Limit periodic perturbations of difference systems with coefficients from commutative groups

Abstract

We study perturbations of homogeneous linear difference systems over infinite fields with absolute values. The coefficient matrices of the treated systems belong to commutative groups which do not need to be bounded. We present a general limit periodic transformation of an arbitrarily given system such that the obtained system has non-almost periodic solutions. We also formulate corollaries which show how the presented construction of the perturbed system improves and extends known results.

Keywords:

• Limit periodicity
• almost periodicity
• limit periodic sequences
• almost periodic solutions
• difference equations
• linear equations

2020 Mathematics Subject Classifications:

• 12H10
• 39A06
• 39A24
• 42A75
• 43A60

1. Introduction

In this paper, we study non-almost periodic solutions of homogeneous linear difference systems (1) $x_{k+1}=A_k\cdot x_k,$(1) where the coefficient matrices $A_k$ are taken from a commutative group $\mathcal{X}$ for all considered k. In addition, we consider these systems over a field F with an absolute value $|\cdot |$. The research of non-almost periodic solutions of general systems in the form (1(1) $x_{k+1}=A_k\cdot x_k,$(1) ) is typically based on special iterative constructions of limit periodic and almost periodic sequences. To prove our results, we use a construction of limit periodic sequences as well. Nevertheless, the presented construction is original and differs from all constructions applied in previous papers. For other constructive methods in this research area, we refer to strongly relevant papers [9, 10, 21–25, 41, 43, 45] (see also [20] for a different usage of constructions of almost periodic sequences). We point out that the basic motivation comes from articles [9, 10, 24, 25] (see Section 4 below).

Now, we mention a short literature overview. We begin with monographs [5, 13, 29, 35], where the fundamental properties of limit periodic and almost periodic sequences and functions are presented. Concerning almost periodicity of solutions of (almost periodic) linear difference equations, we refer to [1, 6–8, 11, 17–19, 46, 48]. We remark that [17] is the first paper, where a construction is applied to prove results about non-almost periodic solutions of homogeneous linear difference equations. Concerning the special complex case of Equation (1(1) $x_{k+1}=A_k\cdot x_k,$(1) ), see, e.g. [3, 26].

If one replaces the commutativity of $\mathcal{X}$ by its boundedness, then one obtains the almost periodic theory of transformable and weakly transformable groups (see [21, 43]). This theory extends results from [36, 37, 39, 40], where Equation (1(1) $x_{k+1}=A_k\cdot x_k,$(1) ) is treated for the unitary (or orthogonal) group $\mathcal{X}$. We highlight that, in [39], there is proved that, in any neighbourhood of any almost periodic unitary Equation (1(1) $x_{k+1}=A_k\cdot x_k,$(1) ), there exists an almost periodic unitary Equation (1(1) $x_{k+1}=A_k\cdot x_k,$(1) ) whose fundamental matrix is not almost periodic. In a certain sense, this result is improved in this paper (see Corollary 5.2 below). We add that the process from [39] cannot be used for a commutative group $\mathcal{X}$.

The corresponding research about almost periodic and non-almost periodic solutions of homogeneous linear differential systems of the form (2) $x^{\mathrm{\prime }}(t)=A(t)\cdot x(t)$(2) is realized as well. We mention papers [27, 28, 38], where the almost periodicity of solutions of Equation (2(2) $x^{\mathrm{\prime }}(t)=A(t)\cdot x(t)$(2) ) with skew-Hermitian and skew-symmetric coefficient matrices A is studied, together with [42, 44] (and also [37]), where non-almost periodic solutions of skew-Hermitian and skew-symmetric Equation (2(2) $x^{\mathrm{\prime }}(t)=A(t)\cdot x(t)$(2) ) are analysed. Constructions of homogeneous linear differential systems with almost periodic coefficients are presented, e.g. in [30, 31, 33, 34].

This paper is organized as follows. Section 2 is devoted to definitions and basic properties of all considered generalizations of periodicity in metric spaces. In Section 3, we state complete notations with respect to the studied difference systems over F with $|\cdot |$. The motivation of our research is explicitly described in Section 4. In the last (and, at the same time, main) section, there are formulated and proved our new results.

2. Periodic, limit periodic, almost periodic, and asymptotically almost periodic sequences

In this section, we present the notion of periodicity, limit periodicity, almost periodicity, and asymptotic almost periodicity in a metric space $(Y,\tau )$. We mention the definitions together with only the properties of considered sequences which are used in the proofs of our results. Note that we put ${\mathbb{N}}_0:=\mathbb{N}\cup \{0\}$.

Definition 2.1

A sequence $\{{\phi }_k{\}}_{k\in \mathbb{Z}}\subseteq Y$ is called periodic if there exists $p\in \mathbb{N}$ such that ${\phi }_k={\phi }_{k\pm p}$ for all $k\in \mathbb{Z}$. A sequence $\{{\phi }_k{\}}_{k\in {\mathbb{N}}_0}\subseteq Y$ is called periodic if there exists $p\in \mathbb{N}$ such that ${\phi }_k={\phi }_{k+p}$ for all $k\in {\mathbb{N}}_0$.

Definition 2.2

A sequence $\{{\phi }_k{\}}_{k\in \mathbb{Z}}\subseteq Y$ or $\{{\phi }_k{\}}_{k\in {\mathbb{N}}_0}\subseteq Y$ is called limit periodic if there exists a sequence of periodic sequences $\{{\phi }_k^n{\}}_{k\in \mathbb{Z}}\subseteq Y$ or $\{{\phi }_k^n{\}}_{k\in {\mathbb{N}}_0}\subseteq Y$ for $n\in \mathbb{N}$ such that $\underset{n\to \mathrm{\infty }}{lim}{\phi }_k^n={\phi }_k$ uniformly with respect to $k\in \mathbb{Z}$ or $k\in {\mathbb{N}}_0$, respectively.

Remark 2.1

In Definition 2.2, the periods of sequences $\{{\phi }_k^n\}$ do not need to be the same. Thus, limit periodic sequences coincide with the so-called semi-periodic sequences. We refer to [4] (and to [2] in the continuous case).

Remark 2.2

In the literature, there is possible to find another definition of limit periodicity which is not equivalent. The different definition leads to a larger class of sequences. See, e.g. [14, 32]. We consider Definition 2.2, because this version is the standard one and we also obtain the strongest results in this case.

Definition 2.3

A sequence $\{{\phi }_k{\}}_{k\in \mathbb{Z}}\subseteq Y$ is called almost periodic if for any $\epsilon >0$, there exists a positive integer $p(\epsilon )$ such that any set consisting of $p(\epsilon )$ consecutive integers contains at least one integer l with the property that $\tau ({\phi }_{k+l},{\phi }_k)<\epsilon ,k\in \mathbb{Z}.$The integer l is called an ε-translation number of $\{{\phi }_k\}$. A sequence $\{{\phi }_k{\}}_{k\in {\mathbb{N}}_0}\subseteq Y$ is called almost periodic if there exists an almost periodic sequence $\{{\psi }_k{\}}_{k\in \mathbb{Z}}\subseteq Y$ such that ${\phi }_k={\psi }_k$ for all $k\in {\mathbb{N}}_0$.

In some books (see, e.g. [16]), the almost periodicity is introduced using the so-called Bochner concept, which is equivalent with the so-called Bohr one (presented in Definition 2.3). This equivalence is the content of the next theorem.

Theorem 2.1

A sequence $\{{\phi }_k{\}}_{k\in \mathbb{Z}}\subseteq Y$ is almost periodic if and only if any sequence $\{s_n{\}}_{n\in \mathbb{N}}\subseteq \mathbb{Z}$ has a subsequence $\{{\overline{s}}_n{\}}_{n\in \mathbb{N}}\subseteq \{s_n{\}}_{n\in \mathbb{N}}$ such that, for all $\vartheta >0$, there exists $K(\vartheta )\in \mathbb{N}$ for which the inequality (3) $\tau ({\phi }_{k+{\overline{s}}_i},{\phi }_{k+{\overline{s}}_j})<\vartheta$(3) holds for all $i,j>K(\vartheta )$, $k\in \mathbb{Z}$.

Proof.

See, e.g. [41, Theorem 2.3].

The result below follows directly from Theorem 2.1.

Corollary 2.1

Let $(Y_1,{\tau }_1),\dots ,(Y_n,{\tau }_n)$ be metric spaces and let $\{{\phi }_k^1{\}}_{k\in \mathbb{Z}},\dots ,\{{\phi }_k^n{\}}_{k\in \mathbb{Z}}$ be sequences in $Y_1,\dots ,Y_n$, respectively. The sequence $\{{\psi }_k{\}}_{k\in \mathbb{Z}}\subseteq Y_1\times \cdots \times Y_n$ given by ${\psi }_k:=({\phi }_k^1,\dots ,{\phi }_k^n),k\in \mathbb{Z},$is almost periodic if and only if all sequences $\{{\phi }_k^1{\}}_{k\in \mathbb{Z}},\dots ,\{{\phi }_k^n{\}}_{k\in \mathbb{Z}}$ are almost periodic.

We will also use the following simple results.

Theorem 2.2

Any almost periodic sequence is bounded.

Proof.

The theorem follows directly from Definition 2.3.

Theorem 2.3

Any uniform limit of almost periodic sequences is almost periodic.

Proof.

