/32 + 4, c1
2 + (4k + 3)(a1
3 − c1
2)/32 + 4,
c1
2 + (4k + 4)(a1
3 − c1
2)/32 + 4, k ∈ {0, 1, . . . , 7},
and
a2
j+1 + 4, b2
j + 4, c2
j + 4,
min{a2
j + 4 + 8−1
/16, b2
j + 4}, max{a2
j + 4, b2
j + 4 − 8−1
/16},
min{c2
j + 4, b2
j + 4 + 8−1
/16}, max{c2
j + 4 − 8−1
/16, b2
j + 4},
min{c2
j + 4 + 8−1
/16, a2
j+1 + 4}, max{c2
j + 4, a2
j+1 + 4 − 8−1
/16}
for j ∈ {1, . . . , 6}. We denote
a3
1 := 2, b3
1 := d2
1, c3
1 := d2
2, a3
2 := d2
3, · · · a3
57 := d2
168.
We remark that, in the sequences of dl
j, l ∈ N, values are a number of time.
In the fourth step, we deﬁne C so that
C(t) − C(t + 23
) <
ε
23
, t ∈ [−23
− 2, −2).
We consider the non-decreasing sequence d3
1, d3
2, . . . , d3
21·82 of
a3
j − 23
, b3
j − 23
, c3
j − 23
,
min{a3
j − 23
+ 8−2
/16, b3
j − 23
}, max{a3
j − 23
, b3
j − 23
− 8−2
/16},
min{c3
j − 23
, b3
j − 23
+ 8−2
/16}, max{c3
j − 23
− 8−2
/16, b3
j − 23
},
min{c3
j − 23
+ 8−2
/16, a3
j+1 − 23
}, max{c3
j − 23
, a3
j+1 − 23
− 8−2
/16}
for j ∈ {1, . . . , 7 · 8}, 144 numbers b1
1 − 23
, and
c1
1 − 23
, min{c1
1 − 23
, b1
1 − 23
+ 8−2
/16}, max{c1
1 − 23
− 8−2
/16, b1
1 − 23
},
6.2 Skew-Hermitian systems without almost periodic solutions 136
c1
2 − 23
, min{c1
2 − 23
, b1
2 − 23
+ 8−2
/16}, max{c1
2 − 23
− 8−2
/16, b1
2 − 23
},
min{a1
1 + (k − 1)(b1
1 − a1
1)/(8 · 4) − 23
+ 8−2
/16, a1
1 + k(b1
1 − a1
1)/(8 · 4) − 23
},
max{a1
1 + (k − 1)(b1
1 − a1
1)/(8 · 4) − 23
, a1
1 + k(b1
1 − a1
1)/(8 · 4) − 23
− 8−2
/16},
a1
1 + k(b1
1 − a1
1)/(8 · 4) − 23
, k ∈ {1, . . . , 8 · 4},
min{c1
1 + (k − 1)(a1
2 − c1
1)/(8 · 4) − 23
+ 8−2
/16, c1
1 + k(a1
2 − c1
1)/(8 · 4) − 23
},
max{c1
1 + (k − 1)(a1
2 − c1
1)/(8 · 4) − 23
, c1
1 + k(a1
2 − c1
1)/(8 · 4) − 23
− 8−2
/16},
c1
1 + k(a1
2 − c1
1)/(8 · 4) − 23
, k ∈ {1, . . . , 8 · 4},
min{a1
2 + (k − 1)(b1
2 − a1
2)/(8 · 4) − 23
+ 8−2
/16, a1
2 + k(b1
2 − a1
2)/(8 · 4) − 23
},
max{a1
2 + (k − 1)(b1
2 − a1
2)/(8 · 4) − 23
, a1
2 + k(b1
2 − a1
2)/(8 · 4) − 23
− 8−2
/16},
a1
2 + k(b1
2 − a1
2)/(8 · 4) − 23
, k ∈ {1, . . . , 8 · 4},
min{c1
2 + (k − 1)(a1
3 − c1
2)/(8 · 4) − 23
+ 8−2
/16, c1
2 + k(a1
3 − c1
2)/(8 · 4) − 23
},
max{c1
2 + (k − 1)(a1
3 − c1
2)/(8 · 4) − 23
, c1
2 + k(a1
3 − c1
2)/(8 · 4) − 23
− 8−2
/16},
c1
2 + k(a1
3 − c1
2)/(8 · 4) − 23
, k ∈ {1, . . . , 8 · 4},
c2
1 − 23
, min{c2
1 − 23
, b2
1 − 23
+ 8−2
/16}, max{c2
1 − 23
− 8−2
/16, b2
1 − 23
},
c2
2 − 23
, min{c2
2 − 23
, b2
2 − 23
+ 8−2
/16}, max{c2
2 − 23
− 8−2
/16, b2
2 − 23
},
c2
3 − 23
, min{c2
3 − 23
, b2
3 − 23
+ 8−2
/16}, max{c2
3 − 23
− 8−2
/16, b2
3 − 23
},
c2
4 − 23
, min{c2
4 − 23
, b2
4 − 23
+ 8−2
/16}, max{c2
4 − 23
− 8−2
/16, b2
4 − 23
},
c2
5 − 23
, min{c2
5 − 23
, b2
5 − 23
+ 8−2
/16}, max{c2
5 − 23
− 8−2
/16, b2
5 − 23
},
c2
6 − 23
, min{c2
6 − 23
, b2
6 − 23
+ 8−2
/16}, max{c2
6 − 23
− 8−2
/16, b2
6 − 23
},
min{a2
1 + (k − 1)(b2
1 − a2
1)/8 − 23
+ 8−2
/16, a2
1 + k(b2
1 − a2
1)/8 − 23
},
max{a2
1 + (k − 1)(b2
1 − a2
1)/8 − 23
, a2
1 + k(b2
1 − a2
1)/8 − 23
− 8−2
/16},
a2
1 + k(b2
1 − a2
1)/8 − 23
, k ∈ {1, . . . , 8},
min{c2
1 + (k − 1)(a2
2 − c2
1)/8 − 23
+ 8−2
/16, c2
1 + k(a2
2 − c2
1)/8 − 23
},
max{c2
1 + (k − 1)(a2
2 − c2
1)/8 − 23
, c2
1 + k(a2
2 − c2
1)/8 − 23
− 8−2
/16},
c2
1 + k(a2
2 − c2
1)/8 − 23
, k ∈ {1, . . . , 8},
...
min{a2
6 + (k − 1)(b2
6 − a2
6)/8 − 23
+ 8−2
/16, a2
6 + k(b2
6 − a2
6)/8 − 23
},
max{a2
6 + (k − 1)(b2
6 − a2
6)/8 − 23
, a2
6 + k(b2
6 − a2
6)/8 − 23
− 8−2
/16},
a2
6 + k(b2
6 − a2
6)/8 − 23
, k ∈ {1, . . . , 8},
min{c2
6 + (k − 1)(a2
7 − c2
6)/8 − 23
+ 8−2
/16, c2
6 + k(a2
7 − c2
6)/8 − 23
},
max{c2
6 + (k − 1)(a2
7 − c2
6)/8 − 23
, c2
6 + k(a2
7 − c2
6)/8 − 23
− 8−2
/16},
c2
6 + k(a2
7 − c2
6)/8 − 23
, k ∈ {1, . . . , 8}.
6.2 Skew-Hermitian systems without almost periodic solutions 137
We put
a4
1 := d3
1, b4
1 := d3
2, c4
1 := d3
3, · · · c4
7·82 := d3
21·82 , a4
7·82+1 := −2.
We recall that C can be increasing or decreasing only on
[a4
j , b4
j ], [c4
j , a4
j+1], j ∈ {1, . . . , 7 · 82
}.
We proceed further in the same way (as in the third and the fourth step). In the 2n-th
step, we deﬁne continuous C so that
C(t) − C(t + 22n−1
) <
ε
2n+1
, t ∈ [−22n−1
− · · · − 2, −22n−3
− · · · − 2).
We get the non-decreasing sequence {d2n−1
l } from
a2n−1
j − 22n−1
, b2n−1
j − 22n−1
, c2n−1
j − 22n−1
,
min{a2n−1
j −22n−1
+82−2n
/16, b2n−1
j −22n−1
}, max{a2n−1
j −22n−1
, b2n−1
j −22n−1
−82−2n
/16},
min{c2n−1
j −22n−1
, b2n−1
j −22n−1
+82−2n
/16}, max{c2n−1
j −22n−1
−82−2n
/16, b2n−1
j −22n−1
},
min{c2n−1
j −22n−1
+82−2n
/16, a2n−1
j+1 −22n−1
}, max{c2n−1
j −22n−1
, a2n−1
j+1 −22n−1
−82−2n
/16}
for j ∈ {1, . . . , 7 · 82n−3
}, from
c1
1 − 22n−1
, min{c1
1 − 22n−1
, b1
1 − 22n−1
+ 82−2n
/16}, max{c1
1 − 22n−1
− 82−2n
/16, b1
1 − 22n−1
},
c1
2 − 22n−1
, min{c1
2 − 22n−1
, b1
2 − 22n−1
+ 82−2n
/16}, max{c1
2 − 22n−1
− 82−2n
/16, b1
2 − 22n−1
},
min{a1
1 + (k − 1)(b1
1 − a1
1)/(8 · 42n−3
) − 22n−1
+ 82−2n
/16, a1
1 + k(b1
1 − a1
1)/(8 · 42n−3
) − 22n−1
},
max{a1
1 +(k −1)(b1
1 −a1
1)/(8·42n−3
)−22n−1
, a1
1 +k(b1
1 −a1
1)/(8·42n−3
)−22n−1
−82−2n
/16},
a1
1 + k(b1
1 − a1
1)/(8 · 42n−3
) − 22n−1
, k ∈ {1, . . . , 8 · 42n−3
},
min{c1
1 + (k − 1)(a1
2 − c1
1)/(8 · 42n−3
) − 22n−1
+ 82−2n
/16, c1
1 + k(a1
2 − c1
1)/(8 · 42n−3
) − 22n−1
},
max{c1
1 + (k − 1)(a1
2 − c1
1)/(8 · 42n−3
) − 22n−1
, c1
1 + k(a1
2 − c1
1)/(8 · 42n−3
) − 22n−1
− 82−2n
/16},
c1
1 + k(a1
2 − c1
1)/(8 · 42n−3
) − 22n−1
, k ∈ {1, . . . , 8 · 42n−3
},
min{a1
2 + (k − 1)(b1
2 − a1
2)/(8 · 42n−3
) − 22n−1
+ 82−2n
/16, a1
2 + k(b1
2 − a1
2)/(8 · 42n−3
) − 22n−1
},
max{a1
2 +(k −1)(b1
2 −a1
2)/(8·42n−3
)−22n−1
, a1
2 +k(b1
2 −a1
2)/(8·42n−3
)−22n−1
−82−2n
/16},
a1
2 + k(b1
2 − a1
2)/(8 · 42n−3
) − 22n−1
, k ∈ {1, . . . , 8 · 42n−3
},
min{c1
2 + (k − 1)(a1
3 − c1
2)/(8 · 42n−3
) − 22n−1
+ 82−2n
/16, c1
2 + k(a1
3 − c1
2)/(8 · 42n−3
) − 22n−1
},
max{c1
2 + (k − 1)(a1
3 − c1
2)/(8 · 42n−3
) − 22n−1
, c1
2 + k(a1
3 − c1
2)/(8 · 42n−3
) − 22n−1
− 82−2n
/16},
c1
2 + k(a1
3 − c1
2)/(8 · 42n−3
) − 22n−1
, k ∈ {1, . . . , 8 · 42n−3
},
c2
1 − 22n−1
, min{c2
1 − 22n−1
, b2
1 − 22n−1
+ 82−2n
/16}, max{c2
1 − 22n−1
− 82−2n
/16, b2
1 − 22n−1
},
...
6.2 Skew-Hermitian systems without almost periodic solutions 138
c2
6 − 22n−1
, min{c2
6 − 22n−1
, b2
6 − 22n−1
+ 82−2n
/16}, max{c2
6 − 22n−1
− 82−2n
/16, b2
6 − 22n−1
},
min{a2
1 + (k − 1)(b2
1 − a2
1)/(8 · 42n−4
) − 22n−1
+ 82−2n
/16, a2
1 + k(b2
1 − a2
1)/(8 · 42n−4
) − 22n−1
},
max{a2
1 +(k −1)(b2
1 −a2
1)/(8·42n−4
)−22n−1
, a2
1 +k(b2
1 −a2
1)/(8·42n−4
)−22n−1
−82−2n
/16},
a2
1 + k(b2
1 − a2
1)/(8 · 42n−4
) − 22n−1
, k ∈ {1, . . . , 8 · 42n−4
},
min{c2
1 + (k − 1)(a2
2 − c2
1)/(8 · 42n−4
) − 22n−1
+ 82−2n
/16, c2
1 + k(a2
2 − c2
1)/(8 · 42n−4
) − 22n−1
},
max{c2
1 + (k − 1)(a2
2 − c2
1)/(8 · 42n−4
) − 22n−1
, c2
1 + k(a2
2 − c2
1)/(8 · 42n−4
) − 22n−1
− 82−2n
/16},
c2
1 + k(a2
2 − c2
1)/(8 · 42n−4
) − 22n−1
, k ∈ {1, . . . , 8 · 42n−4
},
...
min{a2
6 + (k − 1)(b2
6 − a2
6)/(8 · 42n−4
) − 22n−1
+ 82−2n
/16, a2
6 + k(b2
6 − a2
6)/(8 · 42n−4
) − 22n−1
},
max{a2
6 +(k −1)(b2
6 −a2
6)/(8·42n−4
)−22n−1
, a2
6 +k(b2
6 −a2
6)/(8·42n−4
)−22n−1
−82−2n
/16},
a2
6 + k(b2
6 − a2
6)/(8 · 42n−4
) − 22n−1
, k ∈ {1, . . . , 8 · 42n−4
},
min{c2
6 + (k − 1)(a2
7 − c2
6)/(8 · 42n−4
) − 22n−1
+ 82−2n
/16, c2
6 + k(a2
7 − c2
6)/(8 · 42n−4
) − 22n−1
},
max{c2
6 + (k − 1)(a2
7 − c2
6)/(8 · 42n−4
) − 22n−1
, c2
6 + k(a2
7 − c2
6)/(8 · 42n−4
) − 22n−1
− 82−2n
/16},
c2
6 + k(a2
7 − c2
6)/(8 · 42n−4
) − 22n−1
, k ∈ {1, . . . , 8 · 42n−4
},
...
c2n−2
1 − 22n−1
, min{c2n−2
1 − 22n−1
, b2n−2
1 − 22n−1
+ 82−2n
/16},
max{c2n−2
1 − 22n−1
− 82−2n
/16, b2n−2
1 − 22n−1
},
...
c2n−2
7·82n−4 − 22n−1
, min{c2n−2
7·82n−4 − 22n−1
, b2n−2
7·82n−4 − 22n−1
+ 82−2n
},
max{c2n−2
7·82n−4 − 22n−1
− 82−2n
/16, b2n−2
7·82n−4 − 22n−1
},
min{a2n−2
1 +(k−1)(b2n−2
1 −a2n−2
1 )/8−22n−1
+82−2n
/16, a2n−2
1 +k(b2n−2
1 −a2n−2
1 )/8−22n−1
},
max{a2n−2
1 +(k−1)(b2n−2
1 −a2n−2
1 )/8−22n−1
, a2n−2
1 +k(b2n−2
1 −a2n−2
1 )/8−22n−1
−82−2n
/16},
a2n−2
1 + k(b2n−2
1 − a2n−2
1 )/8 − 22n−1
, k ∈ {1, . . . , 8},
min{c2n−2
1 +(k−1)(a2n−2
2 −c2n−2
1 )/8−22n−1
+82−2n
/16, c2n−2
1 +k(a2n−2
2 −c2n−2
1 )/8−22n−1
},
max{c2n−2
1 +(k−1)(a2n−2
2 −c2n−2
1 )/8−22n−1
, c2n−2
1 +k(a2n−2
2 −c2n−2
1 )/8−22n−1
−82−2n
/16},
c2n−2
1 + k(a2n−2
2 − c2n−2
1 )/8 − 22n−1
, k ∈ {1, . . . , 8},
...
min{a2n−2
7·82n−4 + (k − 1)(b2n−2
7·82n−4 − a2n−2
7·82n−4 )/8 − 22n−1
+ 82−2n
/16,
a2n−2
7·82n−4 + k(b2n−2
7·82n−4 − a2n−2
7·82n−4 )/8 − 22n−1
},
max{a2n−2
7·82n−4 + (k − 1)(b2n−2
7·82n−4 − a2n−2
7·82n−4 )/8 − 22n−1
,
a2n−2
7·82n−4 + k(b2n−2
7·82n−4 − a2n−2
7·82n−4 )/8 − 22n−1
− 82−2n
/16},
6.2 Skew-Hermitian systems without almost periodic solutions 139
a2n−2
7·82n−4 + k(b2n−2
7·82n−4 − a2n−2
7·82n−4 )/8 − 22n−1
, k ∈ {1, . . . , 8},
min{c2n−2
7·82n−4 + (k − 1)(a2n−2
7·82n−4+1 − c2n−2
7·82n−4 )/8 − 22n−1
+ 82−2n
/16,
c2n−2
7·82n−4 + k(a2n−2
7·82n−4+1 − c2n−2
7·82n−4 )/8 − 22n−1
},
max{c2n−2
7·82n−4 + (k − 1)(a2n−2
7·82n−4+1 − c2n−2
7·82n−4 )/8 − 22n−1
,
c2n−2
7·82n−4 + k(a2n−2
7·82n−4+1 − c2n−2
7·82n−4 )/8 − 22n−1
− 82−2n
/16},
c2n−2
7·82n−4 + k(a2n−2
7·82n−4+1 − c2n−2
7·82n−4 )/8 − 22n−1
, k ∈ {1, . . . , 8},
and from a number of b1
1 − 22n−1
such that the total number of d2n−1
l is 21 · 82n−2
. We
denote
a2n
1 := d2n−1
1 , b2n
1 := d2n−1
2 , c2n
1 := d2n−1
3 , · · ·
c2n−1
7·82n−2 := d3
21·82n−2 , a2n−1
7·82n−2+1 := −22n−3
− · · · − 2.
