Limit periodic linear difference systems with coefficient matrices from commutative groups

In this paper, limit periodic and almost periodic homogeneous linear difference systems are studied. The coefficient matrices of the considered systems belong to a given commutative group. We 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 problem is 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.


Introduction
For a given commutative group X , we intend to analyse the homogeneous linear difference systems We will consider limit periodic and almost periodic systems (1.1), which means that the sequence of A k will be limit periodic or almost periodic.The basic motivation of this paper comes from [29,35].
In [29] (see also [26]), there are studied systems (1.1) for X being the unitary group and there is proved that, in any neighbourhood of an almost periodic system (1.1), there exist almost periodic systems (1.1) whose fundamental matrices are not almost periodic.The corresponding result about orthogonal difference, skew-Hermitian and skew-symmetric differential systems can be found in [30], [32], and in [34] (see also [27]), respectively.For results concerning almost periodic solutions, we refer to [16,17,28,30], where unitary, orthogonal, skew-Hermitian, and skew-symmetric systems are analysed.In our previous works [13,33], 2 P. Hasil and M. Veselý the above mentioned result of [29] is improved for a general (weakly) transformable group X .We remark that the process from [29] cannot be applied for commutative groups of coefficient matrices which are treated in this paper.
In [35], the study of non-almost periodic solutions of limit periodic systems (1.1) has been initiated and the so-called property P has been introduced.The concept of groups with property P leads to results of the same type as the main results of [13,33].It should be noted that only bounded groups of matrices are treated in [35].The goal of this paper is to prove for other groups of matrices that, in any neighbourhood of a system (1.1), there exist systems (1.1) which have at least one non-almost periodic solution.Moreover, we deal with the corresponding Cauchy problems.For this purpose, we generalize the notion of property P (we introduce property P with respect to a given non-trivial vector) and we use the generalization to obtain the announced results for groups which can be unbounded.Especially, for the used modification of property P, it holds that any group which contains a group with the innovated property has this property as well.
The fundamental properties of limit periodic and almost periodic sequences and functions can be found in a lot of monographs (see, e.g., [4,10,18,24]).Almost periodic solutions of almost periodic linear difference equations are studied in articles [6,7,8,12,14,37].Properties of complex almost periodic systems (1.1) are discussed, e.g., in [3,15,23].In the situation when index k attains only positive values, linear almost periodic equations are treated, e.g., in [1,25].To the best of our knowledge, the first result about non-almost periodic solutions of homogeneous linear difference equations was obtained in [11].
We prove the announced results using constructions of limit periodic sequences.This approach is motivated by the continuous case (special constructions of homogeneous linear differential systems with almost periodic coefficients are used, e.g., in [19,20,21,22,32,34]).Note that the process applied in this paper is substantially different from the ones in all above mentioned papers.Hence, we obtain new results even for almost periodic systems and bounded groups of coefficient matrices.
This paper is organized as follows.In the next section, we mention the notation which is used throughout the whole paper.Then, in Section 3, we define limit periodic, almost periodic, and asymptotically almost periodic sequences and we state their properties which we will need later.In Section 4, we treat the considered homogeneous linear difference systems, where we recall the definitions and results which motivate our recent research and which give the necessary background of the studied problems.In the final section, we formulate and prove our results which are commented by several remarks.

Preliminaries
At first, we mention the used notation which is similar to the one from [35].For arbitrary p ∈ N, we put pN := {pj : j ∈ N}.Let (F, ⊕, ) be an infinite field.Let | • | : F → R be an absolute value on F; i.e., let Theorem 3.6.Let {ϕ k } k∈Z ⊆ S be given.The sequence {ϕ k } is almost periodic if and only if any sequence {l n } n∈N ⊆ Z has a subsequence { ln } n∈N ⊆ {l n } such that, for any ε > 0, there exists K(ε) ∈ N satisfying Proof.See, e.g., [31,Theorem 2.3].
Corollary 3.7.Let p ∈ N be arbitrarily given and let {ϕ k } k∈Z ⊆ S be almost periodic.For any ε > 0, the set of all ε-translation numbers l ∈ pN of {ϕ k } is infinite.
Proof.It suffices to apply Theorem 3.6 for l n := pn, n ∈ N. Indeed, it holds Using Theorem 3.6 n-times, we also obtain the following result.
is almost periodic if and only if all sequences {ϕ 1 k }, . . ., {ϕ n k } are almost periodic.Definition 3.9.We say that a sequence {ϕ k } k∈Z ⊆ S is asymptotically almost periodic if, for every ε > 0, there exist r(ε), R(ε) ∈ N such that any set consisting of r(ε) consecutive integers contains at least one number l satisfying Remark 3.10.Considering Theorem 3.5, we know that any limit periodic sequence is almost periodic.In addition, any almost periodic sequence is evidently asymptotically almost periodic.Note that, in Banach spaces, a sequence is asymptotically almost periodic if and only if it can be expressed as the sum of an almost periodic sequence and a sequence vanishing at infinity (see, e.g., [36,Chapter 5]).