The proof of the theorem can be easily obtained by a modification of the proof of [12, Theorem 6.4].

Finally, we define the asymptotic almost periodicity.

Definition 2.4

A sequence $\{{\phi }_k{\}}_{k\in {\mathbb{N}}_0}\subseteq Y$ is asymptotically almost periodic if for any $\epsilon >0$, there exist positive integers $r(\epsilon )$ and $R(\epsilon )$ such that any set consisting of $r(\epsilon )$ consecutive positive integers contains at least one number l satisfying $\tau ({\phi }_{k+l},{\phi }_k)<\epsilon ,k\ge R(\epsilon ),k\in \mathbb{N}.$

Remark 2.3

Evidently, any periodic sequence is limit periodic. From Theorem 2.3, one can see that any limit periodic sequence is almost periodic. It is also seen that any almost periodic sequence is asymptotically almost periodic. We add that any opposite implication is not generally true.

Remark 2.4

To clarify the asymptotic almost periodicity, we mention the following result which is valid in any Banach space. 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. [47, Chapter 5]).

Remark 2.5

The Bochner concept of asymptotic almost periodicity is presented in [15].

3. Considered systems

In this section, we collect the used notation concerning the studied homogeneous linear difference systems over general infinite fields with absolute values.

Let $(F,\oplus ,\odot )$ be a field with a zero $e_0$ and a unit $e_1$. Of course, with respect to the studied problems, we assume that F is infinite. Let $|\cdot |:F\to \mathbb{R}$ be an absolute value on F, i.e. let

1. $|f|\ge 0$ and $|f|=0$ if and only if $f=e_0$,

2. $|f\odot g|=|f|\cdot |g|$,

3. $|f\oplus g|\le |f|+|g|$,

where $f,g\in F$. Let a positive integer m be arbitrarily given as the dimension of systems under consideration. Then, $Mat(F,m)$ denotes the set of all $m\times m$ matrices with entries from F and $F^m$ the set of all $m\times 1$ vectors with elements from F. In addition, ·, + denote the multiplication and the addition on the spaces $F^m$ and $Mat(F,m)$. We also denote the identity matrix as $I\in Mat(F,m)$, the zero matrix as $O\in Mat(F,m)$, and the zero vector as $o\in F^m$.

The absolute value on F gives the norm on $F^m$ and $Mat(F,m)$ as the sum of m and $m^2$ absolute values of elements, respectively. These norms are denoted by $\Vert \cdot \Vert$. We also have

1. $\Vert M\Vert \ge 0$ and $\Vert M\Vert =0$ if and only if M = O,

2. $\Vert M\cdot N\Vert \le \Vert M\Vert \cdot \Vert N\Vert$,

3. $\Vert M\cdot u\Vert \le \Vert M\Vert \cdot \Vert u\Vert$,

where $M,N\in Mat(F,m)$, $u\in F^m$. The absolute value on F and the norms $\Vert \cdot \Vert$ induce the metrics. For simplicity, each one of these metrics is denoted by ϱ and the ε-neighbourhoods are denoted by ${\mathcal{O}}_{\epsilon }^{\varrho }(\cdot )$ in all considered metric spaces. In contrast with many research papers, we do not assume that the valued field F (with $|\cdot |$) is separable or that the metric space $(F,\varrho )$ is complete, i.e. our main result is valid without such limitations (although it is new for separable and, at the same time, complete spaces).

In the whole paper, $\mathcal{X}\subset Mat(F,m)$ will be an arbitrary infinite commutative group and we will study the homogeneous linear difference systems (4) $x_{k+1}=A_k\cdot x_k,k\in {\mathbb{N}}_0,$(4) where $\{A_k{\}}_{k\in {\mathbb{N}}_0}\subseteq \mathcal{X}$. Let $\mathcal{P}(\mathcal{X})$, $\mathcal{L}\mathcal{P}(\mathcal{X})$, and $\mathcal{A}\mathcal{P}(\mathcal{X})$ denote the set of all Equation (4(4) $x_{k+1}=A_k\cdot x_k,k\in {\mathbb{N}}_0,$(4) ) such that the sequence of matrices $A_k$ is periodic, limit periodic, and almost periodic, respectively. One can identify the sequence $\{A_k{\}}_{k\in {\mathbb{N}}_0}$ with Equation (4(4) $x_{k+1}=A_k\cdot x_k,k\in {\mathbb{N}}_0,$(4) ) which is given by $\{A_k{\}}_{k\in {\mathbb{N}}_0}$. Thus, in $\mathcal{A}\mathcal{P}(\mathcal{X})$ (consider also Theorem 2.2), there is introduced the metric (5) $\sigma (\{A_k\},\{B_k\}):=\underset{k\in {\mathbb{N}}_0}{sup}\varrho (A_k,B_k),\{A_k{\}}_{k\in {\mathbb{N}}_0},\{B_k{\}}_{k\in {\mathbb{N}}_0}\in \mathcal{A}\mathcal{P}(\mathcal{X}).$(5) Analogously, one can consider Equation (4(4) $x_{k+1}=A_k\cdot x_k,k\in {\mathbb{N}}_0,$(4) ) for $k\in \mathbb{Z}$ and define $\sigma (\{A_k\},\{B_k\}):=\underset{k\in \mathbb{Z}}{sup}\varrho (A_k,B_k),\{A_k{\}}_{k\in \mathbb{Z}},\{B_k{\}}_{k\in \mathbb{Z}}\in \mathcal{A}\mathcal{P}(\mathcal{X}).$The symbol ${\mathcal{O}}_{\epsilon }^{\sigma }(\{A_k\})$ means the ε-neighbourhood of $\{A_k{\}}_{k\in {\mathbb{N}}_0}$ or $\{A_k{\}}_{k\in \mathbb{Z}}$ in $\mathcal{A}\mathcal{P}(\mathcal{X})$.

4. Previous results

Non-almost periodic and non-asymptotically almost periodic solutions of perturbations of linear difference equations are studied in many papers. Concerning homogeneous systems given by matrices from a commutative group, we point out at least the following two results about the non-almost periodicity of solutions of initial problems.

Theorem 4.1

Let $\mathcal{X}$ be a commutative group having the property that there exists $\xi >0$ such that, for any $\delta >0$, there exists $l\in \mathbb{N}$ with the property that, for any $u\in F^m$, $\Vert u\Vert \ge 1$, there exist matrices $M_1,M_2,\dots ,M_l\in \mathcal{X}$ satisfying $M_i\in {\mathcal{O}}_{\delta }^{\varrho }(I),i\in \{1,2,\dots ,l\},\Vert M_l\cdots M_2\cdot M_1\cdot u-u\Vert >\xi .$Then, for any $\epsilon >0$, $\{A_k{\}}_{k\in \mathbb{Z}}\in \mathcal{L}\mathcal{P}(\mathcal{X})$, and for any sequence $\{u_n{\}}_{n\in \mathbb{N}}$ of $u_n\in F^m\setminus \{o\}$, there exists $\{B_k{\}}_{k\in \mathbb{Z}}\in {\mathcal{O}}_{\epsilon }^{\sigma }(\{A_k\})$ such that the solution of $x_{k+1}=B_k\cdot x_k,k\in \mathbb{Z},x_0=u_n$is not almost periodic for any $n\in \mathbb{N}.$

Proof.

See [9, Theorem 5.1].

Theorem 4.1 was improved as follows (see Definitions 2.3 and 2.4).

Theorem 4.2

Let $\mathcal{X}$ be a commutative group having the property that there exists $\xi >0$ such that, for any $\delta >0$, there exists $l\in \mathbb{N}$ with the property that, for any $u\in F^m$, $\Vert u\Vert \ge 1$, there exist matrices $M_1,M_2,\dots ,M_l\in \mathcal{X}$ satisfying $M_i\in {\mathcal{O}}_{\delta }^{\varrho }(I),i\in \{1,2,\dots ,l\},\Vert M_l\cdots M_2\cdot M_1\cdot u-u\Vert >\xi .$Then, for any $\epsilon >0$, $\{A_k{\}}_{k\in {\mathbb{N}}_0}\in \mathcal{L}\mathcal{P}(\mathcal{X})$, and for any sequence $\{u_n{\}}_{n\in \mathbb{N}}$ of $u_n\in F^m\setminus \{o\}$, there exists $\{B_k{\}}_{k\in {\mathbb{N}}_0}\in {\mathcal{O}}_{\epsilon }^{\sigma }(\{A_k\})$ such that the solution $\{x_k^n{\}}_{k\in {\mathbb{N}}_0}$ of $x_{k+1}=B_k\cdot x_k,k\in {\mathbb{N}}_0,x_0=u_n$is not asymptotically almost periodic or $\underset{k\to \mathrm{\infty }}{lim inf}\Vert x_k^n\Vert =0$ for any $n\in \mathbb{N}$.

Proof.

See [25, Theorem 4.1].

Remark 4.1

Theorem 4.2 extends also the main result of [22]. For other relevant results, we can refer to [10] as well. Concerning solutions vanishing at infinity in the statement of Theorem 4.2, we add the fact that any non-trivial solution $\{x_k{\}}_{k\in {\mathbb{N}}_0}$ of any almost periodic homogeneous linear difference system cannot be almost periodic if $\underset{k\to \mathrm{\infty }}{lim inf}\Vert x_k\Vert =0.$See [43, Lemma 3.10].