In the (2n + 1)-th step, we deﬁne continuous C so that
C(t) − C(t − 22n
) <
ε
2n+2
, t ∈ (2 + · · · + 22n−2
, 2 + · · · + 22n
].
Now C has constant values on [b2n+1
j , c2n+1
j ], j ∈ {1, . . . , 7 · 82n−1
}, where we put
a2n+1
1 := 2 + 22
+ · · · + 22n−2
and we obtain
b2n+1
1 , c2n+1
1 , a2n+1
2 , · · · c2n+1
7·82n−1 , a2n+1
7·82n−1+1
from the non-decreasing sequence of
a2n
j+1 + 22n
, b2n
j + 22n
, c2n
j + 22n
,
min{a2n
j + 22n
+ 81−2n
/16, b2n
j + 22n
}, max{a2n
j + 22n
, b2n
j + 22n
− 81−2n
/16},
min{c2n
j + 22n
, b2n
j + 22n
+ 81−2n
/16}, max{c2n
j + 22n
− 81−2n
/16, b2n
j + 22n
},
min{c2n
j + 22n
+ 81−2n
/16, a2n
j+1 + 22n
}, max{c2n
j + 22n
, a2n
j+1 + 22n
− 81−2n
/16}
for j ∈ {1, . . . , 7 · 82n−2
} and
c1
1 + 22n
, min{c1
1 + 22n
, b1
1 + 22n
+ 81−2n
/16}, max{c1
1 + 22n
− 81−2n
/16, b1
1 + 22n
},
c1
2 + 22n
, min{c1
2 + 22n
, b1
2 + 22n
+ 81−2n
/16}, max{c1
2 + 22n
− 81−2n
/16, b1
2 + 22n
},
min{a1
1 + (k − 1)(b1
1 − a1
1)/(8 · 42n−2
) + 22n
+ 81−2n
/16, a1
1 + k(b1
1 − a1
1)/(8 · 42n−2
) + 22n
},
max{a1
1 + (k − 1)(b1
1 − a1
1)/(8 · 42n−2
) + 22n
, a1
1 + k(b1
1 − a1
1)/(8 · 42n−2
) + 22n
− 81−2n
/16},
a1
1 + k(b1
1 − a1
1)/(8 · 42n−2
) + 22n
, k ∈ {1, . . . , 8 · 42n−2
},
min{c1
1 + (k − 1)(a1
2 − c1
1)/(8 · 42n−2
) + 22n
+ 81−2n
/16, c1
1 + k(a1
2 − c1
1)/(8 · 42n−2
) + 22n
},
max{c1
1 + (k − 1)(a1
2 − c1
1)/(8 · 42n−2
) + 22n
, c1
1 + k(a1
2 − c1
1)/(8 · 42n−2
) + 22n
− 81−2n
/16},
c1
1 + k(a1
2 − c1
1)/(8 · 42n−2
) + 22n
, k ∈ {1, . . . , 8 · 42n−2
},
6.2 Skew-Hermitian systems without almost periodic solutions 140
min{a1
2 + (k − 1)(b1
2 − a1
2)/(8 · 42n−2
) + 22n
+ 81−2n
/16, a1
2 + k(b1
2 − a1
2)/(8 · 42n−2
) + 22n
},
max{a1
2 + (k − 1)(b1
2 − a1
2)/(8 · 42n−2
) + 22n
, a1
2 + k(b1
2 − a1
2)/(8 · 42n−2
) + 22n
− 81−2n
/16},
a1
2 + k(b1
2 − a1
2)/(8 · 42n−2
) + 22n
, k ∈ {1, . . . , 8 · 42n−2
},
min{c1
2 + (k − 1)(a1
3 − c1
2)/(8 · 42n−2
) + 22n
+ 81−2n
/16, c1
2 + k(a1
3 − c1
2)/(8 · 42n−2
) + 22n
},
max{c1
2 + (k − 1)(a1
3 − c1
2)/(8 · 42n−2
) + 22n
, c1
2 + k(a1
3 − c1
2)/(8 · 42n−2
) + 22n
− 81−2n
/16},
c1
2 + k(a1
3 − c1
2)/(8 · 42n−2
) + 22n
, k ∈ {1, . . . , 8 · 42n−2
},
c2
1 + 22n
, min{c2
1 + 22n
, b2
1 + 22n
+ 81−2n
/16}, max{c2
1 + 22n
− 81−2n
/16, b2
1 + 22n
},
...
c2
6 + 22n
, min{c2
6 + 22n
, b2
6 + 22n
+ 81−2n
/16}, max{c2
6 + 22n
− 81−2n
/16, b2
6 + 22n
},
min{a2
1 + (k − 1)(b2
1 − a2
1)/(8 · 42n−3
) + 22n
+ 81−2n
/16, a2
1 + k(b2
1 − a2
1)/(8 · 42n−3
) + 22n
},
max{a2
1 + (k − 1)(b2
1 − a2
1)/(8 · 42n−3
) + 22n
, a2
1 + k(b2
1 − a2
1)/(8 · 42n−3
) + 22n
− 81−2n
/16},
a2
1 + k(b2
1 − a2
1)/(8 · 42n−3
) + 22n
, k ∈ {1, . . . , 8 · 42n−3
},
min{c2
1 + (k − 1)(a2
2 − c2
1)/(8 · 42n−3
) + 22n
+ 81−2n
/16, c2
1 + k(a2
2 − c2
1)/(8 · 42n−3
) + 22n
},
max{c2
1 + (k − 1)(a2
2 − c2
1)/(8 · 42n−3
) + 22n
, c2
1 + k(a2
2 − c2
1)/(8 · 42n−3
) + 22n
− 81−2n
/16},
c2
1 + k(a2
2 − c2
1)/(8 · 42n−3
) + 22n
, k ∈ {1, . . . , 8 · 42n−3
},
...
min{a2
6 + (k − 1)(b2
6 − a2
6)/(8 · 42n−3
) + 22n
+ 81−2n
/16, a2
6 + k(b2
6 − a2
6)/(8 · 42n−3
) + 22n
},
max{a2
6 + (k − 1)(b2
6 − a2
6)/(8 · 42n−3
) + 22n
, a2
6 + k(b2
6 − a2
6)/(8 · 42n−3
) + 22n
− 81−2n
/16},
a2
6 + k(b2
6 − a2
6)/(8 · 42n−3
) + 22n
, k ∈ {1, . . . , 8 · 42n−3
},
min{c2
6 + (k − 1)(a2
7 − c2
6)/(8 · 42n−3
) + 22n
+ 81−2n
/16, c2
6 + k(a2
7 − c2
6)/(8 · 42n−3
) + 22n
},
max{c2
6 + (k − 1)(a2
7 − c2
6)/(8 · 42n−3
) + 22n
, c2
6 + k(a2
7 − c2
6)/(8 · 42n−3
) + 22n
− 81−2n
/16},
c2
6 + k(a2
7 − c2
6)/(8 · 42n−3
) + 22n
, k ∈ {1, . . . , 8 · 42n−3
},
...
c2n−1
1 + 22n
, min{c2n−1
1 + 22n
, b2n−1
1 + 22n
+ 81−2n
/16},
max{c2n−1
1 + 22n
− 81−2n
/16, b2n−1
1 + 22n
},
...
c2n−1
7·82n−3 + 22n
, min{c2n−1
7·82n−3 + 22n
, b2n−1
7·82n−3 + 22n
+ 81−2n
/16},
max{c2n−1
7·82n−3 + 22n
− 81−2n
/16, b2n−1
7·82n−3 + 22n
},
min{a2n−1
1 + (k − 1)(b2n−1
1 − a2n−1
1 )/8 + 22n
+ 81−2n
/16, a2n−1
1 + k(b2n−1
1 − a2n−1
1 )/8 + 22n
},
max{a2n−1
1 + (k − 1)(b2n−1
1 − a2n−1
1 )/8 + 22n
, a2n−1
1 + k(b2n−1
1 − a2n−1
1 )/8 + 22n
− 81−2n
/16},
a2n−1
1 + k(b2n−1
1 − a2n−1
1 )/8 + 22n
, k ∈ {1, . . . , 8},
6.2 Skew-Hermitian systems without almost periodic solutions 141
min{c2n−1
1 + (k − 1)(a2n−1
2 − c2n−1
1 )/8 + 22n
+ 81−2n
/16, c2n−1
1 + k(a2n−1
2 − c2n−1
1 )/8 + 22n
},
max{c2n−1
1 + (k − 1)(a2n−1
2 − c2n−1
1 )/8 + 22n
, c2n−1
1 + k(a2n−1
2 − c2n−1
1 )/8 + 22n
− 81−2n
/16},
c2n−1
1 + k(a2n−1
2 − c2n−1
1 )/8 + 22n
, k ∈ {1, . . . , 8},
...
min{a2n−1
7·82n−3 + (k − 1)(b2n−1
7·82n−3 − a2n−1
7·82n−3 )/8 + 22n
+ 81−2n
/16,
a2n−1
7·82n−3 + k(b2n−1
7·82n−3 − a2n−1
7·82n−3 )/8 + 22n
},
max{a2n−1
7·82n−3 + (k − 1)(b2n−1
7·82n−3 − a2n−1
7·82n−3 )/8 + 22n
,
a2n−1
7·82n−3 + k(b2n−1
7·82n−3 − a2n−1
7·82n−3 )/8 + 22n
− 81−2n
/16},
a2n−1
7·82n−3 + k(b2n−1
7·82n−3 − a2n−1
7·82n−3 )/8 + 22n
, k ∈ {1, . . . , 8},
min{c2n−1
7·82n−3 + (k − 1)(a2n−1
7·82n−3+1 − c2n−1
7·82n−3 )/8 + 22n
+ 81−2n
/16,
c2n−1
7·82n−3 + k(a2n−1
7·82n−3+1 − c2n−1
7·82n−3 )/8 + 22n
},
max{c2n−1
7·82n−3 + (k − 1)(a2n−1
7·82n−3+1 − c2n−1
7·82n−3 )/8 + 22n
,
c2n−1
7·82n−3 + k(a2n−1
7·82n−3+1 − c2n−1
7·82n−3 )/8 + 22n
− 81−2n
/16},
c2n−1
7·82n−3 + k(a2n−1
7·82n−3+1 − c2n−1
7·82n−3 )/8 + 22n
, k ∈ {1, . . . , 8},
and the corresponding number of b1
1 + 22n
.
Using this construction, we get a continuous function C on R. From Theorem 5.20 it
follows that C is almost periodic. Since
|| C(t) || = 0, t ∈ [0, 1], || C(t) || <
ε
4
, t ∈ (1, 2],
|| C(t) − C(t + 2) || <
ε
4
, t ∈ [−2, 0), || C(t) − C(t − 4) || <
ε
8
, t ∈ (2, 6],
...
C(t) − C(t + 22n−1
) <
ε
2n+1
, t ∈ [−22n−1
− · · · − 23
− 2, −22n−3
− · · · − 23
− 2),
C(t) − C(t − 22n
) <
ε
2n+2
, t ∈ (2 + 22
+ · · · + 22n−2
, 2 + 22
+ · · · + 22n
],
we see that
|| C(t) || <
∞
j=1
2ε
2j+1
= ε, t ∈ R.
We denote
In := [2 + 22
+ · · · + 22n−2
, 2 + 22
+ · · · + 22n
].
We will prove that we can choose constant values of C(t), t ∈ In, on subintervals with the
total length denoted by r2n+1 which is grater than 22n−1
for all n ∈ N. We can choose
values of C on
[4 − 2 + 1/16 + 8−1
/16, 4 − 2 + 1 − 1/16 − 8−1
/16] ⊂ [2, 6],
6.2 Skew-Hermitian systems without almost periodic solutions 142
[4 − 2 + 1 + 1/4 + 1/16 + 8−1
/16, 4 − 2 + 1 + 3/4 − 1/16 − 8−1
/16] ⊂ [2, 6],
[4 + 8−1
/16, 4 + 1 − 8−1
/16], [4 + 1 + 1/4 + 8−1
/16, 4 + 1 + 3/4 − 8−1
/16] ⊂ [2, 6].
Hence,
r3 ≥
55
64
+
23
64
+
63
64
+
31
64
=
43
16
, (6.3)
i.e., the statement is valid for n = 1. We use the induction principle with respect to n.
Assume that the statement is true for 1, 2, . . . , n − 1 and prove it for n. Without loss
of generality (consider the below given process), we can also assume that the estimation
r2j > 22(j−1)
is valid for j ∈ {1, . . . , n} (note that r2 = 5/4 > 20
) if we use the
analogous notation.
In view of the construction, we see that we can choose C on any interval
[s + 22n
+ 81−2n
/16, t + 22n
− 81−2n
/16]
if we can choose C on [s, t], where s = bl
j < cl
j = t, l < 2n + 1. Especially, we can choose
function C on
[22n
+ 81−2n
/16, 1 + 22n
− 81−2n
/16],
[1 + 1/4 + 22n
+ 81−2n
/16, 1 + 3/4 + 22n
− 81−2n
/16],
[−2 + 1/16 + 22n
+ 81−2n
/16, −2 + 1 − 1/16 + 22n
− 81−2n
/16],
[−2 + 1 + 1/4 + 1/16 + 22n
+ 81−2n
/16, −2 + 1 + 3/4 − 1/16 + 22n
− 81−2n
/16]
and on less than 7 · 82n−1
− 4 subintervals of In. Expressing
In = [0 + 22n
, 1 + 22n
] ∪ [1 + 22n
, 2 + 22n
] ∪ [−2 + 22n
, 0 + 22n
] ∪ · · ·
∪ [2 + 22
+ · · · + 22n−4
+ 22n
, 2 + 22
+ · · · + 22n−2
+ 22n
]
∪ [−22n−1
− · · · − 23
− 2 + 22n
, −22n−3
− · · · − 23
− 2 + 22n
]
and using the induction hypothesis, the construction, and (6.3), we obtain that we can
choose C on intervals of the lengths grater than or equal to
1 − 2 · 81−2n
/16, 1/2 − 2 · 81−2n
/16,
1 − 1/8 − 2 · 81−2n
/16, 1/2 − 1/8 − 2 · 81−2n
/16,
43/16 + 22
+ 23
+ · · · + 22n−3
+ 22n−2
− 2 · 81−2n
/16 · 7 · 82n−1
− 4 .
Summing, we get
r2n+1 ≥ 1 +
1
2
+
7
8
+
3
8
+
11
16
+ 22n−1
− 2 −
7
8
> 22n−1
, (6.4)
which is the above statement. Analogously, we can prove
r2n > 22n−2
, n ∈ N. (6.5)
6.2 Skew-Hermitian systems without almost periodic solutions 143
Now we describe the principal fundamental matrix XC on In for arbitrary n ∈ N.
Since C is constant and has the form diag (ia, ia, . . . , ia) for some a ∈ R on each interval
[b2n+1
j , c2n+1
j ], j ∈ {1, . . . , 6 · 42n−1
}, from
XC(t2) − XC(t1) =
t2
t1
C(τ) · XC(τ) dτ, t1, t2 ∈ R,
we obtain
XC(t) − X2n+1
C (t)
≤
k
j=1
b2n+1
j
a2n+1
j
|| C(τ) · XC(τ) || dτ +
a2n+1
j+1
c2n+1
j
|| C(τ) · XC(τ) || dτ
(6.6)
if t ≤ a2n+1
k+1 , t ∈ In, where
X2n+1
C (t) := XC(2 + 22
+ · · · + 22n−2
), t ∈ [2 + 22
+ · · · + 22n−2
, b2n+1
1 ],
X2n+1
C (t) := exp C(b2n+1
1 )(t − b2n+1
1 ) · X2n+1
C (b2n+1
1 ), t ∈ (b2n+1
1 , c2n+1
1 ],
X2n+1
C (t) := XC
2n+1
(c2n+1
1 ), t ∈ (c2n+1
1 , b2n+1
2 ],
...
X2n+1
C (t) := exp C(b2n+1
7·82n−1 )(t − b2n+1
7·82n−1 ) · X2n+1
C (b2n+1
7·82n−1 ), t ∈ (b2n+1
7·82n−1 , c2n+1
7·82n−1 ],
X2n+1
C (t) := X2n+1
C (c2n+1
7·82n−1 ), t ∈ (c2n+1
7·82n−1 , 2 + 22
+ · · · + 22n
].
It is seen that XC is bounded (see also the below given, where it is shown that XC(t) ∈
U(m) for all t) as almost periodic C. Any interval
[2 + · · · + 22n−2
+ l − 1, 2 + · · · + 22n−2
+ l], l ∈ {1, . . . , 22n
}, n ∈ N,
contains at most 42n+1
subintervals, where C can be linear. Indeed, it suﬃces to consider
the construction. We repeat that the length of each one of the considered subintervals is
81−2n
/16 which implies that the total length of them on
Jl
n := [2 + 22
+ · · · + 22n−2
, 2 + 22
+ · · · + 22n−2
+ 22n−l
], l ∈ {1, . . . , n},
is less than 21−l
. Thus (consider also (6.6)), there exists K ∈ R such that
XC(t) − X2n+1
C (t) ≤
K
2l
, t ∈ Jl
n, l ∈ {1, . . . , n}, n ∈ N. (6.7)
From the form diag (ia(t), . . . , ia(t)) of all matrices C(t), we see that
|| C(t) || = | a(t) |, t ∈ R.