Homogeneous linear difference systems over a field
In this section, we describe the studied systems in more details.Let X ⊂ Mat(F, m) be an arbitrarily given group.We recall that we will analyse homogeneous linear difference systems (1.1).Let LP (X ) denotes the set of all systems (1.1) for which the sequence of matrices A k is limit periodic.Analogously, the set of all almost periodic systems (1.1) will be denoted by AP (X ).Especially, we can identify the sequence {A k } with the system in the form (1.1) which is determined by {A k }.In AP (X ), we introduce the metric Henceforth, the symbol O σ ε ({A k }) will denote the ε-neighbourhood of {A k } in AP (X ).Now we recall a definition from [35] which is used in the formulations of the below given Theorems 4.2 and 4.3 (for their proofs, see [35]).We point out that Theorems 4.2 and 4.3 are the basic motivation for our current research.Definition 4.1.We say that 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 ∈ F m satisfying u ≥ 1, one can find matrices N 1 , N 2 , . . . ,N l ∈ X with the property that Theorem 4.2.Let X be bounded and have property P. For any {A k } ∈ LP (X ) and ε > 0, there exists a system {B k } ∈ O σ ε ({A k }) ∩ LP (X ) which does not have any non-zero asymptotically almost periodic solution.
Theorem 4.3.Let X be bounded and have property P. For any {A k } ∈ AP (X ) and ε > 0, there exists a system {B k } ∈ O σ ε ({A k }) which does not have any non-zero asymptotically almost periodic solution.
In this paper, we intend to improve the above theorems.To show how the presented results improve Theorems 4.2 and 4.3, we need to reformulate Definition 4.1 for bounded groups applying the next two lemmas (which we will need later as well).
Lemma 4.4.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 4.5.Let a bounded group X ⊆ Mat(F, m) be given.There exists L > 1 such that Proof.We know that the inequality 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 Hence, the map has the Lipschitz property as well.Let a matrix M ∈ X be given.If we use the expression where m −1 i,j are elements of M −1 ∈ X and M j,i are the algebraic complements of the elements m j,i of M, then it is seen that the map M → M −1 is continuous in the Lipschitz sense on X. Evidently, Lemma 4.4 and the Lipschitz continuity of M → M −1 on X imply the existence of L > 1 for which (4.1) is valid.

P. Hasil and M. Veselý
Using Lemmas 4.4 and 4.5 for bounded X and for we can rewrite Definition 4.1 as follows.
Definition 4.6.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 ∈ F m satisfying u ≥ 1, one can find matrices M 1 , M 2 , . . ., M l ∈ X with the property that To formulate the obtained results in a simple and consistent form, we introduce the following direct generalization of Definition 4.6.Definition 4.7.Let a non-zero vector u ∈ F m 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 Remark 4.8.Since a group with property P has property P with respect to any non-zero vector u (consider ), we can refer to a lot of examples of matrix groups with property P mentioned in our previous paper [35].In [35], there is also proved the following implication.If a complex transformable matrix group contains a matrix M satisfying Mu = u for a vector u ∈ C m , then the group has property P with respect to u.Thus, concerning examples of groups having property P with respect to a given vector, we can also refer to our articles [13,33], where (weakly) transformable groups are studied.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.

