Asymptotically almost periodic solutions of limit and almost periodic linear difference systems

. In this paper, limit periodic and almost periodic homogeneous linear difference systems are considered. We study the systems in which the coefﬁcient matrices are taken from a given bounded group and the elements of the matrices are from an inﬁnite ﬁeld with an absolute value. We show a condition on limit periodic and almost periodic systems which ensures, that the considered systems can be transformed into new systems having certain properties. The new systems possess non-asymptotically almost periodic solutions. The transformation can be done by arbitrarily small changes.


Introduction
We consider the homogeneous linear difference systems of the form where A k ∈ X.We suppose, that X is a bounded group of square matrices over an infinite field.The cases, when sequences {A k } are limit periodic and almost periodic, are studied.We are interested in non-asymptotically almost periodic solutions of the considered systems.Our current research is motivated by the following two facts.The smallest class of systems (1.1), which can have at least one non-asymptotically almost periodic solution and which generalize the pure periodic case, is formed by the limit periodic systems.The most studied class is given by the almost periodic systems.
Our main motivation comes from papers [7,12,13,22,23,26].Papers [22] and [23] (and also [20]) are devoted to unitary and orthogonal homogeneous linear difference systems (1.1).It is shown in [22,23], that, in any neighbourhood of any orthogonal or unitary system, there exists a system of the form (1.1) with a non-almost periodic solution.In papers [7,12,13,26], Email: chvatal.m@mail.muni.czgeneral systems of the form (1.1) are studied.In [7,13], it is supposed, that X is a commutative group.In [12,26], transformable groups are studied.The results of these papers say that, in an arbitrary neighbourhood of any considered system (1.1), there exists a system of the same form without any almost periodic solution other than the trivial one.Our main goal is to complement these results.We investigate more general situations and show, that the systems of the form (1.1) with non-asymptotically almost periodic solutions form a dense subset of the set of all considered systems as well.To prove this result, we improve the method based on constructions introduced in papers [24,25].
The almost periodic (and also limit periodic) systems are studied closely.There are many papers from the field of almost periodic linear systems.In this paragraph, we point out the most relevant of them.In books [4,8,10,19,29], one can find the basic properties of limit periodic and almost periodic sequences and functions.The linear almost periodic equations, with regard to the almost periodicity of their solutions, are analyzed in, e.g., [1,30].For general difference systems, criteria of the existence of almost periodic solutions are presented in [31,32].Concerning linear almost periodic difference systems and their almost periodic solutions, we can refer to [5,6,30] (and also [11,14]).We refer to papers [2,15,18] for other properties of (complex) almost periodic systems.The findings about the skew-Hermitian and skew-symmetric differential systems, which correspond to ones from [22,23], can be found in [25] and [27], respectively.For almost periodic solutions of these systems, we can refer to [16,17,21] as well.Further, if one considers limit periodic homogeneous linear difference systems with respect to their almost periodic solutions, then the properties of such systems can be found in [7,13,28].
This paper is organized as follows.In the next section, we introduce the notation that is used in the whole paper, and we recall some elementary properties of infinite fields with absolute values.In Section 3, we recall the definitions of limit periodicity, almost periodicity, and asymptotic almost periodicity.To define these notions, we recall the Bohr and also the Bochner concept.In the final section, we give the basic motivation explicitly and we formulate and prove the main theorem.

Preliminaries
Let F be an infinite field.Let | • | : F → R be an absolute value on F.Then, the properties hold for every f , g ∈ F, where symbol 0 stands for the real number and, at the same time, for the zero element of F. Note that we will later denote also the zero vector and the zero matrix by the same symbol.Let m ∈ N be arbitrarily given.We denote the set of all square matrices of dimension m with elements in F by symbol Mat m (F) and the set of all m × 1 vectors with elements in F by symbol F m .Using the absolute value, we can define the norms • on F m , Mat m (F) as the sums of the absolute values of the elements.We have We denote the identity matrix in Mat m (F) as I.The absolute value on F and the norms on F m , Mat m (F) induce the corresponding metrics.For simplicity, each of these metrics will be denoted by symbol (•, •).Then, we consider the δ-neighbourhoods in these metrics as Let X ⊂ Mat m (F) be a bounded group.In particular, for every matrix A ∈ X, there exists the inverse matrix A −1 ∈ X and a number H > 0 satisfying A ≤ H for every A ∈ X.Let us denote the set of all limit periodic and almost periodic sequences in X by symbol LP (X) and AP (X), respectively.For the notion of limit and almost periodicity, see Definitions 3.1 and 3.2 below.In AP (X), we consider the metric For the reader's convenience, the δ-neighbourhoods in this set are again denoted by O δ .We put N 0 = N ∪ {0}.

Limit, almost, and asymptotic almost periodicity
We recall the definitions of limit periodic, almost periodic, and asymptotically almost periodic sequences and we mention their properties, which we will need in the proof of the main theorem.The general metric space (M, ) is considered.First, we recall the definition of limit periodicity.Note that it can be defined in another equivalent manner (see [3]).Definition 3.1.We say that a sequence {ϕ k } k∈N 0 is limit periodic if there exists a sequence of periodic sequences {ϕ n k } k∈N 0 ⊆ M, n ∈ N, such that lim n→∞ ϕ n k = ϕ k and the convergence is uniform with respect to k ∈ N 0 .
Next, we recall the concept of almost periodicity.It can be also defined in several equivalent ways.As a definition, we remind the so-called Bohr concept of almost periodicity.We also recall the so-called Bochner concept in the theorem below.
To complete this section, we also recollect the definition of asymptotic almost periodicity.

