Non-almost periodic solutions of limit periodic and almost periodic homogeneous linear difference systems

We study limit periodic and almost periodic homogeneous linear difference systems. The coefficient matrices of the considered systems are taken from a given commutative group. We mention a condition on the group which ensures that, by arbitrarily small changes, the considered systems can be transformed to new systems, which do not possess any almost periodic solution other than the trivial one. The elements of the coefficient matrices are taken from an infinite field with an absolute value.


Introduction
In this paper, for a commutative group X of square matrices over a field, we analyse the homogeneous linear difference systems where {A k } k∈Z ⊆ X.We consider the case, when the sequence {A k } k∈Z is limit periodic or almost periodic.We continue in the research based on the results of papers [8,9,18,22,24].
In [18] (see also [16]), the unitary systems of the form (1.1) are considered.One of the main results of [18] says that the systems with non-almost periodic solutions form a dense subset of the space of all unitary systems.If one is interested in orthogonal difference systems and skew-Hermitian and skew-symmetric differential systems, the corresponding result can be found in [19], [21], and [23], respectively.Concerning almost periodic solutions of these systems, we refer to [12,13,17] as well.
In [8,22], general almost periodic systems (1.1) are examined.There are found groups of matrices such that the homogeneous linear difference systems without any non-trivial almost 2 M. Chvátal periodic solution form a dense subset of the set of all considered systems.Transformable and strongly transformable groups of matrices are introduced.Based on this concept, the above-mentioned result of [18] is generalized for other matrix groups.
In papers [9,24], the limit periodic systems of the form (1.1) are investigated, where matrices A k are taken from a commutative group or from a bounded group.It is shown that any of the systems can be transformed to a new system, which does not possess any non-zero (asymptotically) almost periodic solution.Our goal is to improve the results of [9,24] about systems of the form (1.1) with regard to their non-almost periodic solutions.Furthermore, we recall the corresponding Cauchy problem.Note that the presented results are new even for complex matrix groups.
The fundamental properties of limit periodic and almost periodic sequences or functions have been studied closely.One can easily find many relevant monographs.Here we point out only the books [3,6,15].Concerning almost periodic solutions of linear almost periodic difference systems, we can refer to [4,5,25] (see also [7,10,26]).Other properties of (complex) almost periodic systems can be found in [1,11,14].The properties of limit periodic homogeneous linear difference systems with respect to their almost periodic solutions are mentioned, e.g., in [9,24].
This paper is divided into five sections as follows.First, in the next section, the definitions of limit and almost periodicity are recalled.In Section 3, we introduce the used notations.In Section 4, we collect auxiliary results, which we use in the proof of the main result.Finally, in Section 5, we formulate and prove our main result.

Limit and almost periodicity
In this section, we recall the definitions of limit periodic and almost periodic sequences in a metric space (M, ρ).Definition 2.1.We say that a sequence {ϕ k } k∈Z is limit periodic if there exists a sequence of periodic sequences {ϕ n k } k∈Z ⊆ M, n ∈ N, such that lim n→∞ ϕ n k = ϕ k and the convergence is uniform with respect to k ∈ Z.
Remark 2.2.The limit periodicity can be introduced in another equivalent way (see [2]).Definition 2.3.A sequence {ϕ k } k∈Z ⊆ M is called almost periodic if for any ε > 0 there exists r(ε) ∈ N such that any set consisting of r(ε) consecutive integers contains at least one number l satisfying The definition mentioned above is the so-called Bohr definition of almost periodicity.The almost periodicity can be defined in another equivalent way (see the next theorem).This concept is the so-called Bochner definition.Theorem 2.4.Let {ϕ k } k∈Z ⊆ M be given.The sequence {ϕ k } k∈Z is almost periodic if and only if any sequence {l n } n∈N ⊆ Z has a subsequence { ln } n∈N ⊆ {l n } n∈N such that, for any ε > 0, there exists K(ε) ∈ N satisfying Proof.See, e.g., [20,Theorem 2.3].
Limit periodic linear difference systems 3