Now, we recall the most relevant results whose relevance is demonstrated by corollaries at the end of this paper (see Corollaries 5.1 and 5.2). In these results, arbitrarily small changes of limit periodic and almost periodic systems are considered in such a way that the obtained systems have at least one non-almost periodic solution. We add that these results follow from [24, Theorems 10 and 11], where non-asymptotically almost periodic solutions are analysed (analogously as in Theorem 4.2).

Theorem 4.3

Let the unit ball $\{u\in F^m;\Vert u\Vert \le 1\}$ be compact. Let $\mathcal{X}$ be a commutative group having the property that there exists $\xi >0$ such that, for any $\delta >0$, there exist matrices $M_1,M_2,\dots ,M_l\in \mathcal{X}$ satisfying $M_i\in {\mathcal{O}}_{\delta }^{\varrho }(I),i\in \{1,2,\dots ,l\},\Vert M_l\cdots M_2\cdot M_1-I\Vert >\xi .$Then, for any $\epsilon >0$ and any $\{A_k{\}}_{k\in {\mathbb{N}}_0}\in \mathcal{L}\mathcal{P}(\mathcal{X})$, there exists $\{B_k{\}}_{k\in {\mathbb{N}}_0}\in {\mathcal{O}}_{\epsilon }^{\sigma }(\{A_k\})\cap \mathcal{L}\mathcal{P}(\mathcal{X})$ with the property that the fundamental matrix of $x_{k+1}=B_k\cdot x_k,k\in {\mathbb{N}}_0,$is not almost periodic.

Proof.

See [24, Corollary 1].

Theorem 4.4

Let the unit ball $\{u\in F^m;\Vert u\Vert \le 1\}$ be compact. Let $\mathcal{X}$ be a commutative group having the property that there exists $\xi >0$ such that, for any $\delta >0$, there exist matrices $M_1,M_2,\dots ,M_l\in \mathcal{X}$ satisfying $M_i\in {\mathcal{O}}_{\delta }^{\varrho }(I),i\in \{1,2,\dots ,l\},\Vert M_l\cdots M_2\cdot M_1-I\Vert >\xi .$Then, for any $\epsilon >0$ and any $\{A_k{\}}_{k\in {\mathbb{N}}_0}\in \mathcal{A}\mathcal{P}(\mathcal{X})$, there exists $\{B_k{\}}_{k\in {\mathbb{N}}_0}\in {\mathcal{O}}_{\epsilon }^{\sigma }(\{A_k\})$ with the property that the fundamental matrix of $x_{k+1}=B_k\cdot x_k,k\in {\mathbb{N}}_0,$is not almost periodic.

Proof.

See [24, Corollary 2].

5. Results

We begin with a known result which covers a special case of our main theorem, i.e. Theorem 5.1 formulated below.

Lemma 5.1

Let $\mathcal{X}$ be a commutative group and let $\{A_k{\}}_{k\in {\mathbb{N}}_0}\subseteq \mathcal{X}$ be arbitrarily given. If for any N>0 and $\delta >0$, there exist matrices $M_1,M_2,\dots ,M_l\in {\mathcal{O}}_{\delta }^{\varrho }(I)\cap \mathcal{X}$ satisfying $\Vert M_l\cdots M_2\cdot M_1\Vert >N,$then, for any $\epsilon >0$, there exists a limit periodic sequence $\{D_k{\}}_{k\in {\mathbb{N}}_0}\subseteq \mathcal{X}\cap {\mathcal{O}}_{\epsilon }^{\varrho }(I)$ with the property that the fundamental matrix of $x_{k+1}=A_k\cdot D_k\cdot x_k,k\in {\mathbb{N}}_0,$is not almost periodic.

Proof.

The lemma follows from the proof of [22, Lemma 5.2], where it suffices to put $D_k:=B_k^1\cdot B_k^2\cdots B_k^i\cdots ,k\in {\mathbb{N}}_0.$

Concerning the statement of the main result below, we recall the usual notation (6) $A_{-1}\cdots A_0:=I.$(6) In addition, in its proof, for $k,n\in \mathbb{N}$, we consider the notation $⟨k{⟩}_{2^n}:=k\mathrm{if}k=m\cdot 2^n\mathrm{for}\mathrm{some}m\in {\mathbb{N}}_0$and $⟨k{⟩}_{2^n}:=(m+1)\cdot 2^n\mathrm{if}k=m\cdot 2^n+i\mathrm{for}\mathrm{some}m\in {\mathbb{N}}_0\mathrm{and}i\in \{1,\dots ,2^n-1\},$i.e. $⟨k{⟩}_{2^n}$ denotes the smallest positive integer divisible by $2^n$ which is greater than or equal to k.

Theorem 5.1

Let $\mathcal{X}$ be a commutative group and let $\{A_k{\}}_{k\in {\mathbb{N}}_0}\subseteq \mathcal{X}$ be arbitrarily given. Let there exist $\kappa \in {\mathbb{N}}_0$ and $\xi >0$ such that, for any $\delta >0$, there exist matrices $M_1,M_2,\dots ,M_l\in {\mathcal{O}}_{\delta }^{\varrho }(I)\cap \mathcal{X}$satisfying (7) $\Vert M_l\cdots M_2\cdot M_1\cdot A_{\kappa -1}\cdots A_0-A_{\kappa -1}\cdots A_0\Vert >\xi .$(7) Then, for any $\epsilon >0$, there exists a limit periodic sequence $\{C_k{\}}_{k\in {\mathbb{N}}_0}\subseteq \mathcal{X}\cap {\mathcal{O}}_{\epsilon }^{\varrho }(I)$ with the property that the fundamental matrix of (8) $x_{k+1}=A_k\cdot C_k\cdot x_k,k\in {\mathbb{N}}_0,$(8) is not almost periodic, i.e. Equation (8(8) $x_{k+1}=A_k\cdot C_k\cdot x_k,k\in {\mathbb{N}}_0,$(8) ) has non-almost periodic solutions.

Proof.

If the fundamental matrix (denoted as) $\{X_k^1{\}}_{k\in {\mathbb{N}}_0}$ of (9) $x_{k+1}=A_k\cdot x_k,k\in {\mathbb{N}}_0,$(9) is not almost periodic, then one can consider $\{C_k{\}}_{k\in {\mathbb{N}}_0}\equiv \{I{\}}_{k\in {\mathbb{N}}_0}$. Hence, we assume that the sequence $\{X_k^1{\}}_{k\in {\mathbb{N}}_0}$ is almost periodic, i.e. all solutions of Equation (9(9) $x_{k+1}=A_k\cdot x_k,k\in {\mathbb{N}}_0,$(9) ) are almost periodic. Note that the system has infinitely many non-almost periodic solutions if it has at least one non-almost periodic solution (it suffices to consider its non-zero multiples).

For an arbitrarily given number $\epsilon >0$, we define (10) ${\epsilon }_i:=\frac{\epsilon }{i},i\in \mathbb{N}.$(10) As $l_i$ for $i\in \mathbb{N}$, we denote the integer $l\in \mathbb{N}$ (from the statement of the theorem) which corresponds to $\delta ={\epsilon }_i$. Of course, we can assume that $\epsilon <1$ and that $l_{i+1}\ge l_i$, $i\in \mathbb{N}$. For any $i\in \mathbb{N}$, we consider the product $M_{l_i}^i\cdots M_2^i\cdot M_1^i$ satisfying (7(7) $\Vert M_l\cdots M_2\cdot M_1\cdot A_{\kappa -1}\cdots A_0-A_{\kappa -1}\cdots A_0\Vert >\xi .$(7) ) given by matrices $M_1^i,M_2^i,\dots ,M_{l_i}^i\in {\mathcal{O}}_{{\epsilon }_i}^{\varrho }(I)\cap \mathcal{X}.$Based on Lemma 5.1, we can assume the existence of K>0 with the property that (11) $\Vert M_{l_i}^i\cdots M_2^i\cdot M_1^i\Vert <K,i\in \mathbb{N}.$(11)

For the given $\xi >0$ from the statement of the theorem, we define (see also (11(11) $\Vert M_{l_i}^i\cdots M_2^i\cdot M_1^i\Vert <K,i\in \mathbb{N}.$(11) )) (12) $\zeta :=\frac{\xi }{2K+3}.$(12) It holds (13) $\Vert M_{l_i}^i\cdots M_2^i\cdot M_1^i\cdot B-A\Vert >\zeta ,i\in \mathbb{N},$(13) if (14) $A,B\in \mathcal{X}\cap {\mathcal{O}}_{2\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0).$(14) Indeed, if (14(14) $A,B\in \mathcal{X}\cap {\mathcal{O}}_{2\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0).$(14) ) is valid, then we have (see (7(7) $\Vert M_l\cdots M_2\cdot M_1\cdot A_{\kappa -1}\cdots A_0-A_{\kappa -1}\cdots A_0\Vert >\xi .$(7) ), (11(11) $\Vert M_{l_i}^i\cdots M_2^i\cdot M_1^i\Vert <K,i\in \mathbb{N}.$(11) ), and (12(12) $\zeta :=\frac{\xi }{2K+3}.$(12) )) $\begin{array}{rl}(2K+3)\zeta & =\xi <\Vert M_{l_i}^i\cdots M_2^i\cdot M_1^i\cdot A_{\kappa -1}\cdots A_0-A_{\kappa -1}\cdots A_0\Vert \\ & \le \Vert M_{l_i}^i\cdots M_2^i\cdot M_1^i\cdot A_{\kappa -1}\cdots A_0-M_{l_i}^i\cdots M_2^i\cdot M_1^i\cdot B\Vert \\ & +\Vert M_{l_i}^i\cdots M_2^i\cdot M_1^i\cdot B-A\Vert +\Vert A-A_{\kappa -1}\cdots A_0\Vert \\ & \le \Vert M_{l_i}^i\cdots M_2^i\cdot M_1^i\Vert \cdot \Vert A_{\kappa -1}\cdots A_0-B\Vert \\ & +\Vert M_{l_i}^i\cdots M_2^i\cdot M_1^i\cdot B-A\Vert +\Vert A-A_{\kappa -1}\cdots A_0\Vert \\ & <2K\zeta +\Vert M_{l_i}^i\cdots M_2^i\cdot M_1^i\cdot B-A\Vert +2\zeta ,i\in \mathbb{N},\end{array}$which yields (13(13) $\Vert M_{l_i}^i\cdots M_2^i\cdot M_1^i\cdot B-A\Vert >\zeta ,i\in \mathbb{N},$(13) ).