For simplicity, let a(t) ≥ 0, t ∈ R. Let an
j ∈ R, j ∈ {1, . . . , n}, be arbitrarily chosen.
Considering the construction and combining (6.4) and (6.5), we get that we can choose
constant values of
C(t), t ∈ [2 + · · · + 22n−2
+ (l − 1) 2n
, 2 + · · · + 22n−2
+ l 2n
],
6.2 Skew-Hermitian systems without almost periodic solutions 144
on subintervals with the total length grater than 2n−2
for each l ∈ {1, . . . , 2n
} and all
suﬃciently large n ∈ N. Since we choose C only so that
C(t) − C(t − 22n
) <
ε
2n+2
, t ∈ In,
we see that we can obtain
X2n+1
C (tn
j ) = diag exp ian
j , . . . , exp ian
j
for arbitrary tn
j such that
tn
1 ≥ 2 + 22
+ 24
+ · · · + 22n−2
+ 3n
− 30
, tn
2 ≥ tn
1 + 3n
− 31
,
· · · tn
n ≥ tn
n−1 + 3n
− 3n−1
, 2 + 22
+ 24
+ · · · + 22n
≥ tn
n,
(6.8)
because we have
4n
> n(3n
− 30
) > 3n
− 30
> · · · > 3n
− 3n−1
> 22n−k+1
for suﬃciently large n ∈ N and some k = k(n) ∈ {1, . . . , n} satisfying
22n−k−2
· ε · 2−n−2
> 2π.
We recall that we need to prove the existence of such C, given by the above construction,
for which the vector valued function XA(t) XC(t) u, t ∈ R, is not almost periodic for any
u ∈ Cm
, || u ||1 = 1. Since
(XA(t) · X∗
A(t)) = A(t) · XA(t) · X∗
A(t) − XA(t) · X∗
A(t) · A(t), t ∈ R,
and since the constant function given by I is a solution of X = A X − X A, X(0) = I,
we have XA(t) ∈ U(m) for all t. Thus, XC(t) ∈ U(m), t ∈ R, as well. We add that
XA(t) X∗
A(t) = I, t ∈ R, implies A∗
(t) + A(t) = O, t ∈ R.
Let c ∈ C, | c | = 1, and N ∈ U(m) be arbitrarily given. Obviously, for any M ∈ U(m),
we can choose a number a(M, c) ∈ [0, 2π) so that all eigenvalues of matrix
P := M · diag (exp (ia(M, c)) , . . . , exp (ia(M, c)))
are not in the neighbourhood of c with a given radius which depends only on dimension m.
Indeed, if M has eigenvalues λ1, . . . , λm, then the eigenvalues of P are
λ1 exp (ia(M, c)) , . . . , λm exp (ia(M, c)) .
Considering P u − N u and expressing vectors u ∈ Cm
, || u ||1 = 1, as linear combinations
of the eigenvectors of P, we see that P u cannot be in a neighbourhood of N u for some
c ∈ C, | c | = 1. Thus (the considered multiplication of matrices and vectors is uniformly
continuous), there exist ϑ > 0 and ξ > 0 such that, for any matrices M, N ∈ U(m), one
can ﬁnd a(M, N) ∈ (0, 2π) satisfying
|| M · diag (exp (i˜a) , . . . , exp (i˜a)) · u − N · u ||1 > ϑ (6.9)
6.2 Skew-Hermitian systems without almost periodic solutions 145
for u ∈ Cm
, || u ||1 = 1, ˜a ∈ (a(M, N) − ξ, a(M, N) + ξ).
We can construct C so that we obtain
X2n+1
C (tn
j ) = diag exp ian
j , . . . , exp ian
j
for arbitrarily given an
j ∈ [0, 2π) and any tn
j satisfying (6.8) if n ∈ N is suﬃciently large
and j ∈ {1, . . . , n}. Especially, for suﬃciently large n ∈ N and for
tn
1 := 2 + 22
+ 24
+ · · · + 22n−2
+ 3n
− 30
,
tn
2 := tn
1 + 3n
− 31
, · · · tn
n := tn
n−1 + 3n
− 3n−1
,
(6.10)
we can choose all X2n+1
C (tn
j ) in the form without any conditions. Hence, we obtain diagonal
matrices X2n+1
C (tn
j ), j ∈ {1, . . . , n}, determined by numbers
exp ia XA(tn
j ), XA(tn
j − 3n
+ 3j−1
) · XC(tn
j − 3n
+ 3j−1
)
on their diagonals.
It is seen from (6.10) that each
tn
j ∈ [2 + 22
+ · · · + 22n−2
, 2 + 22
+ · · · + 22n−2
+ n3n
].
Thus (see (6.7)), for any η > 0, we have
XC(tn
j ) − X2n+1
C (tn
j ) < η (6.11)
for suﬃciently large n = n(η) ∈ N and j ∈ {1, . . . , n}. From (6.9) and (6.11) it follows
that
XA(tn
j ) · XC(tn
j ) · u − XA(tn
j − 3n
+ 3j−1
) · XC(tn
j − 3n
+ 3j−1
) · u 1
> ϑ (6.12)
for any u ∈ Cm
, || u ||1 = 1, suﬃciently large n ∈ N, and j ∈ {1, . . . , n}.
By contradiction, suppose that there exists u ∈ Cm
, || u ||1 = 1, with the property that
XA(t) XC(t) u, t ∈ R, is almost periodic. Applying Theorem 5.5 for
ψ(t) = XA(t) · XC(t) · u, t ∈ R, sn = 3n
, n ∈ N, ε = ϑ,
we obtain
|| XA(t + 3n1
) · XC(t + 3n1
) · u − XA(t + 3n2
) · XC(t + 3n2
) · u ||1 < ϑ, t ∈ R, (6.13)
for all n1, n2 from an inﬁnite subset of N. If we rewrite (6.13) into
|| XA(t) · XC(t) · u − XA(t + 3n2
− 3n1
) · XC(t + 3n2
− 3n1
) · u ||1 < ϑ, t ∈ R,
then it is easy to see that (6.12) is not valid for inﬁnitely many n ∈ N. This contradiction
proves the theorem.
6.3 Skew-symmetric systems without almost periodic solutions 146
6.3 Skew-symmetric systems without almost periodic
solutions
Let m = 1. Let us consider systems of m homogeneous linear diﬀerential equations of
the form
x (t) = A(t) · x(t), t ∈ R, (6.14)
where A : R → so(m) is an almost periodic function. Let S denote the set of all systems
(6.14). We can identify the function A with the system (6.14) which is determined
by A. Especially, we write A ∈ S. Let XS = XS (t) denote the principal fundamental
matrix of S ∈ S satisfying XS (0) = I.
In the vector space Rm
, we use the Euclidean norm · 2 (one can also replace it by
the absolute norm or the maximum norm). Let · be the corresponding matrix norm in
Mat (R, m) and let be the metric given by · . Using the boundedness of every almost
periodic function, the distance between two systems A, B ∈ S is deﬁned uniformly on R
by the norm of the matrix valued functions A, B; i.e., we introduce the metric σ by (6.2).
For ε > 0, symbol Oσ
ε (A) denotes the ε-neighbourhood of a system A in S and Oε (M) the
ε-neighbourhood of a matrix M in a given subset of Mat (R, m).
The importance of skew-symmetric systems may be illustrated by the Cameron-Johnson
theorem which states that any almost periodic homogeneous linear diﬀerential system can
be reduced by a Lyapunov transformation to a skew-symmetric system if all solutions
of the given system and all of its limit equations are bounded (see [40]). Further, it
is known (see [170]) that the skew-symmetric systems, all of whose solutions are almost
periodic, form a dense subset in the space of all skew-symmetric systems (special cases are
considered in [113, 114] and the corresponding result about unitary diﬀerence systems is
mentioned in [176]). This fact also motivates the study of skew-symmetric systems without
almost periodic solutions. Concerning basic results about skew-symmetric systems and
their fundamental matrices, we refer to [33, 71, 137].
Now we repeat the basic motivation in an explicit form.
Theorem 6.2. For any A ∈ S and ε > 0, there exists B ∈ Oσ
ε (A) whose all solutions are
almost periodic.
Proof. See [170, Theorem 1, Remark 3].
To prove the announced result, we need the following lemmas.
Lemma 6.3. There exist ξ > 0 and a neighbourhood ˜O (O) of the zero matrix in so(m)
for which the exponential map is a bijection between ˜O (O) and Oξ (I) ∩ SO(m) such that
the maps
A → exp (A) , A ∈ ˜O (O) ; A → ln (A) , A ∈ Oξ (I) ∩ SO(m), (6.15)
are continuous in the Lipschitz sense.
Proof. It is well-known that the exponential map is a bijection between ˜O (O) and Oξ (I)∩
SO(m) for a suﬃciently small ξ > 0 and the corresponding neighbourhood ˜O (O) ⊂ so(m).
6.3 Skew-symmetric systems without almost periodic solutions 147
The fact that the maps in (6.15) are continuous in the Lipschitz sense follows from the
inequality
exp (X + Y ) − exp (X) ≤ Y · exp ( X ) · exp ( Y ) , X, Y ∈ so(m),
and, e.g., from the Richter theorem (see, e.g., [92, Theorem 11.1])
ln (X) =
1
0
(X − I) [t (X − I) + I]−1
dt, X ∈ Oξ (I) ∩ SO(m).
Remark 6.4. Any non-singular matrix has inﬁnitely many logarithms. Here symbol ln (A)
denotes the principal logarithm, which is the unique logarithm whose spectrum lies in the
strip {z ∈ C; Im z ∈ [−π, π)}.
Lemma 6.5. There exists p(ϑ) ∈ N for all ϑ > 0 with the property that, for any sequence
{P0, P1, . . . , Pn, . . . , P2n} ⊂ SO(m), n ≥ p(ϑ),
one can ﬁnd matrices Q2, Q4, . . . , Q2n ∈ SO(m) for which
Q2i ∈ Oϑ (P2i) , i ∈ {1, . . . , n}, P1 · Q2 · P3 · Q4 · · · P2n−1 · Q2n = P0. (6.16)
Proof. First we recall that the group SO(m) is transformable (see Example 2.3). This fact
implies the existence of q(δ) ∈ N for all δ > 0 such that, for any sequence
{P0, P1, . . . , Pq, . . . , Pn} ⊂ SO(m),
there exist T1, . . . , Tq, . . . , Tn ∈ SO(m) satisfying
Ti ∈ Oδ (Pi) , i ∈ {1, . . . , n}, T1 · T2 · · · Tn = P0.
We replace matrices P1, . . . , Pn−1, Pn by P1 ·P2, . . . , P2n−3 ·P2n−2, P2n−1 ·P2n and, using the
transformability of SO(m), we obtain matrices Ti, i ∈ {1, . . . , n}. We put
R1 := (P1 · P2)−1
· T1, . . . , Rn := (P2n−1 · P2n)−1
· Tn.
Since the multiplication of matrices is continuous in the Lipschitz sense on SO(m) as the
map O → OT
, there exists L > 0 such that
Ri ∈ OδL (I) , i ∈ {1, . . . , n},
and, consequently, there exists K > 0 for which
P2 · R1 ∈ OδK (P2) , . . . , P2n · Rn ∈ OδK (P2n) .
We see
T1 = P1 · P2 · R1, . . . , Tn = P2n−1 · P2n · Rn,
i.e., we have (6.16) for Q2 := P2 · R1, . . . , Q2n := P2n · Rn and p(ϑ) := q(ϑ/K).
6.3 Skew-symmetric systems without almost periodic solutions 148
We also need a simple method for constructing almost periodic functions with prescribed
values. This method is a modiﬁcation of the theorems presented in Section 5.3.
Lemma 6.6. If the sequence of non-negative numbers a(i) for i ∈ N has the property that
∞
i=1
a(i) < ∞,
then any continuous function ψ : R → so(m) for which
ψ(t) = ψ (t − 1) , t ∈ (1, 2],
ψ(t) = ψ (t + 2) , t ∈ (−2, 0],
ψ(t) ∈ Oa(1) (ψ (t − 4)) , t ∈ (2, 6],
ψ(t) = ψ (t + 8) , t ∈ (−10, −2] ,
ψ(t) ∈ Oa(2) ψ t − 24
, t ∈ (2 + 22
, 2 + 22
+ 24
],
ψ(t) = ψ t + 25
, t ∈ (−25
− 23
− 2, −23
− 2],
...
ψ(t) ∈ Oa(n) ψ t − 22n
, t ∈ (2 + 22
+ · · · + 22n−2
, 2 + 22
+ · · · + 22n−2
+ 22n
],
ψ(t) = ψ t + 22n+1
, t ∈ (−22n+1
− · · · − 23
− 2, −22n−1
− · · · − 23
− 2],
...
is almost periodic.
Proof. Let ε > 0 be arbitrarily given and let k = k(ε) ∈ N satisfy
∞
i=k
a(i) <
ε
2
. (6.17)
From
ψ(t) ∈ Oa(k) ψ t − 22k
, t ∈ (2 + 22
+ · · · + 2k−2
, 2 + 22
+ · · · + 2k
],
ψ(t) = ψ t + 22k+1
, t ∈ (−22k+1
− · · · − 23
− 2, −22k−1
− · · · − 23
− 2],
ψ(t) ∈ Oa(k+1) ψ t − 22k+2
, t ∈ (2 + 22
+ · · · + 22k
, 2 + 22
+ · · · + 22k+2
],
...
it follows
ψ t + 22k
∈ Oa(k)(ψ(t)), t ∈ (−22k−1
− · · · − 23
− 2, 2 + 22
+ · · · + 22k−2
],
ψ t − 22k
∈ Oa(k)(ψ(t)), t ∈ (−22k−1
− · · · − 23
− 2, 2 + 22
+ · · · + 22k−2
],
6.3 Skew-symmetric systems without almost periodic solutions 149
ψ t − 22k+1
∈ Oa(k)(ψ(t)), t ∈ (−22k−1
− · · · − 23
− 2, 2 + 22
+ · · · + 22k−2
],
ψ t + 22k+1
∈ Oa(k)+a(k+1)(ψ(t)), t ∈ (−22k−1
− · · · − 23
− 2, 2 + 22
+ · · · + 22k−2
],
ψ t + 3 · 22k
∈ Oa(k)+a(k+1)(ψ(t)), t ∈ (−22k−1
− · · · − 23
− 2, 2 + 22
+ · · · + 22k−2
],
ψ t + 22k+2
∈ Oa(k+1)(ψ(t)), t ∈ (−22k−1
− · · · − 23
− 2, 2 + 22
+ · · · + 22k−2
],
ψ t + 22k
+ 22k+2
∈ Oa(k)+a(k+1)(ψ(t)), t ∈ −22k−1
− · · · − 23
− 2, 2 + 22
+ · · · + 22k−2
],
...
Thus (see (6.17)), it holds
ψ t + l · 22k
∈ Oε/2(ψ(t)), t ∈ −22k−1
− · · · − 23
− 2, 2 + 22
+ · · · + 22k−2
, l ∈ Z.
If we express any t ∈ R as t = t1 + t2, where
t1 ∈ −22k−1
− · · · − 23
− 2, 2 + 22
+ · · · + 22k−2
, t2 = j · 22k
for j ∈ Z,
then we have
ψ(t), ψ t + l · 22k
≤ (ψ (t1 + t2) , ψ (t1))
+ ψ (t1) , ψ t1 + (j + l) 22k
<
ε
2
+
ε
2
= ε, t ∈ R, l ∈ Z.
This inequality implies that we can choose l(ε) := 22k(ε)
+1 for any ε > 0 (see Deﬁnition 5.1);
i.e., the resulting function ψ is almost periodic.
Now we can prove the result that the systems having no non-zero almost periodic
solution form an everywhere dense subset of S.
Theorem 6.7. Let A ∈ S and ε > 0 be arbitrary. There exists B ∈ Oσ
ε (A) which does not
have an almost periodic solution other than the trivial one.
Proof. Using Lemma 5.3, the almost periodicity of A implies that there exist δ ∈ (0, 1/3)
and an almost periodic matrix valued function ˜A : R → so(m) satisfying ˜A ∈ Oσ
ε/2(A) and
˜A|[k,k+δ] ≡ const. for any k ∈ Z. Indeed, it suﬃces to deﬁne ˜A by
˜A(t) := A k +
δ
2
, t ∈ [k, k + δ], k ∈ Z,
˜A(t) := A(k − δ) +
t − (k − δ)
δ
A k +
δ
2
− A(k − δ) , t ∈ [k − δ, k), k ∈ Z,
˜A(t) := A k +
δ
2
+
t − (k + δ)
δ
A(k + 2δ) − A k +
δ
2
, t ∈ (k + δ, k + 2δ], k ∈ Z,
˜A(t) := A(t), t /∈
k∈Z
[k − δ, k + 2δ],
where δ > 0 is suﬃciently small. Thus, we assume without loss of generality that A ∈ S is
constant on all interval [k, k + δ], k ∈ Z.