Results
Henceforth, we will assume that X is commutative.To prove the announced result (the below given Theorem 5.3), we use Lemmas 5.1 and 5.2.
Lemma 5.1.Let {A k } ∈ LP (X ) and ε > 0 be arbitrarily given.Let {δ n } n∈N ⊂ R be a decreasing sequence satisfying lim n→∞ δ n = 0 (5.1) ) Proof.Condition (5.3) means that, for any k ∈ Z, there exists i ∈ N such that Especially, the definition of {B k } k∈Z is correct and We show that {B k } is limit periodic.Since {A k } is limit periodic and A k ∈ X , k ∈ Z, there exist periodic sequences {C n k } k∈Z ⊂ X for n ∈ N with the property that Hence (see (5.2), (5.3), (5.6)), we have for all k ∈ Z, n ∈ N. Considering (5.1), we get that {B k } is the uniform limit of the sequence of periodic sequences for some i ∈ N and for all k ∈ Z. Thus (see (5.4)), we obtain (5.7).
Proof.We can assume that all solutions of {A k } are almost periodic.Especially (consider Corollary 3.8), for any ϑ > 0, there exist infinitely many positive integers p with the property that Let {δ n } n∈N ⊂ R be a decreasing sequence satisfying (5.1) and (5.4).For δ n and K n := n, n ∈ N, we consider matrices and Let a sequence of positive numbers ϑ n for n ∈ N be given.Let us consider p 1 1 , p 1 2 ∈ N such that p 1 2 − p 1 1 > 2l 1 and that (see (5.8)) (5.11) In addition, let p 1 1 and p 1 2 be even (consider Corollary 3.7).We define the periodic sequence {B 1 k } k∈Z with period p 1 2 by values Again, we can assume that, for any ϑ > 0, there exist infinitely many positive integers p with the property that (5.12) Otherwise, we obtain the system {B k } ≡ { B1 k } with a non-almost periodic solution.Indeed, it suffices to consider Lemma 5.1 for 2 and (see (5.12)) B1 Especially, for all k ∈ Z, there exists i ∈ {1, 2} such that B2 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 In the n-th step, we consider Bn−1 (5.15) Finally, we put From the construction, we obtain that, for any k ∈ Z, there exists i ∈ N such that B k = A k • B i k .It means that (5.3) is satisfied.Since (5.2) follows from the construction and from (5.9), we can use Lemma 5.1 which guarantees that {B k } ∈ O σ ε ({A k }) ∩ LP (X ).It remains to prove that the fundamental matrix of {B k } is not almost periodic.On contrary, let us assume its almost periodicity.Then, the fundamental matrix is bounded (see Remark 3.4); i.e., there exists K 0 > 0 with the property that (5.16) Let us choose n ∈ N for which n ≥ K 0 + 1.We repeat that the multiplication of matrices is continuous (see also Lemma 4.4).Hence, for given matrix (5.17) We can assume that ϑ n = θ n in (5.14) (see also (5.11), (5.13)).We construct sequences {B (5.18) This contradiction (cf.(5.16) and (5.18)) completes the proof.
Theorem 5.3.Let X have property P with respect to a vector u.For any {A k } ∈ LP (X ) and ε > 0, there exists a system {B k } ∈ O σ ε ({A k }) ∩ LP (X ) whose fundamental matrix is not almost periodic.Proof.Let us consider the solution {x 0 k } k∈Z of the Cauchy problem If {x 0 k } is not almost periodic, then the statement of the theorem is true for B k := A k , k ∈ Z.Hence, we assume that {x 0 k } is almost periodic.We put , n ∈ N.
We continue in the same manner.Before the n-th step, we define Bn−1 Again, we consider that the sequence {x n−1 k } is almost periodic.Otherwise, we can put B k := Bn−1 k , k ∈ Z. Especially, for all p ∈ N, there exist infinitely many numbers j ∈ pN with the property that Denote Let us consider an integer j (1,n−1) ∈ p n N satisfying (5.37) and j (1,n−1) ≥ q n−1 . (5.40) We define {B we put B (1,n) k := I, k ∈ Z.In the other case, we put we consider the solution {x Again, we assume that {x
Especially, for all i = j, i, j ∈ N, there exists l ∈ Z such that This contradiction (consider (5.50) for 2ξ ≤ ϑ) proves that {x k } is not almost periodic.
Remark 5.4.It is seen that the statement of Theorem 5.3 does not change if we replace system {A k } ∈ LP (X ) by a periodic one.Indeed, it follows directly from Definition 3.1.
Remark 5.5.To illustrate Theorem 5.3, let us consider an arbitrary periodic system {M k } in the complex case (i.e., for F = C with the usual absolute value).It means that we have a system for a positive integer p and arbitrarily given non-singular complex matrices M 0 , M 1 , . . ., M p−1 .