We introduce the required sequence $\{C_k{\}}_{k\in {\mathbb{N}}_0}$ by a construction, where we consider auxiliary sequences $\{C_k^i{\}}_{k\in {\mathbb{N}}_0}$ for $i\in \mathbb{N}$. Note that we put $C_k^i:=I$ for all $k\in \{0,1,\dots ,\kappa -1\}$ and $i\in \mathbb{N}$.

We begin with the construction. Since the fundamental matrix $\{X_k^1{\}}_{k\in {\mathbb{N}}_0}$ of Equation (9(9) $x_{k+1}=A_k\cdot x_k,k\in {\mathbb{N}}_0,$(9) ) is almost periodic and $X_{\kappa }^1=A_{\kappa -1}\cdots A_0$, there exists $j(1,1)>\kappa$, $j(1,1)\in \mathbb{N}$, such that (15) $X_{j(1,1)}^1\in \mathcal{X}\cap {\mathcal{O}}_{\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0).$(15) Indeed, it follows directly from Definition 2.3. We recall that, for the given ${\epsilon }_1<1$ (see (10(10) ${\epsilon }_i:=\frac{\epsilon }{i},i\in \mathbb{N}.$(10) )), we consider matrices $M_1^1,M_2^1,\dots ,M_{l_1}^1\in \mathcal{X}$ satisfying (16) $M_j^1\in {\mathcal{O}}_{{\epsilon }_1}^{\varrho }(I),j\in \{1,2,\dots ,l_1\},$(16) $\Vert M_1^1\cdot M_2^1\cdots M_{l_1}^1\cdot A_{\kappa -1}\cdots A_0-A_{\kappa -1}\cdots A_0\Vert >\xi .$We put $r_1:=2(l_1+1)$. We define the periodic sequence $\{C_k^1{\}}_{k\in {\mathbb{N}}_0}$ with the period $p(1,1):=j(1,1)+r_1$by $\begin{array}{r}\begin{array}{c}{C}_0^1:=C_1^1:=\cdots :=C_{⟨j(1,1){⟩}_2}^1:=I,C_{⟨j(1,1){⟩}_2+1}^1:=M_1^1,\\ C_{⟨j(1,1){⟩}_2+2}^1:=I,C_{⟨j(1,1){⟩}_2+3}^1:=M_2^1,\\ C_{⟨j(1,1){⟩}_2+4}^1:=I,C_{⟨j(1,1){⟩}_2+5}^1:=M_3^1,\\ ⋮\\ C_{⟨j(1,1){⟩}_2+2l_1-2}^1:=I,C_{⟨j(1,1){⟩}_2+2l_1-1}^1:=M_{l_1}^1,\\ C_{⟨j(1,1){⟩}_2+2l_1}^1:=C_{⟨j(1,1){⟩}_2+2l_1+1}^1:=\cdots :=C_{p(1,1)-1}^1:=I\end{array}\end{array}$if $\Vert X_{j(1,1)+r_1}^1-X_{j(1,1)}^1\Vert <\zeta ;$and by $\{C_k^1{\}}_{k\in {\mathbb{N}}_0}\equiv \{I{\}}_{k\in {\mathbb{N}}_0}$ if $\Vert X_{j(1,1)+r_1}^1-X_{j(1,1)}^1\Vert \ge \zeta .$Introducing $\{C_k^1{\}}_{k\in {\mathbb{N}}_0}$, we have finished the first step of our construction.

Considering (13(13) $\Vert M_{l_i}^i\cdots M_2^i\cdot M_1^i\cdot B-A\Vert >\zeta ,i\in \mathbb{N},$(13) ) ⇐ (14(14) $A,B\in \mathcal{X}\cap {\mathcal{O}}_{2\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0).$(14) ) and (15(15) $X_{j(1,1)}^1\in \mathcal{X}\cap {\mathcal{O}}_{\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0).$(15) ), we know that the fundamental matrix $\{X_k^{(2,1)}{\}}_{k\in {\mathbb{N}}_0}$ of (17) $x_{k+1}=A_k\cdot C_k^1\cdot x_k,k\in {\mathbb{N}}_0,$(17) satisfies (18) $\Vert X_{j(1,1)+r_1}^{(2,1)}-X_{j(1,1)}^{(2,1)}\Vert \ge \zeta ,$(18) because $X_{j(1,1)+r_1}^{(2,1)}=X_{j(1,1)+r_1}^1\cdot M_1^1\cdot M_2^1\cdots M_{l_1}^1,X_{j(1,1)}^{(2,1)}=X_{j(1,1)}^1$and $X_{j(1,1)}^1,X_{j(1,1)+r_1}^1\in \mathcal{X}\cap {\mathcal{O}}_{2\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0)$if $\Vert X_{j(1,1)+r_1}^1-X_{j(1,1)}^1\Vert <\zeta ,$and because $X_{j(1,1)+r_1}^{(2,1)}=X_{j(1,1)+r_1}^1,X_{j(1,1)}^{(2,1)}=X_{j(1,1)}^1$if $\Vert X_{j(1,1)+r_1}^1-X_{j(1,1)}^1\Vert \ge \zeta .$The second step of our construction has two parts. At first, we repeat that, for ${\epsilon }_2=\epsilon /2$ (see (10(10) ${\epsilon }_i:=\frac{\epsilon }{i},i\in \mathbb{N}.$(10) )), there exist matrices $M_1^2,M_2^2,\dots ,M_{l_2}^2\in \mathcal{X}$ satisfying (19) $\begin{array}{rl}& M_j^2\in {\mathcal{O}}_{{\epsilon }_2}^{\varrho }(I),j\in \{1,2,\dots ,l_2\},\\ & \Vert M_1^2\cdot M_2^2\cdots M_{l_2}^2\cdot A_{\kappa -1}\cdots A_0-A_{\kappa -1}\cdots A_0\Vert >\xi .\end{array}$(19) We put $r_2:=2^{1+2}(l_2+1+r_1).$In the first part of the second step, we consider the fundamental matrix $\{X_k^{(2,1)}{\}}_{k\in {\mathbb{N}}_0}$ of Equation (17(17) $x_{k+1}=A_k\cdot C_k^1\cdot x_k,k\in {\mathbb{N}}_0,$(17) ). If the sequence $\{X_k^{(2,1)}{\}}_{k\in {\mathbb{N}}_0}$ is not almost periodic, then we have the resulting sequence given by the choice $\{C_k{\}}_{k\in {\mathbb{N}}_0}\equiv \{C_k^1{\}}_{k\in {\mathbb{N}}_0}$. Thus, we consider the almost periodicity of $\{X_k^{(2,1)}{\}}_{k\in {\mathbb{N}}_0}$ which implies the existence of an integer $j(2,1)>p(1,1)$ with the property that (20) $X_{j(2,1)}^{(2,1)}\in \mathcal{X}\cap {\mathcal{O}}_{\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0).$(20) We define the periodic sequence $\{C_k^{(2,1)}{\}}_{k\in {\mathbb{N}}_0}$ with the period $p(2,1):=p(1,1)(j(2,1)+r_2)$in the following way.