6.3 Skew-symmetric systems without almost periodic solutions 150
Every almost periodic function is bounded. Hence, there exists η ∈ (0, 1) with the
property that
XS (t + s) − XS (t) < ξ (6.18)
for any t ∈ R, s ∈ [0, η], and S ∈ Oσ
ε (A), where ξ > 0 is taken from Lemma 6.3. We can
also assume that δ < η. Further (see again Lemma 6.3), there exists M ∈ N satisfying
A − B < ϑ if A, B ∈ ˜O (O) , exp (A) ∈ OϑM (exp (B)) ⊆ Oξ (I) ∩ SO(m). (6.19)
We choose an increasing sequence of numbers n(i) ∈ N {1} for i ∈ N arbitrarily so that
2n(i)−1
≥ p
ε
2iM
·
δ
2
, i ∈ N, (6.20)
where p(ϑ) is taken from Lemma 6.5.
Since the sum of skew-symmetric matrices is a skew-symmetric matrix and since the
sum of two almost periodic functions is almost periodic as well (see Corollary 5.9), we have
A1 + A2 ∈ S for any A1, A2 ∈ S. Thus, it suﬃces to ﬁnd C ∈ S ∩ Oσ
ε (O) for which the
system A + C does not have any non-zero almost periodic solution. We construct such
a system C (as continuous function) applying Lemma 6.6 for
a (n(i)) :=
ε
2i
, i ∈ N; a (j) := 0, j /∈ {n(i); i ∈ N}.
Let us denote
ai := 2 + 22
+ · · · + 22n(i)−2
, bi := 2 + 22
+ · · · + 22n(i)−2
+ 22n(i)
,
d1
i :=
1
4
−
1
22n(i)
δ, d2
i :=
3
4
+
1
22n(i)
δ, i ∈ N.
In the ﬁrst step of the construction, we put
C(t) := O, t ∈ −22n(1)−1
− · · · − 23
− 2, 2 + 22
+ · · · + 22n(1)−2
,
C(t) := O, t ∈ (a1, b1]
j∈N
j + d1
1, j + d2
1 ,
C(t) := Cj−a1+1
1 , t ∈ j + d1
2, j + d2
2 ⊂ (a1, b1] ,
for arbitrary matrices
Cj−a1+1
1 ∈ Oε/2 (O) ∩ so(m), j ∈ {a1, . . . , b1 − 1},
and we deﬁne C so that it is linear on the intervals
j + d1
1, j + d1
2 , j + d2
2, j + d2
1 , j ∈ {a1, . . . , b1 − 1}.
In the second step, we put
C(t) := C t + 22n(1)+1
, t ∈ −22n(1)+1
− · · · − 23
− 2, −22n(1)−1
− · · · − 23
− 2 ,
6.3 Skew-symmetric systems without almost periodic solutions 151
C(t) := C t − 22n(1)+2
, t ∈ 2 + 22
+ · · · + 22n(1)
, 2 + 22
+ · · · + 22n(1)+2
,
...
C(t) := C t + 22n(2)−1
, t ∈ −22n(2)−1
− · · · − 23
− 2, −22n(2)−3
− · · · − 23
− 2 ,
C(t) := C t − 22n(2)
, t ∈ (a2, b2]
j∈N
j + d1
2, j + d2
2 ,
and we deﬁne C as linear on the intervals
j + d1
2, j + d1
3 , j + d2
3, j + d2
2 , j ∈ {a2, . . . , b2 − 1}.
At the same time, we deﬁne
C(t) := Cj−a2+1
2 ∈ so(m), t ∈ j + d1
3, j + d2
3 , j ∈ {a2, . . . , b2 − 1},
arbitrarily so that
C(t) − C t − 22n(2)
<
ε
4
, t ∈ (a2, b2] .
We proceed further in the same way. In the i-th step, we put
C(t) := C t + 22n(i−1)+1
, t ∈ −22n(i−1)+1
− · · · − 23
− 2, −22n(i−1)−1
− · · · − 23
− 2 ,
C(t) := C t − 22n(i−1)+2
, t ∈ 2 + 22
+ · · · + 22n(i−1)
, 2 + 22
+ · · · + 22n(i−1)+2
,
...
C(t) := C t + 22n(i)−1
, t ∈ −22n(i)−1
− · · · − 23
− 2, −22n(i)−3
− · · · − 23
− 2 ,
C(t) := C t − 22n(i)
, t ∈ (ai, bi]
j∈N
j + d1
i , j + d2
i , (6.21)
and we deﬁne C as a linear function on the intervals
j + d1
i , j + d1
i+1 , j + d2
i+1, j + d2
i , j ∈ {ai, . . . , bi − 1},
and
C(t) := Cj−ai+1
i ∈ so(m), t ∈ j + d1
i+1, j + d2
i+1 , j ∈ {ai, . . . , bi − 1},
arbitrarily so that
C(t) − C t − 22n(i)
<
ε
2i
, t ∈ (ai, bi] .
For
ζ := max Cj
1 ; j ∈ {1, . . . , 22n(1)
} <
ε
2
,
we have
C(t) ≤ ζ, t ∈ −22n(1)−1
− · · · − 23
− 2, 2 + 22
+ · · · + 22n(1)
,
C(t) < ζ +
ε
4
, t ∈ −22n(2)−1
− · · · − 23
− 2, 2 + 22
+ · · · + 22n(2)
,
6.3 Skew-symmetric systems without almost periodic solutions 152
...
C(t) < ζ +
ε
4
+ · · · +
ε
2i
, t ∈ −22n(i)−1
− · · · − 23
− 2, 2 + 22
+ · · · + 22n(i)
,
...
i.e., there exists ˜ε ∈ (0, ε) with the property that C(t) < ˜ε, t ∈ R. Thus, we obtain an
almost periodic (continuous) function C ∈ S ∩ Oσ
ε (O).
We denote
Ii := [ai, bi] = 2 + 22
+ · · · + 22n(i)−2
, 2 + 22
+ · · · + 22n(i)−2
+ 22n(i)
.
In the construction, we can choose constant values C1
i , . . . , C22n(i)
i on 22n(i)
subintervals
of Ii, where the length of each one of these intervals is
d2
i+1 − d1
i+1 ∈
δ
2
, δ . (6.22)
Each value Cj
i can be chosen arbitrarily from the (ε/2i
)-neighbourhood of a skew-symmetric
matrix, which is given by the previous steps of the construction. Further (see (6.21)), the
function C is determined on intervals
ai, ai + d1
i , ai + d2
i , ai + 1 + d1
i , . . . bi − 2 + d2
i , bi − 1 + d1
i , bi − 1 + d2
i , bi
by C(t) = C t − 22n(i)
.
We repeat that C is linear on the remaining subintervals of Ii. These intervals are
denoted by J1
i , . . . , J22n(i)+1
i , where
J2j−1
i := ai + j − 1 + d1
i , ai + j − 1 + d1
i+1 , j ∈ {1, . . . , 22n(i)
},
J2j
i := ai + j − 1 + d2
i+1, ai + j − 1 + d2
i , j ∈ {1, . . . , 22n(i)
}.
(6.23)
Especially, we see that the length of each Jj
i is less than δ/22n(i)
and that
J1
i , . . . , J2j
i ⊂ (ai, ai + j) , J2j+1
i , . . . , J22n(i)+1
i ⊂ (ai + j, bi) , j ∈ {1, . . . , 22n(i)
− 1},
i.e., the total length lk
i of all subintervals Jj
i ⊂ [ai, ai + k] is
lk
i <
2kδ
22n(i)
, k ∈ {1, . . . , 22n(i)
}. (6.24)
Let us consider S = A+C ∈ Oσ
ε (A). To describe the principal fundamental matrix XS,
we deﬁne
˜Xi
S(t) := XS(t), t ∈ ai, ai + d1
i ,
˜Xi
S(t) := ˜Xi
S ai + d1
i , t ∈ ai + d1
i , ai + d1
i+1 ,
˜Xi
S(t) := exp A + C1
i t − ai − d1
i+1 · ˜Xi
S ai + d1
i+1 , t ∈ ai + d1
i+1, ai + d2
i+1 ,
˜Xi
S(t) := ˜Xi
S ai + d2
i+1 , t ∈ ai + d2
i+1, ai + d2
i ,
6.3 Skew-symmetric systems without almost periodic solutions 153
˜Xi
S(t) := XS (t) · XS ai + d2
i
−1
· ˜Xi
S ai + d2
i , t ∈ ai + d2
i , ai + 1 + d1
i ,
...
˜Xi
S(t) := XS (t) · XS bi − 2 + d2
i
−1
· ˜Xi
S bi − 2 + d2
i ,
t ∈ bi − 2 + d2
i , bi − 1 + d1
i ,
˜Xi
S(t) := ˜Xi
S bi − 1 + d1
i , t ∈ bi − 1 + d1
i , bi − 1 + d1
i+1 ,
˜Xi
S(t) := exp A + C22n(i)
i t − bi + 1 − d1
i+1 · ˜Xi
S bi − 1 + d1
i+1 ,
t ∈ bi − 1 + d1
i+1, bi − 1 + d2
i+1 ,
˜Xi
S(t) := ˜Xi
S bi − 1 + d2
i+1 , t ∈ bi − 1 + d2
i+1, bi − 1 + d2
i ,
˜Xi
S(t) := XS (t) · XS bi − 1 + d2
i
−1
· ˜Xi
S bi − 1 + d2
i , t ∈ bi − 1 + d2
i , bi .
Since
XS (t2) − XS (t1) =
t2
t1
S(s) XS (s) ds, t1, t2 ∈ R,
it is valid that (see also (6.23))
XS (t) − ˜Xi
S (t) ≤
k
j=1
ai+j−1+d1
i+1
ai+j−1+d1
i
S(s) XS (s) ds
+
k
j=1
ai+j−1+d2
i
ai+j−1+d2
i+1
S(s) XS (s) ds
(6.25)
if t ≤ ai + k, k ∈ {1, . . . , 22n(i)
}. Considering S ∈ Oσ
ε (A) and XS (t) , ˜Xi
S (t) ∈ SO(m),
t ∈ R, from (6.24) and (6.25) it follows that there exists N ∈ N satisfying
XS (t) − ˜Xi
S (t) <
Nk
22n(i)−1
(6.26)
for t ∈ [ai, ai + k], k ∈ {1, . . . , 22n(i)
}.
Let n0 ∈ N be such that
N ·
4n(i) 2n(i)
22n(i)−1
<
1
3
, i ≥ n0 (i ∈ N). (6.27)
We put X1 := −I, X2 = −I, when m is even, and
X1 :=







1 0 0 · · · 0
0 −1 0 · · · 0
0 0 −1 · · · 0
...
...
...
...
...
0 0 0 · · · −1







∈ SO(m), X2 :=







−1 · · · 0 0 0
...
...
...
...
...
0 · · · −1 0 0
0 · · · 0 −1 0
0 · · · 0 0 1







∈ SO(m)
6.3 Skew-symmetric systems without almost periodic solutions 154
for odd m. If we express
˜Xi
S ai + d2
i+1 = exp A + C1
i d2
i+1 − d1
i+1 · ˜Xi
S ai + d1
i+1 ,
˜Xi
S ai + 1 + d1
i+1 = XS ai + 1 + d1
i · XS ai + d2
i
−1
· ˜Xi
S ai + d2
i+1 ,
...
˜Xi
S bi − 1 + d1
i+1 = XS bi − 1 + d1
i · XS bi − 2 + d2
i
−1
· ˜Xi
S bi − 2 + d2
i+1 ,
˜Xi
S bi − 1 + d2
i+1 = exp A + C22n(i)
i d2
i+1 − d1
i+1 · ˜Xi
S bi − 1 + d1
i+1 ,
˜Xi
S (bi) = XS (bi) · XS bi − 1 + d2
i
−1
· ˜Xi
S bi − 1 + d2
i+1 ,
then it is seen that we can use Lemma 6.5 to choose values Cj
i on the subintervals
ai + j − 1 + d1
i+1, ai + j − 1 + d2
i+1 , j ∈ {1, . . . , 22n(i)
},
so that we obtain
˜Xi
S ai + 2n(i)
= I, ˜Xi
S ai + 2n(i)
+ 2n(i)
− 1 = X1,
˜Xi
S ai + 3 · 2n(i)
= I, ˜Xi
S ai + 3 · 2n(i)
+ 2n(i)
− 1 = X2,
˜Xi
S ai + 4 · 2n(i)
+ 2n(i)
= I, ˜Xi
S ai + 4 · 2n(i)
+ 2n(i)
+ 2n(i)
− 21
= X1,
˜Xi
S ai + 4 · 2n(i)
+ 3 · 2n(i)
= I, ˜Xi
S ai + 4 · 2n(i)
+ 3 · 2n(i)
+ 2n(i)
− 21
= X2,
...
˜Xi
S ai + 4 (n(i) − 1) 2n(i)
+ 2n(i)
= I,
˜Xi
S ai + 4 (n(i) − 1) 2n(i)
+ 2n(i)
+ 2n(i)
− 2n(i)−1
= X1,
˜Xi
S ai + 4 (n(i) − 1) 2n(i)
+ 3 · 2n(i)
= I,
˜Xi
S ai + 4 (n(i) − 1) 2n(i)
+ 3 · 2n(i)
+ 2n(i)
− 2n(i)−1
= X2.
Indeed, it suﬃces to consider the form of matrices
exp A + Cj
i d2
i+1 − d1
i+1
for which (see (6.18), (6.22))
exp A + Cj
i d2
i+1 − d1
i+1 − I < ξ,
inequality (6.20) with M ∈ N satisfying (6.19) and with
d2
i+1 − d1
i+1 >
δ
2
, 2n(i)
− 1 > 2n(i)
− 21
> · · · > 2n(i)
− 2n(i)−1
= 2n(i)−1
,
and the fact that we can choose all matrix Cj
i from the (ε/2i
)-neighbourhood of a given
skew-symmetric matrix arbitrarily. Note that
ai + 4 (n(i) − 1) 2n(i)
+ 3 · 2n(i)
+ 2n(i)
− 2n(i)−1
< ai + 4n(i) · 2n(i)
(6.28)
6.3 Skew-symmetric systems without almost periodic solutions 155
and ai + 4n(i)2n(i)
< bi for suﬃciently large i ∈ N, i.e., we can construct the resulting
function C with the above mentioned properties on Ii for all i ≥ n0 (see also (6.27)).
Now we use (6.26) and (6.27) in connection with (6.28). For k ∈ {1, . . . , 4n(i)2n(i)
},
where i ≥ n0, we have
XS (t) − ˜Xi
S (t) < N ·
4n(i) 2n(i)
22n(i)−1
<
1
3
, t ∈ [ai, ai + k]. (6.29)
Especially, for all i ≥ n0 (i ∈ N), we obtain
XS si
j − ˜Xi
S si
j <
1
3
, j ∈ {1, . . . , 4n(i)}, (6.30)
where
si
1 := ai + 2n(i)
, si
2 := ai + 2n(i)
+ 2n(i)
− 1 ,
si
3 := ai + 3 · 2n(i)
, si
4 := ai + 3 · 2n(i)
+ 2n(i)
− 1 ,
...
si
4n(i)−3 := ai + 4 (n(i) − 1) 2n(i)
+ 2n(i)
,
si
4n(i)−2 := ai + 4 (n(i) − 1) 2n(i)
+ 2n(i)
+ 2n(i)
− 2n(i)−1
,
si
4n(i)−1 := ai + 4 (n(i) − 1) 2n(i)
+ 3 · 2n(i)
,
si
4n(i) := ai + 4 (n(i) − 1) 2n(i)
+ 3 · 2n(i)
+ 2n(i)
− 2n(i)−1
.
We recall that we need to prove that any non-trivial solution of S is not almost periodic.
By contradiction, suppose that the solution
x(t) = XS(t) · u (6.31)
of the Cauchy problem
x (t) = S(t) · x(t), x(0) = u,
where u ∈ Rm
, || u ||2 = 1, is almost periodic. Applying Theorem 5.5 for ε = 1/3 and
si = 2n(i)
, i ∈ N, we obtain
x t + 2n(i(1))
− x t + 2n(i(2))
2
<
1
3
, t ∈ R, (6.32)
for all i(1), i(2) from an inﬁnite set N0 ⊆ N.
It is immediately seen that
max {|| X1 · u − u ||2, || X2 · u − u ||2} ≥ 1. (6.33)
Thus, from the construction, (6.30), (6.33), and from
˜Xi
S(t) · u − ˜Xi
S(s) · u
2
≤ ˜Xi
S(t) · u − XS(t) · u
2
+ XS(t) · u − XS(s) · u 2 + XS(s) · u − ˜Xi
S(s) · u
2
6.3 Skew-symmetric systems without almost periodic solutions 156
for
t = si
4, s = si
3; t = si
2, s = si
1;
...
t = si
4n(i), s = si
4n(i)−1; t = si
4n(i)−2, s = si
4n(i)−3,
respectively, it follows
1 <
1
3
+ XS si
4j · u − XS si
4j−1 · u 2
+
1
3
or
1 <
1
3
+ XS si
4j−2 · u − XS si
4j−3 · u 2
+
1
3
for j ∈ {1, . . . , n(i)}. Hence, we have
max XS si
4j · u − XS si
4j−1 · u 2
, XS si
4j−2 · u − XS si
4j−3 · u 2
>
1
3
(6.34)
for all j ∈ {1, . . . , n(i)} and i ≥ n0. Since
si
2 − si
1 = 2n(i)
− 1 = si
4 − si
3,
si
6 − si
5 = 2n(i)
− 21
= si
8 − si
7,
...
si
4n(i)−2 − si
4n(i)−3 = 2n(i)
− 2n(i)−1
= si
4n(i) − si
4n(i)−1,
inequality (6.34) implies (see (6.31))
sup
t∈R
x(t) − x t + 2n(i)
− 2j−1
2
>
1
3
(6.35)
for all i ≥ n0 and j ∈ {1, . . . , n(i)}. Of course, we can rewrite (6.32) into
sup
t∈R
x(t) − x t + 2n(i(2))
− 2n(i(1))
2
≤
1
3
, i(1), i(2) ∈ N0.