Preliminaries
In the whole paper, we will consider an infinite field F with an absolute value | • | : F → R.
Let m ∈ N be arbitrarily given.We denote the set of all m × m matrices with elements in F by the symbol Mat m (F).The absolute value gives the norms • on F m and Mat m (F) as the sum of the absolute values of elements.The absolute value and norms induce metrics on F and F m , Mat m (F), respectively.We denote δ-neighbourhoods by symbol O δ in all considered metric spaces.Let X ⊆ Mat m (F).We repeat that X is a commutative group.The set of all limit periodic and almost periodic sequences with values in X will be denoted by LP(X) and AP(X), respectively.In these sets, we consider the metric For the reader's convenience, we also denote δ-neighbourhoods of sequences in LP(X) and AP(X) by symbol O δ .
Instead of {Z k } k∈Z , we will shortly write {Z k }.If index k will be taken from another set, we will specify it at the corresponding place.The identity matrix will be denoted by I.The zero vector will be denoted by 0. Symbol ε stands for a positive real number.
In the definitions given below, we recall (and slightly generalize) the property P from [9].
Definition 3.1.We say that group X has property P if there exists ζ > 0 such that for every δ > 0 there exists l ∈ N such that for every u ∈ F m fulfilling u ≥ 1 there exist matrices M 1 , M 2 , . . . ,M l ∈ X with the property that Definition 3.2.Let u ∈ F m be an arbitrary non-zero vector.We say that group X has property P with respect to u if there exists ζ > 0 such that for every δ > 0 there exist matrices M 1 , M 2 , . . . ,M l ∈ X with the property that

Auxiliary results
Definition 3.1 can be apprehended in a little larger sense using the following lemma.
Lemma 4.1.For any a > 0, there exists f (a) Now we recall two known lemmas, which we need to prove the main result of this paper.
Remark 4.7.Lemma 4.6 remains true if LP(X) is replaced by AP(X) in the statement of this lemma, which gives [9, Lemma 5.8.].

Results
Now, we can prove the main result.We repeat that X ⊆ Mat m (F) is a commutative group.
Theorem 5.1.Let X have property P and ε > 0 be arbitrary.Then, for every {A k } ∈ LP(X) and every sequence {u n } n∈N of non-zero vectors u n ∈ F m , there exists is not almost periodic for any n ∈ N.
Proof.Let ε > 0 be arbitrary.Let ζ be taken from Definition 3.1.We use the following construction.
Again, we denote R , k ∈ Z. Next, we consider the initial problem } be its solution.Then there exists a positive integer j(2, 1, 2) divisible by 16 satis- , there exist matrices taken from Definition 3.1.We define the periodic sequence {S in the following way.Denote a ( ≤ 1/2, then we put . . .

S
(2,1,2) , k ∈ Z.We consider the system with the initial value M.
We continue the construction in the same way.Before the n-th step, we have ) Let us consider the initial problem At the end of this section, we mention some results nearly related with Theorem 5.1.
Corollary 5.2.Let X have property P with respect to a vector u.For any {A k } ∈ LP(X) and ε > 0, there exists a system {S k } ∈ O ε ({A k }) ∩ LP(X) whose fundamental matrix is not almost periodic.
Remark 5.3.The previous corollary is the main result of [9].It shows how our result generalizes the result which is the basic motivation.
The next theorem is a modification of Theorem 5.1, where coefficient matrices are taken from AP(X) instead of LP(X).
Theorem 5.4.Let X have property P and ε > 0 be arbitrary.Then, for every {A k } ∈ AP(X) and every sequence {u n } n∈N of non-zero vectors u n ∈ F m , there exists {S k } ∈ O ε ({A k }) such that the solution of x k+1 = S k x k , x 0 = u n is not almost periodic for any n ∈ N.
Proof.The proof of this theorem can be obtained in the same way as the one of Theorem 5.1.In fact, the same construction can be used.It suffices to modify Lemma 4.6 (see Remark 4.7).
In addition, we obtain the following result (from the proof of Theorem 5.1).
Theorem 5.5.Let ε > 0 be arbitrary.Let X have the property that there exist ζ > 0 and open sets U i , i ∈ N, fulfilling U i ⊆ F m , F m \ O 1 (0) ⊆ i∈N U i ⊆ F m \ {0}, such that, for every δ > 0, there exists l ∈ N such that, for every j ∈ N, there exist matrices M 1 , M 2 , . . ., M l ∈ X with the property that For every {A k } ∈ LP(X), there exists {S k } ∈ O ε ({A k }) ∩ LP(X) such that all non-zero solutions of x k+1 = S k x k are not almost periodic.For every {B k } ∈ AP(X), there exists {R k } ∈ O ε ({B k }) such that all non-zero solutions of x k+1 = R k x k are not almost periodic.
where e denotes the identity element of F. If there exists v ∈ F m , v ≥ a, v < 1, then there exists element g ∈ F such that |g| ≤ v < 1.Thus, lim n→∞ |g n | = lim n→∞ |g| n = 0 and, consequently, we have that limn→∞ |g −n | = lim n→∞ |g −1 | n = ∞.It means that there exists N ∈ N such that |g −N | ≥ 1/a.It suffices to put f (a) = g −N .Remark 4.2.According to Lemma 4.1, Definition 3.1 can be used to vectors satisfying u ≥ a in the following way.Instead of u, we apply the definition for vector f (a) • u.Then the corresponding inequality takes the form We can assume that ω(a) is a non-decreasing function of a.It follows from Remark 4.2 and from the proof of Lemma 4.1.