If $\Vert X_{j(2,1)+r_2}^{(2,1)}-X_{j(2,1)}^{(2,1)}\Vert <\zeta ,$then we put $\begin{array}{r}\begin{array}{c}{C}_0^{(2,1)}:=\cdots :=C_{⟨j(2,1){⟩}_{2^2}+1}^{(2,1)}:=I,C_{⟨j(2,1){⟩}_{2^2}+2}^{(2,1)}:=M_1^2,\\ C_{⟨j(2,1){⟩}_{2^2}+3}^{(2,1)}:=C_{⟨j(2,1){⟩}_{2^2}+4}^{(2,1)}:=C_{⟨j(2,1){⟩}_{2^2}+5}^{(2,1)}:=I,C_{⟨j(2,1){⟩}_{2^2}+6}^{(2,1)}:=M_2^2,\\ ⋮\\ C_{⟨j(2,1){⟩}_{2^2}+4(l_2-1)-1}^{(2,1)}:=C_{⟨j(2,1){⟩}_{2^2}+4(l_2-1)}^{(2,1)}:=C_{⟨j(2,1){⟩}_{2^2}+4(l_2-1)+1}^{(2,1)}:=I,\\ C_{⟨j(2,1){⟩}_{2^2}+4(l_2-1)+2}^{(2,1)}:=M_{l_2}^2,\\ C_{⟨j(2,1){⟩}_{2^2}+4(l_2-1)+3}^{(2,1)}:=\cdots :=C_{p(2,1)-1}^{(2,1)}:=I.\end{array}\end{array}$If $\Vert X_{j(2,1)+r_2}^{(2,1)}-X_{j(2,1)}^{(2,1)}\Vert \ge \zeta ,$then we put $C_0^{(2,1)}:=\cdots :=C_{p(2,1)-1}^{(2,1)}:=I.$Using (13(13) $\Vert M_{l_i}^i\cdots M_2^i\cdot M_1^i\cdot B-A\Vert >\zeta ,i\in \mathbb{N},$(13) ) ⇐ (14(14) $A,B\in \mathcal{X}\cap {\mathcal{O}}_{2\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0).$(14) ) and (20(20) $X_{j(2,1)}^{(2,1)}\in \mathcal{X}\cap {\mathcal{O}}_{\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0).$(20) ), we obtain that the fundamental matrix $\{X_k^{(2,2)}{\}}_{k\in {\mathbb{N}}_0}$ of (21) $x_{k+1}=A_k\cdot C_k^1\cdot C_k^{(2,1)}\cdot x_k,k\in {\mathbb{N}}_0,$(21) has the property that (22) $\Vert X_{j(2,1)+r_2}^{(2,2)}-X_{j(2,1)}^{(2,2)}\Vert \ge \zeta .$(22) Indeed, $X_{j(2,1)+r_2}^{(2,2)}=X_{j(2,1)+r_2}^{(2,1)}\cdot M_1^2\cdot M_2^2\cdots M_{l_2}^2,X_{j(2,1)}^{(2,2)}=X_{j(2,1)}^{(2,1)}$and $X_{j(2,1)}^{(2,1)},X_{j(2,1)+r_2}^{(2,1)}\in \mathcal{X}\cap {\mathcal{O}}_{2\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0)$if $\Vert X_{j(2,1)+r_2}^{(2,1)}-X_{j(2,1)}^{(2,1)}\Vert <\zeta ;$and $X_{j(2,1)+r_2}^{(2,2)}=X_{j(2,1)+r_2}^{(2,1)},X_{j(2,1)}^{(2,2)}=X_{j(2,1)}^{(2,1)}$if $\Vert X_{j(2,1)+r_2}^{(2,1)}-X_{j(2,1)}^{(2,1)}\Vert \ge \zeta .$Now we proceed with the second part of the second step. We consider the fundamental matrix $\{X_k^{(2,2)}{\}}_{k\in {\mathbb{N}}_0}$ of Equation (21(21) $x_{k+1}=A_k\cdot C_k^1\cdot C_k^{(2,1)}\cdot x_k,k\in {\mathbb{N}}_0,$(21) ). Again, we can assume that the sequence $\{X_k^{(2,2)}{\}}_{k\in {\mathbb{N}}_0}$ is almost periodic. Therefore, there exists an integer $j(2,2)>p(2,1)$ such that (23) $X_{j(2,2)}^{(2,2)}\in \mathcal{X}\cap {\mathcal{O}}_{\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0).$(23) We define the periodic sequence $\{C_k^{(2,2)}{\}}_{k\in {\mathbb{N}}_0}$ with the period $p(2,2):=p(2,1)(j(2,2)+r_2)$in the following way.

If $\Vert X_{j(2,2)+r_2-r_1}^{(2,2)}-X_{j(2,2)}^{(2,2)}\Vert <\zeta ,$then we put $\begin{array}{r}\begin{array}{c}{C}_0^{(2,2)}:=\cdots :=C_{⟨j(2,2){⟩}_{2^3}+3}^{(2,2)}:=I,C_{⟨j(2,2){⟩}_{2^3}+4}^{(2,2)}:=M_1^2,\\ C_{⟨j(2,2){⟩}_{2^3}+5}^{(2,2)}:=\cdots :=C_{⟨j(2,2){⟩}_{2^3}+11}^{(2,2)}:=I,C_{⟨j(2,2){⟩}_{2^3}+12}^{(2,2)}:=M_2^2,\\ ⋮\\ C_{⟨j(2,2){⟩}_{2^3}+8(l_2-1)-3}^{(2,2)}:=\cdots :=C_{⟨j(2,2){⟩}_{2^3}+8(l_2-1)+3}^{(2,2)}:=I,C_{⟨j(2,2){⟩}_{2^3}+8(l_2-1)+4}^{(2,2)}:=M_{l_2}^2,\\ C_{⟨j(2,2){⟩}_{2^3}+8(l_2-1)+5}^{(2,2)}:=\cdots :=C_{p(2,2)-1}^{(2,2)}:=I.\end{array}\end{array}$If $\Vert X_{j(2,2)+r_2-r_1}^{(2,2)}-X_{j(2,2)}^{(2,2)}\Vert \ge \zeta ,$then we put $C_0^{(2,2)}:=\cdots :=C_{p(2,2)-1}^{(2,2)}:=I.$Finally, in the second step, we introduce $C_k^2:=C_k^1\cdot C_k^{(2,1)}\cdot C_k^{(2,2)},k\in {\mathbb{N}}_0.$Evidently, the sequence $\{C_k^2{\}}_{k\in {\mathbb{N}}_0}\subseteq \mathcal{X}\cap {\mathcal{O}}_{\epsilon }^{\varrho }(I)$ is periodic. Applying (13(13) $\Vert M_{l_i}^i\cdots M_2^i\cdot M_1^i\cdot B-A\Vert >\zeta ,i\in \mathbb{N},$(13) ) ⇐ (14(14) $A,B\in \mathcal{X}\cap {\mathcal{O}}_{2\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0).$(14) ) and (23(23) $X_{j(2,2)}^{(2,2)}\in \mathcal{X}\cap {\mathcal{O}}_{\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0).$(23) ), we have that the fundamental matrix $\{X_k^{(3,1)}{\}}_{k\in {\mathbb{N}}_0}$ of $x_{k+1}=A_k\cdot C_k^2\cdot x_k,k\in {\mathbb{N}}_0,$satisfies (24) $\Vert X_{j(2,2)+r_2-r_1}^{(3,1)}-X_{j(2,2)}^{(3,1)}\Vert \ge \zeta .$(24) Indeed, $X_{j(2,2)+r_2-r_1}^{(3,1)}=X_{j(2,2)+r_2-r_1}^{(2,2)}\cdot M_1^2\cdot M_2^2\cdots M_{l_2}^2,X_{j(2,2)}^{(3,1)}=X_{j(2,2)}^{(2,2)}$and $X_{j(2,2)}^{(2,2)},X_{j(2,2)+r_2-r_1}^{(2,2)}\in \mathcal{X}\cap {\mathcal{O}}_{2\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0)$if $\Vert X_{j(2,2)+r_2-r_1}^{(2,2)}-X_{j(2,2)}^{(2,2)}\Vert <\zeta ;$and $X_{j(2,2)+r_2-r_1}^{(3,1)}=X_{j(2,2)+r_2-r_1}^{(2,2)},X_{j(2,2)}^{(3,1)}=X_{j(2,2)}^{(2,2)}$if $\Vert X_{j(2,2)+r_2-r_1}^{(2,2)}-X_{j(2,2)}^{(2,2)}\Vert \ge \zeta .$We continue in the construction in the same manner. Before the step no. i for an arbitrary integer $i\ge 3$, we have the sequence ${\left\{C_k^{i-1}\right\}}_{k\in {\mathbb{N}}_0}\equiv {\{C_k^{i-2}\cdot C_k^{(i-1,1)}\cdot C_k^{(i-1,2)}\cdots C_k^{(i-1,i-1)}\}}_{k\in {\mathbb{N}}_0},$which is periodic with the period $p(i-1,i-1)$. We point out that the step no. i has i parts.

For ${\epsilon }_i$ (see (10(10) ${\epsilon }_i:=\frac{\epsilon }{i},i\in \mathbb{N}.$(10) )), we have matrices $M_1^i,M_2^i,\dots ,M_{l_i}^i\in \mathcal{X}$ satisfying (25) $M_j^i\in {\mathcal{O}}_{{\epsilon }_i}^{\varrho }(I),j\in \{1,2,\dots ,l_i\},$(25) $\Vert M_1^i\cdot M_2^i\cdots M_{l_i}^i\cdot A_{\kappa -1}\cdots A_0-A_{\kappa -1}\cdots A_0\Vert >\xi .$We put $r_i:=2^{1+2+\cdots +i}(l_i+1+r_{i-1}).$At first, in the step no. i, we consider the fundamental matrix $\{X_k^{(i,1)}{\}}_{k\in {\mathbb{N}}_0}$ of $x_{k+1}=A_k\cdot C_k^{i-1}\cdot x_k,k\in {\mathbb{N}}_0.$If $\{X_k^{(i,1)}{\}}_{k\in {\mathbb{N}}_0}$ is not almost periodic, then it suffices to choose $\{C_k{\}}_{k\in {\mathbb{N}}_0}\equiv \{C_k^{i-1}{\}}_{k\in {\mathbb{N}}_0}$. Let $\{X_k^{(i,1)}{\}}_{k\in {\mathbb{N}}_0}$ be almost periodic. Then, there exists an integer $j(i,1)>p(i-1,i-1)$ such that (26) $X_{j(i,1)}^{(i,1)}\in \mathcal{X}\cap {\mathcal{O}}_{\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0).$(26) We put $S(i):=1+(1+2+\cdots +i-1),i\ge 3,i\in \mathbb{N}.$We define the periodic sequence $\{C_k^{(i,1)}{\}}_{k\in {\mathbb{N}}_0}$ with the period $p(i,1):=p(i-1,i-1)(j(i,1)+r_i)$in the following way.