Considering (6.35), we see that (6.32) cannot be true for all i(1), i(2) from an inﬁnite
set N0. This contradiction proves the theorem.
The presented process can be applied to prove the existence of systems from S with
several properties. For example, we mention the following result.
Theorem 6.8. Let A ∈ S and ε > 0 be arbitrarily given. There exists B ∈ Oσ
ε (A) with
the property that
{XB(t); t ∈ R} = SO(m).
6.3 Skew-symmetric systems without almost periodic solutions 157
Proof. Let a sequence {Xk}k∈N ⊂ SO(m) be dense in SO(m). In the proof of Theorem 6.7,
we can replace considered matrices X1, X2 by arbitrary matrices Xk, Xk+1. Thus, there is
shown the existence of a system S = A + C ∈ Oσ
ε (A) with the property that (see (6.29))
XS si
j − Xj < N ·
4n(i) 2n(i)
22n(i)−1
for some si
j ∈ R and all j ∈ {1, . . . , 2n(i)}, i ≥ n0. Now it suﬃces to consider that
lim
i→∞
N ·
4n(i) 2n(i)
22n(i)−1
= 0.
At the end, we remark that the question of generalizations of Theorems 6.1 and 6.7
concerning other homogeneous linear diﬀerential systems, which can have only almost periodic
solutions, remains open (contrary to the corresponding discrete case, see Chapters 2
and 4).
Chapter 7
Values of almost periodic and limit
periodic functions
In this chapter, we prove two theorems about almost periodic and limit periodic functions
having given values. These theorems correspond to Theorem 3.1, where it is shown
that, for any countable and totally bounded set, there exists a limit periodic sequence
whose range is this set. In the statements of the presented results, we need that the
totally bounded set is the range of a uniformly continuous function ϕ for which the
set {ϕ(k); k ∈ Z} is ﬁnite (in the almost periodic case) or the uniformly continuous function
takes a value periodically (in the limit periodic case). We also construct limit periodic
functions whose ranges contain arbitrarily given totally bounded sequences if one requires
the connection by arcs of the space of values.
7.1 Preliminaries
We put R+
0 := [0, ∞). Let X = ∅ be an arbitrary set and let : X × X → R+
0 be
a pseudometric on X. For given ε > 0 and x ∈ X, the ε-neighbourhood of x is denoted
by Oε(x).
7.2 Functions with given values
At ﬁrst, we construct an almost periodic function with given values. Concerning a continuous
counterpart of Theorem 3.1 (or directly Deﬁnition 5.1 and Lemma 5.4), the given
set of values has to be the totally bounded range of a continuous function. In addition,
any almost periodic function is uniformly continuous (see Lemma 5.3). Considering these
facts, we formulate the following theorem.
Theorem 7.1. Let ϕ : R → X be a uniformly continuous function such that the set
{ϕ(k); k ∈ Z} is ﬁnite and the set {ϕ(t); t ∈ R} is totally bounded. Then, there exists an
158
7.2 Functions with given values 159
almost periodic function ψ with the property that
{ψ(k); k ∈ Z} = {ϕ(k); k ∈ Z}, {ψ(t); t ∈ R} = {ϕ(t); t ∈ R} (7.1)
and that, for any l ∈ Z, there exists q(l) ∈ N for which
ψ(l + s) = ψ(l + s + jq(l)), j ∈ Z, s ∈ [0, 1). (7.2)
Proof. We construct ψ : R → X applying Corollary 5.23 similarly as {ψk} applying Corollary
1.28 in the proof of Theorem 3.1. Considering that the set {ϕ(k); k ∈ Z} is ﬁnite, let
suﬃciently large M, N ∈ Z have the property that ϕ(M) = ϕ(N) and that, for any l ∈ Z,
there exists j(l) ∈ {N, N + 1, . . . , M − 1} for which
ϕ(l) = ϕ(j(l)), ϕ(l + 1) = ϕ(j(l) + 1). (7.3)
Without loss of generality, we can assume that N = 0 because, if N < 0, then we can
redeﬁne ﬁnitely many the below given εi and put ψ ≡ ϕ on a suﬃciently large interval.
Since ϕ is uniformly continuous with totally bounded range (see also (7.3)), for arbitrarily
small ε > 0, there exist l1(ε), . . . , lm(ε)(ε) ∈ Z such that, for any l ∈ Z, we have
(ϕ(l + s), ϕ(li + s)) < ε, s ∈ [0, 1],
for at least one integer li ∈ {l1(ε), . . . , lm(ε)(ε)}. We put εi := 2−i
, i ∈ N, i.e.,
li
1 := l1(2−i
), . . . , li
m(i) := lm(2−i) 2−i
, i ∈ N.
In addition, we assume that
{li
j; j ∈ {1, . . . , m(i)}, i ∈ N} = Z. (7.4)
First we deﬁne
ψ(t) := ϕ(t), t ∈ [0, M]. (7.5)
We choose arbitrary n(1) ∈ N for which 22n(1)
M > m(1). There exist (see (7.3))
j1
1, j1
2, . . . , j1
m(1) ∈ {0, 1, . . . , M − 1}
such that
ϕ(l1
1) = ψ(j1
1), ϕ(l1
1 + 1) = ψ(j1
1 + 1),
ϕ(l1
2) = ψ(j1
2), ϕ(l1
2 + 1) = ψ(j1
2 + 1),
...
ϕ(l1
m(1)) = ψ(j1
m(1)), ϕ(l1
m(1) + 1) = ψ(j1
m(1) + 1).
We deﬁne
ψ(s + M + j1
1) := ϕ(s + l1
1), s ∈ [0, 1],
ψ(t) := ψ(t − M), t ∈ (M, 2M] [M + j1
1, M + j1
1 + 1],
ψ(s + 2M + j1
2) := ϕ(s + l1
2), s ∈ [0, 1],
7.2 Functions with given values 160
ψ(t) := ψ(t − 2M), t ∈ (2M, 3M] [2M + j1
2, 2M + j1
2 + 1],
...
ψ(s + m(1)M + j1
m(1)) := ϕ(s + l1
m(1)), s ∈ [0, 1],
ψ(t) := ψ(t − m(1)M), t ∈ (m(1)M, (m(1) + 1)M] [m(1)M + j1
m(1), m(1)M + j1
m(1) + 1],
and we deﬁne ψ as periodic with period M on
[−(22n(1)−1
+ · · · + 23
+ 2)M, (2 + 22
+ · · · + 22n(1)
)M] (M, (m(1) + 1)M).
It is easily to see that we construct ψ as in Corollary 5.23 for
εi := L, i ∈ {1, . . . , 2n(1) + 1}, (7.6)
if L > 0 is suﬃciently large.
In the second step, we choose n(2) > n(1) + m(2) (n(2) ∈ N) and we put
ψ(t) := ψ(t + 22n(1)+1
M), t ∈ [−(22n(1)+1
+ · · · + 2)M, · · · , −(22n(1)−1
+ · · · + 2)M),
ψ(t) := ψ(t − 22n(1)+2
M), t ∈ ((2 + · · · + 22n(1)
)M, . . . , (2 + · · · + 22n(1)+2
)M],
...
ψ(t) := ψ(t + 22n(2)−1
M), t ∈ [−(22n(2)−1
+ · · · + 2)M, · · · , −(22n(2)−3
+ · · · + 2)M),
and
εi := 0, i ∈ {2n(1) + 2, . . . , 2n(2)}, ε2n(2)+1 := 2−1
. (7.7)
From n(2) > n(1) + m(2) and the above construction, we see that, for each integer j,
1 ≤ j ≤ m(1), there exist at least 2m(2) + 2 intervals of the form
[a, a + 1] ⊂ [−(22n(2)−1
+ · · · + 2)M, . . . , (22n(2)−2
+ · · · + 2)M]
such that a ∈ Z and
ψ|[a,a+1] ≡ ϕ|[l1
j ,l1
j +1], i.e., ψ(s + a) = ϕ(s + l1
j ), s ∈ [0, 1].
Hence, we can deﬁne continuous
ψ(t) ∈ Oε2n(2)+1
ψ(t − 22n(2)
M) , t ∈ ((2 + · · · + 22n(2)−2
)M, . . . , (2 + · · · + 22n(2)
)M],
for which
ψ|[22n(2)M,22n(2)M+1] ≡ ψ|[0,1],
ψ|[k,k+1] ≡ ψ|[0,1] for some k,
k ∈ {(2 + · · · + 22n(2)−2
)M, . . . , (2 + · · · + 22n(2)
)M − 1} {22n(2)
M},
and
ψ|[l,l+1] ≡ ϕ|[j(l),j(l)+1], l ∈ {(2 + · · · + 22n(2)−2
)M, . . . , (2 + · · · + 22n(2)
)M − 1},
some j(l) ∈ {0, . . . , M − 1, l1
1, . . . , l1
m(1), l2
1, . . . , l2
m(2)},
7.2 Functions with given values 161
{ϕ(t); t ∈ [l1
1, l1
1 + 1] ∪ · · · ∪ [l1
m(1), l1
m(1) + 1] ∪ [l2
1, l2
1 + 1] ∪ · · · ∪ [l2
m(2), l2
m(2) + 1]}
⊆ {ψ(t); t ∈ [(2 + · · · + 22n(2)−2
)M, . . . , (2 + · · · + 22n(2)
)M]}.
In the third step, we choose n(3) > n(2) + m(3) (n(3) ∈ N) and we construct ψ for
εi := 0, i ∈ {2n(2) + 2, . . . , 2n(3)}, ε2n(3)+1 := 2−2
. (7.8)
We have continuous
ψ(t) ∈ Oε2n(3)+1
ψ(t − 22n(3)
M) , t ∈ ((2 + · · · + 22n(3)−2
)M, . . . , (2 + · · · + 22n(3)
)M],
satisfying
ψ|[l,l+1] ≡ ϕ|[j(l),j(l)+1], l ∈ {(2 + · · · + 22n(3)−2
)M, . . . , (2 + · · · + 22n(3)
)M − 1},
at least one j(l) ∈ {0, . . . , M − 1, l1
1, l1
2, . . . , l3
m(3)},
{ϕ(t); t ∈ [l1
1, l1
1 + 1] ∪ [l1
2, l1
2 + 1] ∪ · · · ∪ [l3
m(3), l3
m(3) + 1]}
⊆ {ψ(t); t ∈ [(2 + · · · + 22n(3)−2
)M, . . . , (2 + · · · + 22n(3)
)M]}.
In addition, we have
ψ|[l,l+1] ≡ ψ|[0,1], l ∈ {j 22n(2)
M; j ∈ Z}
∩ {−(22n(3)−1
+ · · · + 2)M, . . . , (2 + · · · + 22n(3)
)M − 1},
ψ|[22n(3)M+1,22n(3)M+2] ≡ ψ|[1,2], ψ|[22n(3)M−1,22n(3)M] ≡ ψ|[−1,0],
ψ|[k,k+1] ≡ ψ|[1,2] for some k,
k ∈ {(2 + · · · + 22n(3)−2
)M, . . . , (2 + · · · + 22n(3)
)M − 1} {22n(3)
M + 1},
ψ|[k,k+1] ≡ ψ|[−1,0] for some k,
k ∈ {(2 + · · · + 22n(3)−2
)M, . . . , (2 + · · · + 22n(3)
)M − 1} {22n(3)
M − 1}.
Continuing in the same manner, in the i-th step, we choose n(i) > n(i − 1) + m(i)
(n(i) ∈ N) and we construct ψ for
εk := 0, k ∈ {2n(i − 1) + 2, . . . , 2n(i)}, ε2n(i)+1 := 2−i+1
. (7.9)
For simplicity, let i − 2 < 22n(2)
M (see also the proof of Theorem 3.1 for j 22n(2)
replaced
by [j 22n(2)
M, j 22n(2)
M + 1], 1 + j 22n(3)
by [1 + j 22n(3)
M, 1 + j 22n(3)
M + 1], and so on).
Again, for each j(1) ∈ {1, . . . , i−1}, j(2) ∈ {1, . . . , m(j(1))}, there exist at least 2m(i)+2
integers
l ∈ {−(22n(i)−1
+ · · · + 2)M, . . . , (2 + · · · + 22n(i)−2
)M − 1}
({j 22n(2)
M; j ∈ Z} ∪ {1 + j 22n(3)
M; j ∈ Z} ∪ {−1 + j 22n(3)
M; j ∈ Z}∪
· · · ∪ {i − 3 + j 22n(i−1)
M; j ∈ Z} ∪ {3 − i + j 22n(i−1)
M; j ∈ Z})
such that
ψ|[l,l+1] ≡ ϕ| l
j(1)
j(2)
,l
j(1)
j(2)
+1
.
7.2 Functions with given values 162
Thus, we can deﬁne continuous
ψ(t) ∈ Oε2n(i)+1
ψ(t − 22n(i)
M) , t ∈ ((2 + · · · + 22n(i)−2
)M, . . . , (2 + · · · + 22n(i)
)M],
satisfying
ψ|[l,l+1] ≡ ϕ|[j(l),j(l)+1], l ∈ {(2 + · · · + 22n(i)−2
)M, . . . , (2 + · · · + 22n(i)
)M − 1},
at least one j(l) ∈ {0, . . . , M − 1, l1
1, l1
2, . . . , li
m(i)},
{ϕ(t); t ∈ [l1
1, l1
1 + 1] ∪ [l1
2, l1
2 + 1] ∪ · · · ∪ [li
m(i), li
m(i) + 1]}
⊆ {ψ(t); t ∈ [(2 + · · · + 22n(i)−2
)M, . . . , (2 + · · · + 22n(i)
)M]}.
(7.10)
In addition, we can deﬁne ψ so that
ψ|[l,l+1] ≡ ψ|[0,1], l ∈ {j 22n(2)
M; j ∈ Z}
∩ {−(22n(i)−1
+ · · · + 2)M, . . . , (2 + · · · + 22n(i)
)M − 1},
ψ|[l,l+1] ≡ ψ|[1,2], l ∈ {1 + j 22n(3)
M; j ∈ Z}
∩ {−(22n(i)−1
+ · · · + 2)M, . . . , (2 + · · · + 22n(i)
)M − 1},
ψ|[l,l+1] ≡ ψ|[−1,0], l ∈ {−1 + j 22n(3)
M; j ∈ Z}
∩ {−(22n(i)−1
+ · · · + 2)M, . . . , (2 + · · · + 22n(i)
)M − 1},
...
ψ|[l,l+1] ≡ ψ|[i−3,i−2], l ∈ {i − 3 + j 22n(i−1)
M; j ∈ Z}
∩ {−(22n(i)−1
+ · · · + 2)M, . . . , (2 + · · · + 22n(i)
)M − 1},
ψ|[l,l+1] ≡ ψ|[3−i,4−i], l ∈ {3 − i + j 22n(i−1)
M; j ∈ Z}
∩ {−(22n(i)−1
+ · · · + 2)M, . . . , (2 + · · · + 22n(i)
)M − 1},
ψ|[22n(i)M+i−2,22n(i)M+i−1] ≡ ψ|[i−2,i−1], ψ|[22n(i)M+2−i,22n(i)M+3−i] ≡ ψ|[2−i,3−i],
ψ|[k,k+1] ≡ ψ|[i−2,i−1] for some k,
k ∈ {(2 + · · · + 22n(i)−2
)M, . . . , (2 + · · · + 22n(i)
)M − 1} {22n(i)
M + i − 2},
ψ|[k,k+1] ≡ ψ|[2−i,3−i] for some k,
k ∈ {(2 + · · · + 22n(i)−2
)M, . . . , (2 + · · · + 22n(i)
)M − 1} {22n(i)
M + 2 − i}.
Evidently, it is valid
(ϕ(i), ϕ(j)) ≥ 2−K
or (ϕ(i), ϕ(j)) = 0
for all i, j ∈ Z and some K ∈ N. If we begin the construction by
lK
1 := l1 2−K
, . . . , lK
m(K) := lm(2−K ) 2−K
,
then we have to obtain
ψ(k) = ψ(k + M), k ∈ Z.
7.2 Functions with given values 163
Hence, we can construct the above ψ in order that the sequence {ψ(k)}k∈Z is periodic with
period M which gives (5.14) and the continuity of ψ. We construct ψ using the process
from Corollary 5.23 for all i ∈ N and we obtain an almost periodic function ψ : R → X.
Indeed, we have (7.5) and, summarizing (7.6), (7.7), (7.8), . . . , (7.9), . . . , we get (5.13)
(see also (3.22)). For periodic {ψ(k)}k∈Z, the ﬁrst identity in (7.1) follows from (7.3)
and (7.5) and the second one from the construction, (7.4), and (7.10). As in the proof of
Theorem 3.1, we see that, for any l ∈ Z, there exists i(l) ∈ N satisfying
ψ|[k,k+1] ≡ ψ|[l,l+1], k ∈ {l + j 22n(i(l))
M; j ∈ Z}.
Thus, we obtain (7.2) for q(l) = 22n(i(l))
M.
As an example which illustrates the previous theorem, we mention the following result.
Corollary 7.2. For any continuous function F : [0, 1] → X, there exists an almost periodic
function ψ with the property that
{ψ(t); t ∈ R} = {F(t); t ∈ (0, 1)}.