Definition 3.4.
A sequence {ϕ k } k∈N 0 ⊆ M is called asymptotically almost periodic if, for any ε > 0, there exists r(ε) ∈ N and m(ε) ∈ N such that any set consisting of r(ε) consecutive positive integers contains at least one number l ∈ N satisfying Note that, in Banach spaces, any asymptotically almost periodic sequence is the sum of an almost periodic sequence and a sequence, which vanishes at infinity.Similarly, as in the case of almost periodicity, we remind the equivalent concept of asymptotic almost periodicity.

Results
In the beginning of this section, we call up the most relevant known results.By doing this, one can see, how our result complements our motivations.
Theorem 4.1.Let X ⊆ Mat m (F) be a commutative group.Let, for every non-zero vector u ∈ F m , there exist ξ > 0 such that, for every δ > 0, there exist matrices M 1 , . . ., M l ∈ X satisfying Let ε > 0 and a non-zero vector u ∈ F m be arbitrary.For any {A k } k∈N 0 ∈ LP (X ), there exists Proof.See [13].
Theorem 4.2.Let X ⊆ Mat m (F) be a commutative group.Let there exist ξ > 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 l ∈ X with the property that (4.1) is valid.Let ε > 0 be arbitrary.Then, for every {A k } k∈N 0 ∈ 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.
Theorem 4.3.Let (F, ) be separable.Let X ⊆ Mat m (F) be a bounded group.Let there exist ξ > 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 l ∈ X with the property that Let ε > 0 be arbitrary.Then, for every {A k } k∈N 0 ∈ LP (X ), there exists does not have any non-zero asymptotically almost periodic solution.
Theorem 4.4.Let (F, ) be separable.Let X ⊆ Mat m (F) be a bounded group.Let there exist ξ > 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 l ∈ X with the property that (4.2) is valid.Let ε > 0 be arbitrary.Then, for every {A k } k∈Z ∈ AP (X ), there exists does not have any non-zero asymptotically almost periodic solution.
For the reader's convenience (see Theorem 4.6 below), we recall the definitions of transformable and weakly transformable groups (for further informations, see, e.g., [12,26]).Definition 4.5.We say that an infinite set X ⊆ Mat m (F) is transformable, if it meets the following conditions: (ii) for any L ∈ (0, ∞) and ε > 0, there exists p = p(L, ε) ∈ N such that, for any n ≥ p (n ∈ N) and any sequence {C 0 , C 1 , . . ., C n } ⊂ X , L ≤ (C i , 0), i ∈ {0, . . ., n}, one can find a sequence {D 1 , . . ., D n } ⊂ X for which (iii) the multiplication of matrices is uniformly continuous on X and has the Lipschitz property on a neighbourhood of I in X ; (iv) for any L ∈ (0, ∞), there exists The group X is weakly transformable if there exist a transformable group X 0 ⊂ X , matrices X 1 , . . . ,X l ∈ X , and δ X > 0 such that the following conditions hold: (i) any U ∈ X can be expressed as U = C(U) • X j for some C(U) ∈ X 0 , j ∈ {1, . . ., l}; Theorem 4.6.Let (F, ) be complete.Let X ⊆ Mat m (F) be weakly transformable.Let there exist a sequence {M i } n∈N ⊆ X 0 such that, for any non-zero vector u ∈ F m , one can find i does not possess a non-trivial almost periodic solution.
Before we formulate the main result of this paper, we recall some elementary properties of the bounded group X.We use them in the proof of the main theorem.Lemma 4.7.Let V k ∈ X and M k ∈ X, k ∈ {0, . . ., K}, be given matrices.Then, there exist matrices T k ∈ X, k ∈ {0, . . ., K}, such that: Proof.It is seen, that the matrices . . .
satisfy the equality in the part (i).Analogously, the matrices satisfy the equality in (ii).It holds It can be directly verified by the simple computation Now, we can prove the announced result.We recall, that X is a bounded group.
Theorem 4.9.Let X have the property that there exists ξ > 0 such that, for every δ > 0, there exist matrices M 1 , . . ., M l ∈ X with the property that Then, for every {A k } k∈N 0 ∈ LP (X) and an arbitrary positive number ε, there exists is not asymptotically almost periodic.
Let us define the sequence It follows directly from the construction that, for every k ∈ N 0 , there exists q In other words, Moreover, {A k } k∈N 0 is limit periodic.Thus, we know that there exist periodic sequences {B n k } k∈N 0 ⊆ X satisfying (see Definition 3.1) We denote the period of {T n k } k∈N 0 as p T n and the period of {B n k } k∈N 0 as p B n for any n ∈ N.Then, the sequence {B where j ∈ N, and for all k ∈ N 0 , n ∈ N. From it follows (consider lim j→∞ δ jj = 0 or see directly (4.7), (4.8)) that {T k } k∈N 0 is the uniform limit of the sequence of periodic sequences It means that {T k } k∈N 0 ∈ LP (X).Moreover (see (4.16) and also (4.8)), T n k ∈ O δ 11 (I) for every k ∈ N 0 , n ∈ N. From (4.15), we get (see (4.5), (4.8)) for all k ∈ N 0 .Hence, sup i.e., {T k } k∈N 0 ∈ O ε ({A k } k∈N 0 ).Since X is bounded, we know that inf k∈N 0 X k > 0. We show that the fundamental matrix {X k } k∈N 0 , X 0 = I of the system is not asymptotically almost periodic.By contradiction, we suppose that the fundamental matrix is asymptotically almost periodic.From inf k∈N 0 X k > 0, it follows that, there exists b ∈ N satisfying