If $\Vert X_{j(i,1)+r_i}^{(i,1)}-X_{j(i,1)}^{(i,1)}\Vert <\zeta ,$then we put $\begin{array}{r}\begin{array}{c}{C}_0^{(i,1)}:=\cdots :=C_{⟨j(i,1){⟩}_{2^{S(i)}}+2^{S(i)-1}-1}^{(i,1)}:=I,C_{⟨j(i,1){⟩}_{2^{S(i)}}+2^{S(i)-1}}^{(i,1)}:=M_1^i,\\ C_{⟨j(i,1){⟩}_{2^{S(i)}}+2^{S(i)-1}+1}^{(i,1)}:=\cdots :=C_{⟨j(i,1){⟩}_{2^{S(i)}}+2^{S(i)}+2^{S(i)-1}-1}^{(i,1)}:=I,\\ C_{⟨j(i,1){⟩}_{2^{S(i)}}+2^{S(i)}+2^{S(i)-1}}^{(i,1)}:=M_2^i,\\ ⋮\\ C_{⟨j(i,1){⟩}_{2^{S(i)}}+2^{S(i)}(l_i-2)+2^{S(i)-1}+1}^{(i,1)}:=\cdots :=C_{⟨j(i,1){⟩}_{2^{S(i)}}+2^{S(i)}(l_i-1)+2^{S(i)-1}-1}^{(i,1)}:=I,\\ C_{⟨j(i,1){⟩}_{2^{S(i)}}+2^{S(i)}(l_i-1)+2^{S(i)-1}}^{(i,1)}:=M_{l_i}^i,\\ C_{⟨j(i,1){⟩}_{2^{S(i)}}+2^{S(i)}(l_i-1)+2^{S(i)-1}+1}^{(i,1)}:=\cdots :=C_{p(i,1)-1}^{(i,1)}:=I.\end{array}\end{array}$If $\Vert X_{j(i,1)+r_i}^{(i,1)}-X_{j(i,1)}^{(i,1)}\Vert \ge \zeta ,$then we put $C_0^{(i,1)}:=\cdots :=C_{p(i,1)-1}^{(i,1)}:=I.$From (13(13) $\Vert M_{l_i}^i\cdots M_2^i\cdot M_1^i\cdot B-A\Vert >\zeta ,i\in \mathbb{N},$(13) ) ⇐ (14(14) $A,B\in \mathcal{X}\cap {\mathcal{O}}_{2\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0).$(14) ) and (26(26) $X_{j(i,1)}^{(i,1)}\in \mathcal{X}\cap {\mathcal{O}}_{\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0).$(26) ), we know that the fundamental matrix $\{X_k^{(i,2)}{\}}_{k\in {\mathbb{N}}_0}$ of (27) $x_{k+1}=A_k\cdot C_k^{i-1}\cdot C_k^{(i,1)}\cdot x_k,k\in {\mathbb{N}}_0,$(27) has the property that (28) $\Vert X_{j(i,1)+r_i}^{(i,2)}-X_{j(i,1)}^{(i,2)}\Vert \ge \zeta .$(28) Indeed, $X_{j(i,1)+r_i}^{(i,2)}=X_{j(i,1)+r_i}^{(i,1)}\cdot M_1^i\cdot M_2^i\cdots M_{l_i}^i,X_{j(i,1)}^{(i,2)}=X_{j(i,1)}^{(i,1)}$and $X_{j(i,1)}^{(i,1)},X_{j(i,1)+r_i}^{(i,1)}\in \mathcal{X}\cap {\mathcal{O}}_{2\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0)$if $\Vert X_{j(i,1)+r_i}^{(i,1)}-X_{j(i,1)}^{(i,1)}\Vert <\zeta ;$and $X_{j(i,1)+r_i}^{(i,2)}=X_{j(i,1)+r_i}^{(i,1)},X_{j(i,1)}^{(i,2)}=X_{j(i,1)}^{(i,1)}$if $\Vert X_{j(i,1)+r_i}^{(i,1)}-X_{j(i,1)}^{(i,1)}\Vert \ge \zeta .$Next, we consider the second part of step no. i, where we consider the fundamental matrix $\{X_k^{(i,2)}{\}}_{k\in {\mathbb{N}}_0}$ of Equation (27(27) $x_{k+1}=A_k\cdot C_k^{i-1}\cdot C_k^{(i,1)}\cdot x_k,k\in {\mathbb{N}}_0,$(27) ). We assume that the sequence $\{X_k^{(i,2)}{\}}_{k\in {\mathbb{N}}_0}$ is almost periodic which gives an integer $j(i,2)>p(i,1)$ such that (29) $X_{j(i,2)}^{(i,2)}\in \mathcal{X}\cap {\mathcal{O}}_{\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0).$(29) We define the periodic sequence $\{C_k^{(i,2)}{\}}_{k\in {\mathbb{N}}_0}$ with the period $p(i,2):=p(i,1)(j(i,2)+r_i)$in the following way.

If $\Vert X_{j(i,2)+r_i-r_1}^{(i,2)}-X_{j(i,2)}^{(i,2)}\Vert <\zeta ,$then we put $\begin{array}{r}\begin{array}{c}{C}_0^{(i,2)}:=\cdots :=C_{⟨j(i,2){⟩}_{2^{S(i)+1}}+2^{S(i)}-1}^{(i,2)}:=I,C_{⟨j(i,2){⟩}_{2^{S(i)+1}}+2^{S(i)}}^{(i,2)}:=M_1^i,\\ C_{⟨j(i,2){⟩}_{2^{S(i)+1}}+2^{S(i)}+1}^{(i,2)}:=\cdots :=C_{⟨j(i,2){⟩}_{2^{S(i)+1}}+2^{S(i)+1}+2^{S(i)}-1}^{(i,2)}:=I,\\ C_{⟨j(i,2){⟩}_{2^{S(i)+1}}+2^{S(i)+1}+2^{S(i)}}^{(i,2)}:=M_2^i,\\ ⋮\\ C_{⟨j(i,2){⟩}_{2^{S(i)+1}}+2^{S(i)+1}(l_i-2)+2^{S(i)}+1}^{(i,2)}:=\cdots :=C_{⟨j(i,2){⟩}_{2^{S(i)+1}}+2^{S(i)+1}(l_i-1)+2^{S(i)}-1}^{(i,2)}:=I,\\ C_{⟨j(i,2){⟩}_{2^{S(i)+1}}+2^{S(i)+1}(l_i-1)+2^{S(i)}}^{(i,2)}:=M_{l_i}^i,\\ C_{⟨j(i,2){⟩}_{2^{S(i)+1}}+2^{S(i)+1}(l_i-1)+2^{S(i)}+1}^{(i,2)}:=\cdots :=C_{p(i,2)-1}^{(i,2)}:=I.\end{array}\end{array}$If $\Vert X_{j(i,2)+r_i-r_1}^{(i,2)}-X_{j(i,2)}^{(i,2)}\Vert \ge \zeta ,$then we put $C_0^{(i,2)}:=\cdots :=C_{p(i,2)-1}^{(i,2)}:=I.$Considering (13(13) $\Vert M_{l_i}^i\cdots M_2^i\cdot M_1^i\cdot B-A\Vert >\zeta ,i\in \mathbb{N},$(13) ) ⇐ (14(14) $A,B\in \mathcal{X}\cap {\mathcal{O}}_{2\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0).$(14) ) and (29(29) $X_{j(i,2)}^{(i,2)}\in \mathcal{X}\cap {\mathcal{O}}_{\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0).$(29) ), we obtain that the fundamental matrix $\{X_k^{(i,3)}{\}}_{k\in {\mathbb{N}}_0}$ of $x_{k+1}=A_k\cdot C_k^{i-1}\cdot C_k^{(i,1)}\cdot C_k^{(i,2)}\cdot x_k,k\in {\mathbb{N}}_0,$satisfies (30) $\Vert X_{j(i,2)+r_i-r_1}^{(i,3)}-X_{j(i,2)}^{(i,3)}\Vert \ge \zeta .$(30) Indeed, $X_{j(i,2)+r_i-r_1}^{(i,3)}=X_{j(i,2)+r_i-r_1}^{(i,2)}\cdot M_1^i\cdot M_2^i\cdots M_{l_i}^i,X_{j(i,2)}^{(i,3)}=X_{j(i,2)}^{(i,2)}$and $X_{j(i,2)}^{(i,2)},X_{j(i,2)+r_i-r_1}^{(i,2)}\in \mathcal{X}\cap {\mathcal{O}}_{2\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0)$if $\Vert X_{j(i,2)+r_i-r_1}^{(i,2)}-X_{j(i,2)}^{(i,2)}\Vert <\zeta ;$and $X_{j(i,2)+r_i-r_1}^{(i,3)}=X_{j(i,2)+r_i-r_1}^{(i,2)},X_{j(i,2)}^{(i,3)}=X_{j(i,2)}^{(i,2)}$if $\Vert X_{j(i,2)+r_i-r_1}^{(i,2)}-X_{j(i,2)}^{(i,2)}\Vert \ge \zeta .$We continue in the step no. i. Before its last part, we consider the fundamental matrix $\{X_k^{(i,i)}{\}}_{k\in {\mathbb{N}}_0}$ of $x_{k+1}=A_k\cdot C_k^{i-1}\cdot C_k^{(i,1)}\cdot C_k^{(i,2)}\cdots C_k^{(i,i-1)}\cdot x_k,k\in {\mathbb{N}}_0.$Again, we assume that the sequence $\{X_k^{(i,i)}{\}}_{k\in {\mathbb{N}}_0}$ is almost periodic. Therefore, there exists an integer $j(i,i)>p(i,i-1)$ such that (31) $X_{j(i,i)}^{(i,i)}\in \mathcal{X}\cap {\mathcal{O}}_{\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0).$(31) We define the periodic sequence $\{C_k^{(i,i)}{\}}_{k\in {\mathbb{N}}_0}$ with the period $p(i,i):=p(i,i-1)(j(i,i)+r_i)$in the following way.