Proof. It suﬃces to show, that there exists a uniformly continuous function ϕ : R → X
for which {ϕ(k); k ∈ Z} = {F(1/2)} and {ϕ(t); t ∈ R} = {F(t); t ∈ (0, 1)}, and to apply
Theorem 7.1. For example, one can put
ϕ(k + s) := F
1
2
+ s , k ∈ N, s ∈ 0,
k
2k + 1
,
ϕ(k + s) := F
1
2
+
k
2k + 1
, k ∈ N, s ∈
k
2k + 1
, 1 −
k
2k + 1
,
ϕ(k + s) := F
1
2
+ 1 − s , k ∈ N, s ∈ 1 −
k
2k + 1
, 1 ;
ϕ(k + s) := F
1
2
− s , k ∈ Z N, s ∈ 0,
k
2k − 1
,
ϕ(k + s) := F
1
2
−
k
2k − 1
, k ∈ Z N, s ∈
k
2k − 1
, 1 −
k
2k − 1
,
ϕ(k + s) := F
1
2
+ s − 1 , k ∈ Z N, s ∈ 1 −
k
2k − 1
, 1 .
In the limit periodic case, we obtain:
Theorem 7.3. Let F : R → X be a uniformly continuous function. If the range of F is
totally bounded and F(p) = F(p + kr) for all k ∈ Z and some p ∈ R, r > 0, then there
exists a limit periodic function f : R → X such that
{f(t); t ∈ R} = {F(t); t ∈ R} (7.11)
and that, for every l ∈ Z, one can ﬁnd q(l) ∈ N for which
f(l + s) = f (l + s + jq(l)) , j ∈ Z, s ∈ [0, 1]. (7.12)
7.2 Functions with given values 164
Proof. We consider the function G : R → X given by the formula
G(t) := F (rt + p) , t ∈ R.
We see that G(0) = G(k), k ∈ Z, and that
{G(t); t ∈ R} = {F(t); t ∈ R}. (7.13)
Since G is a uniformly continuous function whose range is totally bounded, for all ε > 0,
there exists m = m(ε) ∈ N {1} satisfying
(G(l + s), G(j + s)) < ε (7.14)
for all l ∈ Z, s ∈ [0, 1], and at least one j = j(l) ∈ {−m, . . . , 0, . . . , m−1}. We can assume
that
m(ε) → ∞ if ε → 0+
. (7.15)
Firstly, we put
X := {G(t); t ∈ R}, (7.16)
m(n) := m 2−n
, n ∈ N,
and
f1
(s + 2lm(1)) := G(s), s ∈ [−m(1), m(1)), l ∈ Z.
In the second step, we deﬁne a periodic continuous function f2
: R → X with period
22
m(1)m(2) arbitrarily so that
f2
(s) = f1
(s), s ∈ [−m(1), m(1)),
f2
(s + 2m(1)m(2)) = f1
(s), s ∈ [−1, 1),
f2
(s); s ∈ [−2m(1)m(2) + 1, −1) ∪ [1, 2m(1)m(2) − 1) = {G(s); s ∈ [−m(2), m(2)]} ,
and
f2
(t), f1
(t) <
1
2
, t ∈ R. (7.17)
In fact, using (7.14), one can choose this function f2
in such a way that, for each
j ∈ {−2m(1)m(2), . . . , 2m(1)m(2) − 1}, there exists
i = i(j) ∈ {−m(2), . . . , m(2) − 1}
having the property
f2
(s + j) = G(s + i), s ∈ [0, 1].
Henceforth, we assume that we choose all functions fn
in this way.
In the third step, we deﬁne a periodic continuous function f3
: R → X with period
23
m(1)m(2)m(3) arbitrarily so that
f3
(s) = f2
(s), s ∈ [−2m(1)m(2), 2m(1)m(2)),
f3
(s + 2m(1)m(2)k) = f2
(s), s ∈ [−1, 1), k ∈ Z,
7.2 Functions with given values 165
f3
s + 22
m(1)m(2)m(3) = f2
(s), s ∈ [−2, 2),
and
f3
(t), f2
(t) <
1
22
, t ∈ R, (7.18)
f3
(s); s ∈ I3 = {G(s); s ∈ [−m(3), m(3)]} ,
where
I3 := −22
m(1)m(2)m(3) + 2, −2 ∪ 2, 22
m(1)m(2)m(3) − 2
k∈Z
[2m(1)m(2)k − 1, 2m(1)m(2)k + 1) .
In the general n-th step, we deﬁne a periodic continuous function fn
: R → X with
period 2n
m(1)m(2) · · · m(n) arbitrarily so that
fn
(s) = fn−1
(s), s ∈ −2n−2
m(1)m(2) · · · m(n − 1), 2n−2
m(1)m(2) · · · m(n − 1) ,
fn
(s + 2m(1)m(2)k) = fn−1
(s), s ∈ [−1, 1), k ∈ Z,
fn
s + 22
m(1)m(2)m(3)k = fn−1
(s), s ∈ [−2, 2), k ∈ Z,
...
fn
s + 2n−2
m(1)m(2) · · · m(n − 1)k = fn−1
(s), s ∈ [−n + 2, n − 2), k ∈ Z,
fn
s + 2n−1
m(1)m(2) · · · m(n) = fn−1
(s), s ∈ [−n + 1, n − 1),
and
fn
(t), fn−1
(t) <
1
2n−1
, t ∈ R, (7.19)
{fn
(s); s ∈ In} = {G(s); s ∈ [−m(n), m(n)]} , (7.20)
where
In := −2n−1
· · · m(n) + n − 1, −n + 1 ∪ n − 1, 2n−1
· · · m(n) − n + 1
k∈Z
[2m(1)m(2)k − 1, 2m(1)m(2)k + 1) ∪
k∈Z
22
m(1)m(2)m(3)k − 2, 22
m(1)m(2)m(3)k + 2 ∪ · · · ∪
k∈Z
2n−2
m(1) · · · m(n − 1)k − n + 2, 2n−2
m(1) · · · m(n − 1)k + n − 2 .
We can deﬁne function f : R → X by
f(t) := lim
n→∞
fn
(t), t ∈ R, (7.21)
because
fn
(s) = fn+i
(s), n, i ∈ N, s ∈ −2n−1
m(1) · · · m(n), 2n−1
m(1) · · · m(n) . (7.22)
7.2 Functions with given values 166
The inequality
(f(t), fn
(t)) ≤ fn+1
(t), fn
(t) + fn+2
(t), fn+1
(t) + · · ·
<
1
2n
+
1
2n+1
+ · · · =
1
2n−1
, t ∈ R, n ∈ N,
which follows from (7.19) (consider together with (7.17), (7.18)), implies the limit periodicity
of f deﬁned in (7.21).
The resulting function f satisﬁes (7.11) and (7.12). Immediately, we obtain (7.11) from
(7.15), (7.20) (see also (7.13), (7.16)), and (7.22). Let n ∈ N be arbitrary. Considering the
construction above, we have
fn+i
(s + 2n
m(1) · · · m(n + 1)j) = fn+1
(s), s ∈ [−n, n], j ∈ Z, i ∈ N.
Finally, we get (7.12) for
q(l) = 2| l |+1
m(1)m(2) · · · m(| l | + 2), l ∈ Z.
Remark 7.4. As in Section 3.2, we can formulate Theorem 7.3 in the form when X is
a uniform space with a countable fundamental system of entourages. We refer to the
references mentioned in Section 3.2.
Remark 7.5. Here, we comment the periodic condition on F in the theorem above. Without
the requirement that there exist p ∈ R and r > 0 such that F(p) = F(p + kr), k ∈ Z,
Theorem 7.3 is not valid.
For example, there exists a uniformly continuous function F : R → R2
whose range is
{F(t); t ∈ R} = x, sin
1
x
; x ∈ (0, 1] . (7.23)
Suppose that a limit periodic function f : R → R2
satisﬁes (7.11) and (7.12). There exist
j ∈ Z and a ∈ (0, 1] for which (see (7.11) and (7.23))
{f(t); t ∈ [j, j + 1]} = x, sin
1
x
; x ∈ [a, 1] . (7.24)
Let us consider q(j) from the statement of Theorem 7.3. Since f is uniformly continuous,
there exists b ∈ (0, a) such that
{f(t); t ∈ [i, i + 1 + q(j)]} ⊆ x, sin
1
x
; x ∈ [b, 1]
for all i ∈ Z if
[1, sin 1] ∈ {f(t); t ∈ [i, i + 1 + q(j)]}. (7.25)
Nevertheless, (7.25) follows from (7.12) and (7.24). Thus, we have
{f(t); t ∈ R} ⊆ x, sin
1
x
; x ∈ [b, 1] . (7.26)
This contradiction (given by (7.11) and (7.26)) shows that the periodic condition on F
cannot be omitted.
7.2 Functions with given values 167
Immediately, from Theorem 7.3, we get the following consequence.
Corollary 7.6. Let F : R → X be a uniformly continuous function. If the range of F is
totally bounded and F(p) = F(p + kr) for all k ∈ Z and some p ∈ R, r > 0, then there
exists an almost periodic function f : R → X such that (7.11) is valid and that, for every
l ∈ Z, one can ﬁnd q(l) ∈ N for which (7.12) is valid.
Important cases of pseudometric spaces are covered by the spaces whose elements can
be connected by arcs.
Corollary 7.7. Let F : R → X be a uniformly continuous function whose range X :=
{F(t); t ∈ R} is totally bounded. If all x, y ∈ X can be connected in X by continuous
curves which depend uniformly continuously on x and y, then there exists a limit periodic
function f : R → X such that
{F(t); t ∈ R} ⊆ {f(t); t ∈ R} (7.27)
and that, for every l ∈ Z, one can ﬁnd q(l) ∈ N for which (7.12) is valid.
Proof. Let us choose t0 ∈ R arbitrarily and consider continuous functions gk : [0, 1] → X,
k ∈ Z, for which
gk(0) = F(k), gk
1
2
= F(t0), gk(1) = F(k).
We deﬁne G : R → X as
G(2k + s) := F(k + s), G(2k + 1 + s) := gk+1(s), k ∈ Z, s ∈ [0, 1).
We can assume (consider directly the statement of the corollary) that, for any ε > 0, there
exists δ(ε) > 0 such that
(gk(s), gl(s)) < ε, s ∈ [0, 1],
if
(gk(0), gl(0)) = (F(k), F(l)) < δ(ε)
for arbitrary k, l ∈ Z.
Thus, there exists a uniformly continuous function G : R → X with a totally bounded
range and with the properties
G
3
2
+ 2k = F(t0), k ∈ Z,
{F(t); t ∈ R} ⊆ {G(t); t ∈ R}. (7.28)
Using Theorem 7.3 for this function G, we get a limit periodic function f : R → X satisfying
{f(t); t ∈ R} = {G(t); t ∈ R} (7.29)
and (7.12). Finally, (7.28) and (7.29) give (7.27).
Especially, from the proof of Corollary 7.7, we obtain the last result.
Corollary 7.8. Let F : R → X be a uniformly continuous function whose range X :=
{F(t); t ∈ R} is totally bounded. If there exists x ∈ X such that x and F(k) for k ∈ Z
can be connected in X by continuous curves which depend uniformly continuously on F(k),
then one can construct a limit periodic function f : R → X satisfying (7.11) and (7.12).
Author’s papers
All mentioned papers can be found in MathSciNet (18 of them in WoS). The status is on
the date 2015/09/21.
(1) Vesel´y, M.: On orthogonal and unitary almost periodic homogeneous linear difference
systems. In: Proceedings of Colloquium on Diﬀerential and Diﬀerence Equations
(Brno, 2006), 179–184. Folia Fac. Sci. Natur. Univ. Masaryk. Brun. Math.,
Vol. 16, Masaryk Univ., Brno, 2007.
(2) Vesel´y, M.: Construction of almost periodic sequences with given properties. Electron.
J. Diﬀerential Equations 2008, no. 126, 1–22.
(3) Vesel´y, M.: Almost periodic sequences and functions with given values. Arch. Math.
(Brno) 47 (2011), no. 1, 1–16.
(4) Vesel´y, M.: Construction of almost periodic functions with given properties. Electron.
J. Diﬀerential Equations 2011, no. 29, 1–25.
(5) Vesel´y, M.: Almost periodic homogeneous linear diﬀerence systems without almost
periodic solutions. J. Diﬀerence Equ. Appl. 18 (2012), no. 10, 1623–1647.
(6) Hasil, P.; Vesel´y, M.: Criticality of one term 2n-order self-adjoined diﬀerential equations.
In: Proceedings of the 9th Colloquium on the Qualitative Theory on Differential
Equations, no. 18, 1–12, Electron. J. Qual. Theory Diﬀer. Equ., Szeged,
2012.
(7) Hasil, P.; Vesel´y, M.: Almost periodic transformable diﬀerence systems. Appl. Math.
Comput. 218 (2012), no. 9, 5562–5579.
(8) Hasil, P.; Vesel´y, M.: Critical oscillation constant for diﬀerence equations with almost
periodic coeﬃcients. Abstr. Appl. Anal. 2012, art. ID 471435, 1–19.
(9) Vesel´y, M.: Almost periodic skew-symmetric diﬀerential systems. Electron. J. Qual.
Theory Diﬀer. Equ. 2012, no. 72, 1–16.
(10) Hasil, P.; Vesel´y, M.: Oscillation of half-linear diﬀerential equations with asymptotically
almost periodic coeﬃcients. Adv. Diﬀerence Equ. 2013, no. 122, 1–15.
(11) Hasil, P.; Vesel´y, M.: Oscillation and non-oscillation of asymptotically almost periodic
half-linear diﬀerence equations. Abstr. Appl. Anal. 2013, atr. ID 432936,
1–12.
168
(12) Hasil, P.; Vesel´y, M.: Conditional oscillation of Riemann–Weber half-linear differential
equations with asymptotically almost periodic coeﬃcients. Studia Sci. Math.
Hungar. 51 (2014), no. 3, 303–321.
(13) Hasil, P.; Vesel´y, M.: Limit periodic linear diﬀerence systems with coeﬃcient matrices
from commutative groups. Electron. J. Qual. Theory Diﬀer. Equ. 2014, no. 23,
1–25.
(14) Hasil, P.; Maˇr´ık, R.; Vesel´y, M.: Conditional oscillation of half-linear differential
equations with coeﬃcients having mean values. Abstr. Appl. Anal. 2014, atr. ID
258159, 1–14.
(15) Hasil, P.; Vesel´y, M.: Non-oscillation of half-linear diﬀerential equations with periodic
coeﬃcients. Electron. J. Qual. Theory Diﬀer. Equ. 2015, no. 1, 1–21.
(16) Hasil, P.; Vesel´y, M.: Non-oscillation of perturbed half-linear diﬀerential equations
with sums of periodic coeﬃcients. Adv. Diﬀerence Equ. 2015, no. 190, 1–17.
(17) Hasil, P.; Vesel´y, M.: Oscillation constants for half-linear diﬀerence equations with
coeﬃcients having mean values. Adv. Diﬀerence Equ. 2015, no. 210, 1–18.
(18) Hasil, P.; Vesel´y, M.: Limit periodic homogeneous linear diﬀerence systems. Appl.
Math. Comput. 265 (2015), 958–972.
(19) Doˇsl´y, O.; Vesel´y, M.: Oscillation and non-oscillation of Euler type half-linear differential
equations. J. Math. Anal. Appl. 429 (2015), no. 1, 602–621.
(20) Hasil, P.; Vesel´y, M.: Oscillation constant for modiﬁed Euler type half-linear equations.
Electron. J. Diﬀerential Equations 2015, no. 220, 1–14.
(21) Hasil, P.; Vesel´y, M.: Values of limit periodic sequences and functions. Math. Slovaca,
in press.
(PhD) Vesel´y, M.: Constructions of almost periodic sequences and functions and homogeneous
linear diﬀerence and diﬀerential systems. Ph.D. thesis. Brno, 2011.
169
Bibliography
[1] Agarwal, R. P.: Diﬀerence equations and inequalities. Theory, methods, and applications.
Marcel Dekker, Inc., New York, 2000.
[2] Akhmet, M. U.: Almost periodic solutions of the linear diﬀerential equation with
piecewise constant argument. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal.
16 (2009), 743–753.
[3] Aldrovandi, R.: Special matrices of mathematical physics. Stochastic, circulant and
Bell matrices. World Scientiﬁc Publishing Co., Inc., Singapore, 2001.
[4] Alonso, A. I.; Hong, J.; Obaya, G. R.: Almost periodic type solutions of diﬀerential
equations with piecewise constant argument via almost periodic type sequences. Appl.
Math. Lett. 13 (2000), no. 2, 131–137.
[5] AlSharawi, Z.; Angelos, J.: Linear almost periodic diﬀerence equations. J. Comput.
Math. Optim. 4 (2008), 61–91.
[6] Amerio, L.: Abstract almost-periodic functions and functional equations. Boll. U.M.I.
21 (1966), 287–334.
[7] Amerio, L.; Prouse, G.: Almost-periodic functions and functional equations. Van
Nostrand Reinhold Company, New York, 1971.
[8] Andres, J.; Pennequin, D.: Semi-periodic solutions of diﬀerence and diﬀerential equations.
Bound. Value Probl. 141 (2012), 1–16.
[9] Auslander, J.; Markley, N.: Locally almost periodic minimal ﬂows. J. Diﬀerence Equ.
Appl. 15 (2009), no. 1, 97–109.
[10] Bˆanzaru, T.: Multivalued almost periodic mappings. Bul. S¸ti. Tehn. Inst. Politehn.
Timi¸soara—Ser. Mat.-Fiz.-Mec. Teoret. Apl. 19(33) (1974), no. 1, 25–28.