It is well-known that a solution of {M k } is almost periodic if and only if it is bounded (see, e.g., [33,Corollary 3.9] or [35,Theorem 5]).The fundamental matrix Φ(k, 0) of {M k } satisfying Φ(0, 0) = I is given by Thus, to describe the structure of almost periodic solutions, it suffices to consider the multiples and, in fact, the constant system 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 5.3, 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 5.6.We repeat that the basic motivation of this paper comes from [35], where nonasymptotically almost periodic solutions of limit periodic systems are considered.Of course, systems with coefficient matrices from bounded groups are analysed in [35].For general groups, it is not possible to prove the main results of [35], i.e., Theorems 4.2 and 4.3.It suffices to consider the constant system given by matrix I/2 in the complex case.Any solution {x k } k∈Z of this system has the property that Thus, there exists a neighbourhood of the system such that, for any solution {y k } k∈Z of an almost periodic system from the neighbourhood, we obtain lim k→∞ y k = 0, which gives the asymptotic almost periodicity of {y k } (see Remark 3.10).At the same time, in [35], 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 5.3, we can apply this theorem for groups of matrices in the following form , where X is taken from a commutative matrix group having property P with respect to a concrete vector.In this sense, Theorem 5.3 generalizes Theorem 4.2 as well.
The construction from the proof of Theorem 5.3 can be applied for the Cauchy (initial) problem.Especially, we immediately obtain the following result.Theorem 5.7.Let a non-zero vector u ∈ F m 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 M 1 , M 2 , . . ., M l ∈ X satisfying For any {A k } ∈ LP (X ) and ε > 0, there exists a system {B k } ∈ O σ ε ({A k }) ∩ LP (X ) for which the solution of x k+1 = B k • x k , k ∈ Z, x 0 = u is not almost periodic.
P. Hasil and M. Veselý Proof.The theorem follows from the proof of Theorem 5.3, where (5.61) is satisfied (i.e., the case, which is covered by Lemma 5.2, does not happen).
Similarly to Theorem 4.3 which is the almost periodic version of Theorem 4.2, we formulate the below given Theorem 5.10 as the almost periodic version of Theorem 5.3.To prove it, we need the next two lemmas.Lemma 5.8.Let {A k } ∈ AP (X ) and ε > 0 be arbitrarily given.Let {δ n } n∈N ⊂ R be a decreasing sequence satisfying (5.1) and let {B n k } k∈Z ⊂ X be periodic sequences for n ∈ N such that (5.2) and (5.3) are valid.Then, {B k } ∈ AP (X ) if In addition, {B k } ∈ O σ ε ({A k }) if (5.4) is fulfilled.Proof.The lemma can be proved analogously as Lemma 5.1.In the proof of Lemma 5. Using the same way which is applied in the proof of Lemma 5.2, we can prove its almost periodic counterpart.Indeed, we do not use the limit periodicity of {A k } in the proof (consider also Lemma 5.8).Lemma 5.9.If for any δ > 0 and K > 0, there exist matrices M 1 , M 2 , . . ., M l ∈ X such that then, for any {A k } ∈ AP (X ) and ε > 0, there exists a system {B k } ∈ O σ ε ({A k }) whose fundamental matrix is not almost periodic.Theorem 5.10.Let X have property P with respect to a vector.For any {A k } ∈ AP (X ) and ε > 0, there exists a system {B k } ∈ O σ ε ({A k }) whose fundamental matrix is not almost periodic.Proof.The theorem can be proved using the same construction as Theorem 5.3.It suffices to replace Lemma 5.1 by Lemma 5.8 and Lemma 5.2 by Lemma 5.9.
Analogously, we get the following result as well.
Theorem 5.11.Let a non-zero vector u ∈ F m 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 M 1 , M 2 , . . ., M l ∈ X satisfying For any {A k } ∈ AP (X ) and ε > 0, there exists a system {B k } ∈ O σ ε ({A k }) for which the solution of x k+1 = B k • x k , k ∈ Z, x 0 = u is not almost periodic.
Remark 5.12.We add that Theorems 5.10 and 5.11 do not follow from Theorems 5.3 and 5.7.Indeed, in [5], there is proved that there exist systems which are almost periodic and which are not limit periodic (e.g., the sequence {e ik } k∈Z is almost periodic and, at the same time, it is not limit periodic).It means that there exist almost periodic systems which have neighbourhoods without limit periodic systems.

. 14 ) 10 P
We define the periodic sequence {B n k } k∈Z with period p n 2 by values B n 0 := I, B n 1 := I, . . ., B n p n