If $\Vert X_{j(i,i)+r_i-r_{i-1}}^{(i,i)}-X_{j(i,i)}^{(i,i)}\Vert <\zeta ,$then we put $\begin{array}{r}\begin{array}{c}{C}_0^{(i,i)}:=\cdots :=C_{⟨j(i,i){⟩}_{2^{S(i)+i-1}}+2^{S(i)+i-2}-1}^{(i,i)}:=I,C_{⟨j(i,i){⟩}_{2^{S(i)+i-1}}+2^{S(i)+i-2}}^{(i,i)}:=M_1^i,\\ C_{⟨j(i,i){⟩}_{2^{S(i)+i-1}}+2^{S(i)+i-2}+1}^{(i,i)}:=\cdots :=C_{⟨j(i,i){⟩}_{2^{S(i)+i-1}}+2^{S(i)+i-1}+2^{S(i)+i-2}-1}^{(i,i)}:=I,\\ C_{⟨j(i,i){⟩}_{2^{S(i)+i-1}}+2^{S(i)+i-1}+2^{S(i)+i-2}}^{(i,i)}:=M_2^i,\\ ⋮\\ C_{⟨j(i,i){⟩}_{2^{S(i)+i-1}}+2^{S(i)+i-1}(l_i-2)+2^{S(i)+i-2}+1}^{(i,i)}:=\cdots :=C_{⟨j(i,i){⟩}_{2^{S(i)+i-1}}+2^{S(i)+i-1}(l_i-1)+2^{S(i)+i-2}-1}^{(i,i)}:=I,\\ C_{⟨j(i,i){⟩}_{2^{S(i)+i-1}}+2^{S(i)+i-1}(l_i-1)+2^{S(i)+i-2}}^{(i,i)}:=M_{l_i}^i,\\ C_{⟨j(i,i){⟩}_{2^{S(i)+i-1}}+2^{S(i)+i-1}(l_i-1)+2^{S(i)+i-2}+1}^{(i,i)}:=\cdots :=C_{p(i,i)-1}^{(i,i)}:=I.\end{array}\end{array}$

If $\Vert X_{j(i,i)+r_i-r_{i-1}}^{(i,i)}-X_{j(i,i)}^{(i,i)}\Vert \ge \zeta ,$then we put $C_0^{(i,i)}:=\cdots :=C_{p(i,i)-1}^{(i,i)}:=I.$Finally, in this step, we define $C_k^i:=C_k^{i-1}\cdot C_k^{(i,1)}\cdot C_k^{(i,2)}\cdots C_k^{(i,i)},k\in {\mathbb{N}}_0.$Considering (13(13) $\Vert M_{l_i}^i\cdots M_2^i\cdot M_1^i\cdot B-A\Vert >\zeta ,i\in \mathbb{N},$(13) ) ⇐ (14(14) $A,B\in \mathcal{X}\cap {\mathcal{O}}_{2\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0).$(14) ) and (31(31) $X_{j(i,i)}^{(i,i)}\in \mathcal{X}\cap {\mathcal{O}}_{\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0).$(31) ), we obtain that the fundamental matrix $\{X_k^{(i+1,1)}{\}}_{k\in {\mathbb{N}}_0}$ of $x_{k+1}=A_k\cdot C_k^i\cdot x_k,k\in {\mathbb{N}}_0,$satisfies (32) $\Vert X_{j(i,i)+r_i-r_{i-1}}^{(i+1,1)}-X_{j(i,i)}^{(i+1,1)}\Vert \ge \zeta .$(32) Indeed, $X_{j(i,i)+r_i-r_{i-1}}^{(i+1,1)}=X_{j(i,i)+r_i-r_{i-1}}^{(i,i)}\cdot M_1^i\cdot M_2^i\cdots M_{l_i}^i,X_{j(i,i)}^{(i+1,1)}=X_{j(i,i)}^{(i,i)}$and $X_{j(i,i)}^{(i,i)},X_{j(i,i)+r_i-r_{i-1}}^{(i,i)}\in \mathcal{X}\cap {\mathcal{O}}_{2\zeta }^{\varrho }(A_{\kappa -1}\cdots A_0)$if $\Vert X_{j(i,i)+r_i-r_{i-1}}^{(i,i)}-X_{j(i,i)}^{(i,i)}\Vert <\zeta ;$and $X_{j(i,i)+r_i-r_{i-1}}^{(i+1,1)}=X_{j(i,i)+r_i-r_{i-1}}^{(i,i)},X_{j(i,i)}^{(i+1,1)}=X_{j(i,i)}^{(i,i)}$if $\Vert X_{j(i,i)+r_i-r_{i-1}}^{(i,i)}-X_{j(i,i)}^{(i,i)}\Vert \ge \zeta .$We continue in the construction for $i+1,i+2,\dots$ We obtain the resulting sequence of $C_k$ by the limit $C_k:=\underset{i\to \mathrm{\infty }}{lim}{C}_k^i,k\in {\mathbb{N}}_0.$This definition is correct, because $C_0^i=I$ for all $i\in \mathbb{N}$ and $C_k^i=C_k^k$ for all $i\ge k$, $i,k\in \mathbb{N}$. In addition, from the construction, one can see that (see also (16(16) $M_j^1\in {\mathcal{O}}_{{\epsilon }_1}^{\varrho }(I),j\in \{1,2,\dots ,l_1\},$(16) ), (19(19) $\begin{array}{rl}& M_j^2\in {\mathcal{O}}_{{\epsilon }_2}^{\varrho }(I),j\in \{1,2,\dots ,l_2\},\\ & \Vert M_1^2\cdot M_2^2\cdots M_{l_2}^2\cdot A_{\kappa -1}\cdots A_0-A_{\kappa -1}\cdots A_0\Vert >\xi .\end{array}$(19) ), …, (25(25) $M_j^i\in {\mathcal{O}}_{{\epsilon }_i}^{\varrho }(I),j\in \{1,2,\dots ,l_i\},$(25) ), …) (33) $\Vert C_k-C_k^i\Vert <{\epsilon }_{i+1},k\in {\mathbb{N}}_0,i\in \mathbb{N},$(33) and that, for any $k\in {\mathbb{N}}_0$, there exist uniquely determined $i_1(k)\in \mathbb{N}$, $i_2(k)\in \{1,\dots ,i_1(k)\}$ for which (see (10(10) ${\epsilon }_i:=\frac{\epsilon }{i},i\in \mathbb{N}.$(10) )) (34) $C_k=C_k^{(i_1(k),i_2(k))}\in {\mathcal{O}}_{{\epsilon }_{i_1(k)}}^{\varrho }(I)\subseteq {\mathcal{O}}_{\epsilon }^{\varrho }(I),$(34) where we put $C_k^{(1,1)}:=C_k^1$, $k\in {\mathbb{N}}_0$. In particular, from (33(33) $\Vert C_k-C_k^i\Vert <{\epsilon }_{i+1},k\in {\mathbb{N}}_0,i\in \mathbb{N},$(33) ), we obtain that $\{C_k{\}}_{k\in {\mathbb{N}}_0}$ is the uniform limit of the sequences $\{C_k^i{\}}_{k\in {\mathbb{N}}_0}$ for $i\to \mathrm{\infty }$; and, from (34(34) $C_k=C_k^{(i_1(k),i_2(k))}\in {\mathcal{O}}_{{\epsilon }_{i_1(k)}}^{\varrho }(I)\subseteq {\mathcal{O}}_{\epsilon }^{\varrho }(I),$(34) ), we have that $\{C_k{\}}_{k\in {\mathbb{N}}_0}\subseteq \mathcal{X}\cap {\mathcal{O}}_{\epsilon }^{\varrho }(I)$. Since all sequences $\{C_k^i{\}}_{k\in {\mathbb{N}}_0}$ are periodic, the sequence $\{C_k{\}}_{k\in {\mathbb{N}}_0}$ is limit periodic (see Definition 2.2).