[11] Bˆanzaru, T.; Criv˘at¸, N.: On almost periodic multivalued maps with values in uniform
spaces. Bul. S¸tiint¸. Tehn. Inst. Politehn. “Traian Vuia” Timi¸soara Ser. Mat. Fiz.
26(40) (1981), no. 2, 47–51.
[12] Basit, B. R.; G¨unzler, H.: Generalized vector valued almost periodic and ergodic
distributions. J. Math. Anal. Appl. 314 (2006), no. 1, 363–381.
170
BIBLIOGRAPHY 171
[13] Bede, B.; Gal, S. G.: Almost periodic fuzzy-number-valued functions. Fuzzy Sets and
Systems 147 (2004), no. 3, 385–403.
[14] Behrouzi, F.: Almost periodic functions on groupoids. Rend. Semin. Mat. Univ.
Padova 132 (2014), 45–59.
[15] Bel’gart, L. V.; Romanovski˘ı, R. K.: The exponential dichotomy of solutions to systems
of linear diﬀerence equations with almost periodic coeﬃcients. Russ. Math.
54 (2010), 44–51.
[16] Bernstein, D.S .: Matrix mathematics. Theory, facts, and formulas. Princeton University
Press, Princeton, 2009.
[17] Berg, I. D.; Wilansky, A.: Periodic, almost-periodic, and semiperiodic sequences.
Michigan Math. J. 9 (1962), 363–368.
[18] Besicovitch, A. S.: Almost periodic functions. Dover Publications, Inc., New York,
1955.
[19] Bhatti, M. I.; N’Gu´er´ekata, G. M.; Latif, M. A.: Almost periodic functions deﬁned
on Rn
with values in p-Fr´echet spaces, 0 < p < 1. Libertas Math. 29 (2009), 83–100.
[20] Bhatti, M. I.; Latif, M. A.: Almost periodic functions deﬁned on Rn
with values in
fuzzy setting. Punjab Univ. J. Math. (Lahore) 39 (2007), 19–27.
[21] Blot, J.; Pennequin, D.: Existence and structure results on almost periodic solutions
of diﬀerence equations. J. Diﬀer. Equations Appl. 7 (2001), no. 3, 383–402.
[22] Bochner, S.: Abstrakte fastperiodische Funktionen. Acta Math. 61 (1933), no. 1,
149–184.
[23] Bochner, S.: A new approach to almost periodicity. Proc. Nat. Acad. Sci. 48 (1962),
2039–2043.
[24] Bochner, S.; von Neumann, J.: Almost periodic functions in groups. II. Trans. Amer.
Math. Soc. 37 (1935), no. 1, 21–50.
[25] Bochner, S.; von Neumann, J.: On compact solutions of operational-diﬀerential equations.
I. Ann. Math. (2) 36 (1935), no. 1, 255–291.
[26] Bogatyrev, B. M.; Eremenko, V. A.: Solution of implicit ﬁnite diﬀerence systems. In:
Asymptotic methods of nonlinear mechanics, 8–13, 193. Akad. Nauk Ukrain. SSR,
Inst. Mat., Kiev, 1979.
[27] Bohr, H.: Zur Theorie der fastperiodischen Funktionen. I. Eine Verallgemeinerung
der Theorie der Fourierreihen. Acta Math. 45 (1925), no. 1, 29–127.
[28] Bohr, H.: Zur Theorie der fastperiodischen Funktionen. II. Zusammenhang der fastperiodischen
Funktionen mit Funktionen von unendlich vielen Variabeln; gleichm¨assige
Approximation durch trigonometrische Summen. Acta Math. 46 (1925), no. 1-2,
101–214.
BIBLIOGRAPHY 172
[29] Bohr, H.; Neugebauer, O.: ¨Uber lineare Diﬀerentialgleichungen mit konstanten Koefﬁzienten
und fastperiodischer rechter Seite. Math.-Phys. Klasse 1926, C 17, 8–22.
[30] Brudnyi, A.; Kinzebulatov, D.: Holomorphic semi-almost periodic functions. Integral
Equations Operator Theory 66 (2010), no. 3, 293–325.
[31] Bugajewski, D.; N’Gu´er´ekata, G. M.: On some classes of almost periodic functions
in abstract spaces. Int. J. Math. Sci. 2004, no. 61-64, 3237–3247.
[32] Burd, V.: Method of averaging for diﬀerential equations on an inﬁnite interval. Theory
and applications. Lecture Notes in Pure and Applied Mathematics, Vol. 255. Chapman
& Hall/CRC, Boca Raton, 2007.
[33] Burton, T. A.: Linear diﬀerential equations with periodic coeﬃcients. Proc. Amer.
Math. Soc. 17 (1966), 327–329.
[34] Campos, J.; Tarallo, M. Almost automorphic linear dynamics by Favard theory. J.
Diﬀerential Equations 256 (2014), no. 4, 1350–1367.
[35] Caraballo, T.; Cheban, D.: Almost periodic and almost automorphic solutions of
linear differential/diﬀerence equations without Favard’s separation condition. I. J.
Diﬀerential Equations 246 (2009), no. 1, 108–128.
[36] Caraballo, T.; Cheban, D.: Almost periodic and almost automorphic solutions of
linear differential/diﬀerence equations without Favard’s separation condition. II. J.
Diﬀerential Equations 246 (2009), no. 3, 1164–1186.
[37] Caraballo, T.; Cheban, D.: Almost periodic and almost automorphic solutions of
linear diﬀerential equations. Discrete Contin. Dyn. Syst. 33 (2013), no. 5, 1857–1882.
[38] Cenusa, G.: Random vector functions -almost periodic. Bull. Math. Soc. Sci. Math.
R. S. Roumanie (N.S.) 25(73) (1981), no. 1, 15–31.
[39] Cheban, D. N.: Asymptotically almost periodic solutions of diﬀerential equations.
Hindawi Publishing Corporation, New York, 2009.
[40] Cheban, D. N.: Bounded solutions of linear almost periodic systems of diﬀerential
equations. Izv. Ross. Akad. Nauk Ser. Mat. 62 (1998), no. 3, 155–174; translation in:
Izv. Math. 62 (1998), no. 3, 581–600.
[41] Chv´atal, M.: Non-almost periodic solutions of limit periodic and almost periodic
homogeneous linear diﬀerence systems. Electron. J. Qual. Theory Diﬀer. Equ. 2014,
no. 76, 1–20.
[42] Cieutat, P.; Haraux, A.: Exponential decay and existence of almost periodic solutions
for some linear forced diﬀerential equations. Port. Math. (N.S.) 59 (2002),
no. 2, 141–159.
[43] Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A.: Atlas of ﬁnite
groups. Maximal subgroups and ordinary characters for simple groups. Clarendon
Press, Oxford, 1985.
BIBLIOGRAPHY 173
[44] Coppel, W. A.: Almost periodic properties of ordinary diﬀerential equations. Ann.
Mat. Pura Appl. 76 (1967), no. 4, 27–49.
[45] Corduneanu, C.: Almost periodic discrete processes. Libertas Mathematica 2 (1982),
159–169.
[46] Corduneanu, C.: Almost periodic functions. John Wiley and Sons, New York, 1968.
[47] Corduneanu, C.: Almost periodic oscillations and waves. Springer, New York, 2009.
[48] Corduneanu, C.: Almost periodic solutions for a class of functional diﬀerential equations.
Funct. Diﬀer. Equ. 14 (2007), no. 2-4, 223–229.
[49] Corduneanu, C.: Some comments on almost periodicity and related topics. Commun.
Math. Anal. 8 (2010), no. 2, 5–15.
[50] Criv˘at¸, N.: Almost periodic multifunctions with values in generalized uniform spaces.
Bul. S¸tiint¸. Univ. Politeh. Timi¸s. Ser. Mat. Fiz. 52(66) (2007), no. 1, 69–75.
[51] Criv˘at¸, N.: Almost periodic multifunctions with values in generalized uniform spaces
(II). Bul. S¸tiint¸. Univ. Politeh. Timi¸s. Ser. Mat. Fiz. 55(69) (2010), no. 2, 16–21.
[52] Demenchuk, A. K.: Irregular forced almost periodic solutions of ordinary linear differential
systems. Mathematica 46(69) (2004), no. 1, 67–74.
[53] Demenchuk, A. K.: On irregular forced almost periodic solutions of linear systems
in a critical nonresonance case. Vestsi Nats. Akad. Navuk Belarusi Ser. Fiz.-Mat.
Navuk 2005, no. 1, 23–27, 125.
[54] Demenchuk, A. K.: On the existence of almost periodic solutions of linear diﬀerential
systems in a noncritical case. Vestsi Nats. Akad. Navuk Belarusi Ser. Fiz.-Mat. Navuk
2003, no. 3, 11–16, 124.
[55] Demenchuk, A. K.: On the existence of partially irregular almost periodic solutions
of linear nonhomogeneous diﬀerential systems in a critical nonresonance case. Diﬀer.
Uravn. 40 (2004), no. 5, 590–596, 717; translation in: Diﬀer. Equ. 40 (2004), no. 5,
634–640.
[56] Diagana, T.: Almost automorphic type and almost periodic type functions in abstract
spaces. Springer, Cham, 2013.
[57] Diagana, T.; Elaydi, S.; Yakubu, A.-A.: Population models in almost periodic environments.
J. Diﬀerence Equ. Appl. 13 (2007), 239–260.
[58] Dunlavy, D. M.; Mackey, D. S.; Mackey, N.: Structure preserving algorithms for perplectic
eigenproblems. Electron. J. Linear Algebra 13 (2005), 10–39.
[59] Duy, T. K.: Limit-periodic arithmetical functions and the ring of ﬁnite integral adeles.
Lith. Math. J. 51 (2011), no. 4, 486–506.
BIBLIOGRAPHY 174
[60] Dˇzafarov, A. S.; Gasanov, G. M.: Almost periodic functions with respect to the Hausdorﬀ
metric, and their properties. Izv. Akad. Nauk Azerba˘ıdˇzan. SSR Ser. Fiz.-Tehn.
Mat. Nauk 1977, no. 1, 57–62.
[61] Dˇzafarov, A. S.; Gasanov, G. M.: Some problems of almost-periodic functions with
respect to the Hausdorﬀ metric. In: Theory of functions and approximations, Part 2
(Saratov, 1986), 45–47. Saratov. Gos. Univ., Saratov, 1988.
[62] Eliasson, L. H.: Reducibility and point spectrum for linear quasi-periodic skew-products.
In: Proceedings of the International Congress of Mathematicians (Berlin,
1998). Doc. Math. 1998, Extra Vol. II, 779–787.
[63] Ellis, R.: The construction of minimal discrete ﬂows. Amer. J. Math. 87 (1965),
564–574.
[64] Esclangon, E.: Sur les int´egrales quasi-p´eriodiques d’´equations diﬀ´erentielles lin´eaires.
C. R. Acad. Sci. Paris 158 (1914), 1254–1256.
[65] Fan, K.: Les fonctions asymptotiquement presque-p´eriodiques d’une variable enti`ere
et leur application `a l’´etude de l’it´eration des transformations continues. Math.
Z. 48 (1943), 685–711.
[66] Favard, J.: Sur certains syst`emes diﬀ´erentiels scalaires lin´eaires et homog`enes `a coefﬁcients
presque-p´eriodiques. Ann. Mat. Pura Appl. 61 (1963), no. 4, 297–316.
[67] Favard, J.: Sur les ´equations diﬀ´erentielles lin´eaires `a coeﬃcients presque-p´eriodiques.
Acta Math. 51 (1928), no. 1, 31–81.
[68] Favorov, S.; Parfyonova, N.: Almost periodic mappings to complex manifolds. In:
Functional analysis and its applications, 81–84. North-Holland Math. Stud., 197,
Elsevier, Amsterdam, 2004.
[69] Faxing, L.: The existence of almost-automorphic solutions of almost-automorphic
systems. Ann. Diﬀerential Equations 3 (1987), no. 3, 329–349.
[70] Feng, Q.; Yuan, R.: Existence of almost periodic solutions for neutral delay diﬀerence
systems. Front. Math. China 4 (2009), no. 3, 437–462.
[71] Filippov, M. G.: Reducibility of linear almost periodic systems of diﬀerential equations
with a skew-symmetric matrix. In: Analytic methods for studying nonlinear
diﬀerential systems, 120–127. Akad. Nauk Ukrainy, Inst. Mat., Kiev, 1992.
[72] Fink, A. M.: Almost periodic diﬀerential equations. Lecture Notes in Math., Vol. 377.
Springer-Verlag, Berlin, 1974.
[73] Flor, P.: ¨Uber die Wertmengen fastperiodischer Folgen. Monatsh. Math. 67 (1963),
12–17.
[74] Fr´echet, M.: Les fonctions asymptotiquement presque-p´eriodiques. Revue Sci. (Rev.
Rose Illus.) 79 (1941), 341–354.
BIBLIOGRAPHY 175
[75] Fr´echet, M.: Les fonctions asymptotiquement presque-p´eriodiques continues. C. R.
Acad. Sci. Paris 213 (1941), 520–522.
[76] Fulton, W.; Harris, J.: Representation theory. A ﬁrst course. Springer-Verlag, New
York, 1991.
[77] Funakosi, S.: On extension of almost periodic functions. Proc. Japan Acad. 43 (1967),
739–742.
[78] Furstenberg, H.: Strict ergodicity and transformation of the torus. Amer. J. Math.
83 (1961), 573–601.
[79] Gal, C. G.; Gal, S. G.; N’Gu´er´ekata, G. M.: Almost automorphic functions with values
in p-Fr´echet spaces. Electron. J. Diﬀerential Equations 2008, no. 21, 1–18.
[80] Gantmacher, F. R.: The theory of matrices. (Vol. 1. Reprint of the 1959 translation.)
AMS Chelsea Publishing, Providence, 1998.
[81] Georgi, H.; Glashow, S. L.: Unity of all elementary-particle forces. Phys. Rev. Lett.
32 (1974), 438–441.
[82] Gockenbach, M. S.: Finite-dimensional linear algebra. Discrete Mathematics and its
Applications (Boca Raton). CRC Press, Boca Raton, 2010.
[83] Gopalsamy, K.; Liu, P.; Zhang, S.: Almost periodic solutions of nonautonomous
linear diﬀerence equations. Appl. Anal. 81 (2002), no. 2, 281–301.
[84] Gottschalk, W. H.: Orbit-closure decompositions and almost periodic properties. Bull.
Amer. Math. Soc. 50 (1944), 915–919.
[85] Grande, R. F.: Hierarchy of almost-periodic function spaces. Rend. Mat. Appl. (7)
26 (2006), no. 2, 121–188.
[86] Guan, Y.; Wang, K.: Translation properties of time scales and almost periodic functions.
Math. Comput. Modelling 57 (2013), no. 5-6, 1165–1174.
[87] Hamaya, Y.: Existence of an almost periodic solution in a diﬀerence equation by
Lyapunov functions. Nonlinear Stud. 8 (2001), no. 3, 373–379.
[88] Han, Y.; Hong, J.: Almost periodic random sequences in probability. J. Math. Anal.
Appl. 336 (2007), no. 2, 962–974.
[89] Haraux, A.: A simple almost-periodicity criterion and applications. J. Diﬀerential
Equations 66 (1987), no. 1, 51–61.
[90] Haraux, A.: Nonlinear evolution equations. Global behavior of solutions. Lecture
Notes in Math., Vol. 841. Springer-Verlag, New York, 1981.
[91] Harmer, M.: Hermitian symplectic geometry and extension theory. J. Phys. A-Math.
Theor. 33 (2000), 9193–9203.
BIBLIOGRAPHY 176
[92] Higham, N. J.: Functions of matrices. Theory and computation. Society for Industrial
and Applied Mathematics, Philadelphia, 2008.
[93] Hong, J.; Yuan, R.: The existence of almost periodic solutions for a class of differential
equations with piecewise constant argument. Nonlinear Anal. 28 (1997),
no. 8, 1439–1450.
[94] Horn, R. A.; Johnson, C. R.: Matrix analysis. Cambridge University Press, Cambridge,
1985.
[95] Hu, Z. S.; Mingarelli, A. B.: Favard’s theorem for almost periodic processes on Banach
space. Dynam. Systems Appl. 14 (2005), no. 3-4, 615–631.
[96] Hu, Z. S.; Mingarelli, A. B.: On a question in the theory of almost periodic diﬀerential
equations. Proc. Amer. Math. Soc. 127 (1999), 2665–2670.
[97] Hu, Z. S.; Mingarelli, A. B.: On a theorem of Favard. Proc. Amer. Math. Soc.
132 (2004), no. 2, 417–428.
[98] Ishii, H.: On the existence of almost periodic complete trajectories for contractive
almost periodic processes. J. Diﬀerential Equations 43 (1982), no. 1, 66–72.
[99] Jaiswal, A.; Sharma, P. L.; Singh, B.: A note on almost periodic functions. Math.
Japon. 19 (1974), no. 3, 279–282.
[100] Jajte, R.: On almost-periodic sequences. Colloq. Math. 13 (1964/1965), 265–267.
[101] Janeczko, S.; Zajac, M.: Critical points of almost periodic functions. Bull. Polish
Acad. Sci. Math. 51 (2003), no. 1, 107–120.
[102] Johnson, R. A.: A linear almost periodic equation with an almost automorphic solution.
Proc. Amer. Math. Soc. 82 (1981), no. 2, 199–205.
[103] Johnson, R. A.: Bounded solutions of scalar almost periodic linear equations. Illinois
J. Math. 25 (1981), no. 4, 632–643.
[104] Johnson, R. M.: On a Floquet theory for almost periodic, two-dimensional linear
system. J. Diﬀerential Equations 37 (1980), no. 2, 184–205.
[105] Kacaran, T. K.; Perov, A. I.: Theorems of Favard and Bohr-Neugebauer for multidimensional
diﬀerential equations. Izv. Vysˇs. Uˇcebn. Zaved. Matematika 1968,
no. 5(72), 62–70.
[106] Katznelson, Y. M.; Katznelson, Y. R.: A (terse) introduction to linear algebra. Student
Mathematical Library, Vol. 44. American Mathematical Society, Providence,
2008.
[107] Kelley, J. L.: General topology. Graduate texts in Mathematics, Vol. 27. Springer,
New York, 1975.
BIBLIOGRAPHY 177
[108] Kelley, W. G.; Peterson, A. C.: Diﬀerence equations. An introduction with applications.
Harcourt/Academic Press, San Diego, 2001.
[109] Khan, L. A.; Alsulami, S. M.: Almost periodicity in linear topological spacesrevisited.
Commun. Math. Anal. 13 (2012), no. 1, 54–63.
[110] Kirichenova, O. V.; Kotyurgina, A. S.; Romanovski˘ı, R. K.: The method of Lyapunov
functions for systems of linear diﬀerence equations with almost periodic coeﬃcients.
Siberian Math. J. 37 (1996), 147–150.
[111] Kope´c, J.: On vector-valued almost periodic functions. Ann. Soc. Polon. Math.
25 (1952), 100–105.
[112] Kultaev, T.: Construction of particular solutions of a system of linear diﬀerential
equations with almost periodic coeﬃcients. Ukra¨ın. Mat. Zh. 39 (1987), no. 4,
526–529, 544; translation in: Ukrainian Math. J. 39 (1987), no. 4, 428–431.
[113] Kurzweil, J.; Vencovsk´a, A.: Linear differential equations with quasiperiodic coefﬁcients.
Czechoslovak Math. J. 37(112) (1987), no. 3, 424–470.
[114] Kurzweil, J.; Vencovsk´a, A.: On a problem in the theory of linear diﬀerential equations
with quasiperiodic coeﬃcients. In: Ninth international conference on nonlinear
oscillations, Vol. 1 (Kiev, 1981), 214–217, 444. Naukova Dumka, Kiev, 1984.
[115] Lakshmikantham, V.; Trigiante, D.: Theory of diﬀerence equations. Numerical methods
and applications. Monographs and Textbooks in Pure and Applied Mathematics,
Vol. 251. Marcel Dekker, Inc., New York, 2002.
[116] Lan, N. T.: On the almost periodic solutions of diﬀerential equations on Hilbert
spaces. Int. J. Diﬀer. Equ. 2009, art. ID 575939, 1–11.
[117] Levitan, B. M.: Poˇcti-periodiˇceskie funkcii. Gosudarstv. Izdat. Tehn.-Teor. Lit.,
Moscow, 1953.
[118] Levitan, B. M.; ˇZikov, V. V.: Favard theory. Uspekhi Mat. Nauk 32 (1977),
no. 2(194), 123–171, 263; translation in: Russian Math. Surveys 32 (1977), no. 2,
129–180.
[119] Li, Y.; Li, B.: Almost periodic time scales and almost periodic functions on time
scales. J. Appl. Math. 2015, art. ID 730672, 1–8.
[120] Li, Y.; Wang, C.: Almost periodic functions on time scales and applications. Discrete
Dyn. Nat. Soc. 2011, art. ID 727068, 1–20.
[121] Li, Y.; Wang, C.: Almost periodic solutions to dynamic equations on time scales and
applications. J. Appl. Math. 2012, art. ID 463913, 1–19.
[122] Lin, F.; Ni, H.: The existence and stability of almost periodic solution. Ann. Differential
Equations 23 (2007), no. 2, 173–179.
BIBLIOGRAPHY 178
[123] Lipnitski˘ı, A. V.: Lower bounds for the norm of solutions of linear diﬀerential systems
with a linear parameter. Diﬀer. Uravn. 50 (2014), no. 3, 412–416; translation in:
Diﬀer. Equ. 50 (2014), no. 3, 410–414.
[124] Lipnitski˘ı, A. V.: On the discontinuity of Lyapunov exponents of an almost periodic
linear diﬀerential system aﬃnely dependent on a parameter. Diﬀer. Uravn. 44 (2008),
no. 8, 1041–1049; translation in: Diﬀer. Equ. 44 (2008), no. 8, 1072–1081.
[125] Lipnitski˘ı, A. V.: On the singular and higher characteristic exponents of an almost
periodic linear diﬀerential system that depends aﬃnely on a parameter. Diﬀer. Uravn.
42 (2006), no. 3, 347–355, 430; translation in: Diﬀer. Equ. 42 (2006), no. 3, 369–379.
[126] Lizama, C.; Mesquita, J. G.: Almost automorphic solutions of non-autonomous difference
equations. J. Math. Anal. Appl. 407 (2013), no. 2, 339–349.
[127] Lizama, C.; Mesquita, J. G.; Ponce, R.: A connection between almost periodic functions
deﬁned on timescales and R. Appl. Anal. 93 (2014), no. 12, 2547–2558.
[128] Maak, W.: Fastperiodische Funktionen. Springer-Verlag, Berlin, 1950.
[129] Malkin, I. G.: Some problems of the theory of nonlinear oscillations. Springer-Verlag,
New York, 1991.
[130] Mauclaire, J.-L.: Suites limite-p´eriodiques et th´eorie des nombres. I. Proc. Japan
Acad. Ser. A Math. Sci. 56 (1980), no. 4, 180–182.
[131] Mauclaire, J.-L.; Sur la th´eorie des suites presque-p´eriodiques. I. Proc. Japan Acad.
Ser. A Math. Sci. 61 (1985), no. 5, 153–155.
[132] Mauclaire, J.-L.; Sur la th´eorie des suites presque-p´eriodiques. II. Proc. Japan Acad.
Ser. A Math. Sci. 61 (1985), no. 6, 190–192.
[133] Meisters, G. H.: On almost periodic solutions of a class of diﬀerential equations.
Proc. Amer. Math. Soc. 10 (1959), 113–119.
[134] Miller, A.: Almost periodicity. Old and new results. Real Anal. Exchange 2006, 30th
Summer Symposium Conference, 69–83.
[135] Minh, N. V.; Naito, T.; Shin, J. S.: Almost periodic solutions of diﬀerential equations
in Banach spaces: some new results and methods. Vietnam J. Math. 29 (2001), no. 4,
295–330.
[136] Moody, R. V.; Nesterenko, M.; Patera, J.: Computing with almost periodic functions.
Acta Crystallogr. Sect. A 64 (2008), no. 6, 654–669.
[137] Moshchevitin, N. G.: Diﬀerential equations with almost periodic and conditionally
periodic coeﬃcients: recurrence and reducibility. Mat. Zametki 64 (1998), no. 2,
229–237; translation in: Math. Notes 64 (1998), no. 1-2, 194–201.
[138] Muchnik, A.; Semenov, A.; Ushakov, M.: Almost periodic sequences. Theoret. Comput.
Sci. 304 (2003), no. 1-3, 1–33.
BIBLIOGRAPHY 179
[139] Muni, G.: Functions of S almost periodic with respect to a homeomorphic group.
Boll. Un. Mat. Ital. B (5) 17 (1980), no. 1, 232–243.
[140] Nerurkar, M. G.; Sussmann, H. J.: Construction of ergodic cocycles that are fundamental
solutions to linear systems of a special form. J. Mod. Dyn. 1 (2007), 205–253.
[141] Nerurkar, M. G.; Sussmann, H. J.: Construction of minimal cocycles arising from
speciﬁc diﬀerential equations. Israel J. Math. 100 (1997), 309–326.
[142] N’Gu´er´ekata, G. M.: Almost automorphic and almost periodic functions in abstract
spaces. Kluwer Academic/Plenum Publishers, New York, 2001.
[143] N’Gu´er´ekata, G. M.: Topics in almost automorphy. Springer-Verlag, New York, 2005.
[144] Opial, Z.: Sur les solutions presque-p´eriodiques d’une classe d’´equations diff´erentielles.
Ann. Polon. Math. 9 (1960/1961), 157–181.
[145] Ortega, R.; Tarallo, M.: Almost periodic linear diﬀerential equations with non-separated
solutions. J. Funct. Anal. 237 (2006), no. 2, 402–426.
[146] Palmer, K. J.: On bounded solutions of almost periodic linear diﬀerential systems. J.
Math. Anal. Appl. 103 (1984), no. 1, 16–25.
[147] Pankov, A. A.; Tverdokhleb, Y. S.: On almost periodic diﬀerence operators. Dopov.
Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki 2000, no. 2, 24–26.
[148] Papaschinopoulos, G.: Exponential separation, exponential dichotomy, and almost
periodicity of linear diﬀerence equations. J. Math. Anal. Appl. 120 (1986), 276–287.
[149] Petukhov, A. P.: The space of almost periodic functions with the Hausdorﬀ metric.
Mat. Sb. 185 (1994), no. 3, 69–92; translation in: Russian Acad. Sci. Sb. Math.
81 (1995), no. 2, 321–341.
[150] Pinto, M.; Robledo, G.: Cauchy matrix for linear almost periodic systems and some
consequences. Nonlinear Anal. 74 (2011), no. 16, 5426–5439.
[151] Precupanu, A.: Relations entre les fonctions et les suites presque-automorphes. An.
S¸ti. Univ. “Al. I. Cuza” Ia¸si Sect¸. I a Mat. (N.S.) 17 (1971), 37–41.
[152] Radov´a, L.: Theorems of Bohr-Neugebauer-type for almost-periodic diﬀerential equations.
Math. Slovaca 54 (2004), no. 2, 191–207.
[153] Ragimov, M. B.: On a certain theorem of Bohr. Azerba˘ıdˇzan. Gos. Univ. Uˇcen. Zap.
Ser. Fiz.-Mat. Nauk 1968, no. 4, 61–65.
[154] Ritter, G. X.: A characterization of almost periodic homeomorphisms on the 2-sphere
and the annulus. General Topology Appl. 9 (1978), no. 3, 185–191.
[155] Sell, G. R.: A remark on an example of R. A. Johnson. Proc. Amer. Math. Soc.
82 (1981), no. 2, 206–208.
BIBLIOGRAPHY 180
[156] Seynsche, I.: Zur Theorie der fastperiodischen Zahlfolgen. Rend. Circ. Mat. Palermo
55 (1931), 395–421.
[157] Sharma, N.: Almost periodic functions on fuzzy metric space. J˜n¯an¯abha 23 (1993),
129–131.
[158] Sharma, P. L.; Anki Reddy, K. C.; Funakosi, S.: Remarks on uniform space valued
almost periodic functions depending on parameters. Math. Sem. Notes Kobe Univ.
4 (1976), no. 1, 87–90.
[159] Shcherbakov, B. A.: The nature of the recurrence of the solutions of linear diﬀerential
systems. An. S¸ti. Univ. “Al. I. Cuza” Ia¸si Sect¸. I a Mat. (N.S.) 21 (1975), 57–59.
[160] Shtern, A. I.: Almost periodic functions and representations in locally convex spaces.
Uspekhi Mat. Nauk 60 (2005), no. 3(363), 97–168; translation in: Russian Math.
Surveys 60 (2005), no. 3, 489–557.
[161] Shubin, M. A.: Local Favard theory. Moscow Univ. Math. Bul. 34 (1978), no. 2,
32–37.
[162] Sljusarˇcuk, V. E.: Bounded and almost periodic solutions of diﬀerence equations in
a Banach space. In: Analytic methods for the study of the solutions of nonlinear
diﬀerential equations, 147–156. Izdanie Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev,
1975.
[163] Song, Y.: Periodic and almost periodic solutions of functional diﬀerence equations
with ﬁnite delay. Adv. Diﬀerence Equ. 2007, art. ID 68023, 1–15.
[164] Stamov, G. T.: Almost periodic solutions of impulsive diﬀerential equations. Lecture
Notes in Math., Vol. 2047. Springer, Heidelberg, 2012.
[165] Sule˘ımanov, S. P.: Almost periodic functions of several variables with respect to Hausdorﬀ
metrics and some of their properties. Akad. Nauk Azerba˘ıdˇzan. SSR Dokl.
36 (1980), no. 9, 8–11.
[166] Tarallo, M.: Fredholm’s alternative for a class of almost periodic linear systems.
Discrete Contin. Dyn. Syst. 32 (2012), no. 6, 2301–2313.
[167] Tarallo, M.: The Favard separation condition for almost periodic linear systems. J.
Dynam. Diﬀerential Equations 25 (2013), no. 2, 291–304.
[168] Thanh, N. T.: Asymptotically almost periodic solutions on the half-line. J. Diﬀerence
Equ. Appl. 11 (2005), no. 15, 1231–1243.
[169] Tkachenko, V. I.: Linear almost periodic diﬀerence equations with bounded solutions.
In: Asymptotic solutions of nonlinear equations with a small parameter, 121–124.
Akad. Nauk Ukrain., Inst. Mat., Kiev, 1991.
[170] Tkachenko, V. I.: On linear almost periodic systems with bounded solutions. Bull.
Austral. Math. Soc. 55 (1997), no. 2, 177–184.
BIBLIOGRAPHY 181
[171] Tkachenko, V. I.: On linear homogeneous almost periodic systems that satisfy the
Favard condition. Ukra¨ın. Mat. Zh. 50 (1998), no. 3, 409–413; translation in:
Ukrainian Math. J. 50 (1998), no. 3, 464–469.
[172] Tkachenko, V. I.: On linear systems with quasiperiodic coeﬃcients and bounded solutions.
Ukra¨ın. Mat. Zh. 48 (1996), no. 1, 109–115; translation in: Ukrainian Math.
J. 48 (1996), no. 1, 122–129.
[173] Tkachenko, V. I.: On reducibility of linear quasiperiodic systems with bounded solutions.
In: Proceedings of the 6th Colloquium on the Qualitative Theory of Diﬀerential
Equations (Szeged, 1999), no. 29, 1–11. Proc. Colloq. Qual. Theory Diﬀer. Equ.,
Electron. J. Qual. Theory Diﬀer. Equ., Szeged, 2000.
[174] Tkachenko, V. I.: On reducibility of systems of linear diﬀerential equations with
quasiperiodic skew-adjoint matrices. Ukra¨ın. Mat. Zh. 54 (2002), no. 3, 419–424;
translation in: Ukrainian Math. J. 54 (2002), no. 3, 519–526.
[175] Tkachenko, V. I.: On uniformly stable linear quasiperiodic systems. Ukra¨ın. Mat.
Zh. 49 (1997), no. 7, 981–987; translation in: Ukrainian Math. J. 49 (1997), no. 7,
1102–1108.
[176] Tkachenko, V. I.: On unitary almost periodic diﬀerence systems. In: Advances in
diﬀerence equations (Veszpr´em, 1995), 589–596. Gordon and Breach, Amsterdam,
1997.
[177] Tornehave, H.: On almost periodic movements. Danske Vid. Selsk. Mat.-Fysiske
Medd. 28 (1954), no. 13, 1–42.
[178] Varadarajan, V. S.: Supersymmetry for mathematicians. An introduction. In:
Courant Lecture Notes in Mathematics, Vol. 11. New York University, Courant Institute
of Mathematical Sciences, New York, 2004.
[179] Walther, A.: Fastperiodische Folgen und Potenzreihen mit fastperiodischen Koefﬁzienten.
Abh. Math. Sem. Hamburg Universit¨at VI (1928), 217–234.
[180] Wang, C.; Agarwal, R. P.: A further study of almost periodic time scales with some
notes and applications. Abstr. Appl. Anal. 2014, art. ID 267384, 1–11.
[181] Weston, J. D.: Almost periodic functions. Mathematika 2 (1955), 128–131.
[182] Yang, X. T.: Stability and boundedness of solutions and existence of almost periodic
solutions for diﬀerence equations. Acta Math. Sci. Ser. A Chin. Ed. 28 (2008), no. 5,
870–878.
[183] Yoshizawa, T.: Stability theory and the existence of periodic solutions and almost
periodic solutions. Applied Mathematical Sciences, Vol. 14. Springer-Verlag, New
York, 1975.
[184] Zaidman, S.: Almost-periodic functions in abstract spaces. Research Notes in
Mathematics, Vol. 126. Pitman Advanced Publishing Program, Boston, 1985.
BIBLIOGRAPHY 182
[185] Zhang, H.; Li, Y.: Almost periodic solutions to dynamic equations on time scales. J.
Egyptian Math. Soc. 21 (2013), no. 1, 3–10.
[186] Zhang, S.: Almost periodicity in diﬀerence systems. In: New trends in diﬀerence
equations (Temuco, 2000), 291–306. Taylor & Francis, London, 2002.
[187] Zhang, S.: Almost periodic solutions for diﬀerence systems and Lyapunov functions.
In: Differential equations and computational simulations (Chengdu, 1999), 476–481.
World Sci. Publ., River Edge, 2000.
[188] Zhang, S.: Almost periodic solutions of diﬀerence systems. Chinese Sci. Bull.
43 (1998), no. 24, 2041–2046.
[189] Zhang, S.: Existence of almost periodic solutions for diﬀerence systems. Ann. Differential
Equations 16 (2000), no. 2, 184–206.
[190] Zhang, S.; Zheng, G.: Almost periodic solutions of delay diﬀerence systems. Appl.
Math. Comput. 131 (2002), no. 2-3, 497–516.
[191] Zhang, S.; Zheng, G.: Existence of almost periodic solutions of neutral delay difference
systems. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 9 (2002),
no. 4, 523–540.