It remains to show that the fundamental matrix $\{X_k{\}}_{k\in {\mathbb{N}}_0}$ of Equation (8(8) $x_{k+1}=A_k\cdot C_k\cdot x_k,k\in {\mathbb{N}}_0,$(8) ) is not almost periodic. Using (18(18) $\Vert X_{j(1,1)+r_1}^{(2,1)}-X_{j(1,1)}^{(2,1)}\Vert \ge \zeta ,$(18) ) from the first step, (22(22) $\Vert X_{j(2,1)+r_2}^{(2,2)}-X_{j(2,1)}^{(2,2)}\Vert \ge \zeta .$(22) ) and (24(24) $\Vert X_{j(2,2)+r_2-r_1}^{(3,1)}-X_{j(2,2)}^{(3,1)}\Vert \ge \zeta .$(24) ) from the second step, …, (28(28) $\Vert X_{j(i,1)+r_i}^{(i,2)}-X_{j(i,1)}^{(i,2)}\Vert \ge \zeta .$(28) ), (30(30) $\Vert X_{j(i,2)+r_i-r_1}^{(i,3)}-X_{j(i,2)}^{(i,3)}\Vert \ge \zeta .$(30) ), …, (32(32) $\Vert X_{j(i,i)+r_i-r_{i-1}}^{(i+1,1)}-X_{j(i,i)}^{(i+1,1)}\Vert \ge \zeta .$(32) ) from the step no. i, …, we have (35) $\begin{array}{rl}\Vert X_{j(1,1)+r_1}-X_{j(1,1)}\Vert & =\Vert X_{j(1,1)+r_1}^{(2,1)}-X_{j(1,1)}^{(2,1)}\Vert \ge \zeta ,\end{array}$(35) (36) $\begin{array}{rl}\Vert X_{j(2,1)+r_2}-X_{j(2,1)}\Vert & =\Vert X_{j(2,1)+r_2}^{(2,2)}-X_{j(2,1)}^{(2,2)}\Vert \ge \zeta ,\end{array}$(36) (37) $\begin{array}{rl}\Vert X_{j(2,2)+r_2-r_1}-X_{j(2,2)}\Vert & =\Vert X_{j(2,2)+r_2-r_1}^{(3,1)}-X_{j(2,2)}^{(3,1)}\Vert \ge \zeta ,\end{array}$(37) (38) $\begin{array}{rl}& ⋮\\ \Vert X_{j(i,1)+r_i}-X_{j(i,1)}\Vert & =\Vert X_{j(i,1)+r_i}^{(i,2)}-X_{j(i,1)}^{(i,2)}\Vert \ge \zeta ,\end{array}$(38) (39) $\begin{array}{rl}\Vert X_{j(i,2)+r_i-r_1}-X_{j(i,2)}\Vert & =\Vert X_{j(i,2)+r_i-r_1}^{(i,3)}-X_{j(i,2)}^{(i,3)}\Vert \ge \zeta ,\end{array}$(39) (40) $\begin{array}{rl}& ⋮\\ \Vert X_{j(i,i)+r_i-r_{i-1}}-X_{j(i,i)}\Vert & =\Vert X_{j(i,i)+r_i-r_{i-1}}^{(i+1,1)}-X_{j(i,i)}^{(i+1,1)}\Vert \ge \zeta ,\\ & ⋮\end{array}$(40) We apply Theorem 2.1 (together with the definition of almost periodicity for variable $k\in {\mathbb{N}}_0$ in Definition 2.3), where we put $s_1:=0,s_{i+1}:=r_i,i\in \mathbb{N}.$Evidently, (35(35) $\begin{array}{rl}\Vert X_{j(1,1)+r_1}-X_{j(1,1)}\Vert & =\Vert X_{j(1,1)+r_1}^{(2,1)}-X_{j(1,1)}^{(2,1)}\Vert \ge \zeta ,\end{array}$(35) )–(37(37) $\begin{array}{rl}\Vert X_{j(2,2)+r_2-r_1}-X_{j(2,2)}\Vert & =\Vert X_{j(2,2)+r_2-r_1}^{(3,1)}-X_{j(2,2)}^{(3,1)}\Vert \ge \zeta ,\end{array}$(37) ), …, (38(38) $\begin{array}{rl}& ⋮\\ \Vert X_{j(i,1)+r_i}-X_{j(i,1)}\Vert & =\Vert X_{j(i,1)+r_i}^{(i,2)}-X_{j(i,1)}^{(i,2)}\Vert \ge \zeta ,\end{array}$(38) ), (39(39) $\begin{array}{rl}\Vert X_{j(i,2)+r_i-r_1}-X_{j(i,2)}\Vert & =\Vert X_{j(i,2)+r_i-r_1}^{(i,3)}-X_{j(i,2)}^{(i,3)}\Vert \ge \zeta ,\end{array}$(39) ), …, (40(40) $\begin{array}{rl}& ⋮\\ \Vert X_{j(i,i)+r_i-r_{i-1}}-X_{j(i,i)}\Vert & =\Vert X_{j(i,i)+r_i-r_{i-1}}^{(i+1,1)}-X_{j(i,i)}^{(i+1,1)}\Vert \ge \zeta ,\\ & ⋮\end{array}$(40) ), … imply a contradiction with (3(3) $\tau ({\phi }_{k+{\overline{s}}_i},{\phi }_{k+{\overline{s}}_j})<\vartheta$(3) ) for $\vartheta =\zeta$. Therefore, we have proved that $\{X_k{\}}_{k\in {\mathbb{N}}_0}$ cannot be almost periodic.

Remark 5.1

Let us briefly comment the main condition in Theorem 5.1, i.e. the validity of (7(7) $\Vert M_l\cdots M_2\cdot M_1\cdot A_{\kappa -1}\cdots A_0-A_{\kappa -1}\cdots A_0\Vert >\xi .$(7) ). This assumption is motivated by the statement of the theorem, where we obtain non-almost periodic solutions using arbitrarily small changes of the given system $\{A_k{\}}_{k\in {\mathbb{N}}_0}$. In addition, we consider limit periodic transformations, because the result is not valid for only periodic ones (see, e.g. $\{A_k{\}}_{k\in {\mathbb{N}}_0}\equiv \{I{\}}_{k\in {\mathbb{N}}_0}$ and the unitary group $\mathcal{X}$ in the scalar case together with [23, Theorem 5]).

Example 5.1

Theorems 4.3 and 4.4 can be used, e.g. for any transformable commutative group $\mathcal{X}$ (see [21, 23]). Let us consider the complex or real case (i.e. $F=\mathbb{C}$ or $F=\mathbb{R}$ with the usual absolute value) and an arbitrary non-singular matrix S. Then, the groups $\begin{array}{rl}{S}_{\mathbb{Z}}& =\{S^{-1}\cdot \mathrm{diag}[{\mathrm{e}}^{\mathrm{i}{\lambda }_1},\dots ,{\mathrm{e}}^{\mathrm{i}{\lambda }_m}]\cdot S;{\lambda }_1,\dots ,{\lambda }_m\in \mathbb{Z}\},\\ S_{\mathbb{Q}}& =\{S^{-1}\cdot \mathrm{diag}[{\mathrm{e}}^{\mathrm{i}{\lambda }_1},\dots ,{\mathrm{e}}^{\mathrm{i}{\lambda }_m}]\cdot S;{\lambda }_1,\dots ,{\lambda }_m\in \mathbb{Q}\},\\ S_{\mathbb{R}}& =\{S^{-1}\cdot \mathrm{diag}[{\mathrm{e}}^{\mathrm{i}{\lambda }_1},\dots ,{\mathrm{e}}^{\mathrm{i}{\lambda }_m}]\cdot S;{\lambda }_1,\dots ,{\lambda }_m\in \mathbb{R}\}\end{array}$are transformable and commutative, i.e. they satisfy all conditions of Theorems 4.3 and 4.4. In particular, in the complex case, we can consider the matrix group $CU(m)$ which is the intersection of the group of unitary matrices and the set of circulant matrices $\left(\begin{array}{ccccc}{a}_1& a_2& a_3& \cdots & a_m\\ a_m& a_1& a_2& \cdots & a_{m-1}\\ a_{m-1}& a_m& a_1& \cdots & a_{m-2}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ a_2& a_3& a_4& \cdots & a_1\end{array}\right).$Indeed, it is well-known that $CU(m)$ is a special case of $S_{\mathbb{R}}$. In the real case, we point out that the group $SO(m)$ of all $m\times m$ orthogonal matrices with determinant 1 is transformable. Therefore, for m = 2 and $\alpha \in [-\pi ,\pi )$, all matrices in the form $\left(\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \phantom{-}\cos \alpha \end{array}\right)$determine the transformable and commutative group $SO(2)$, i.e. $SO(2)$ satisfies all conditions of Theorems 4.3 and 4.4. Concerning other concrete examples, we refer again to [21, 23].

In addition, we can consider direct sums. We recall that, for groups ${\mathcal{X}}_j\subset Mat(F,m_j)$, $j\in \{1,2,\dots ,n\}$, the direct sum is defined as $\underset{j=1}{\overset{n}{⨁}}{\mathcal{X}}_j={\mathcal{X}}_1\oplus {\mathcal{X}}_2\oplus \cdots \oplus {\mathcal{X}}_n,$where $M\in \underset{j=1}{\overset{n}{⨁}}{\mathcal{X}}_j$ if it has the form $M=\left(\begin{array}{cccc}{M}_1& O& \cdots & O\\ O& M_2& \cdots & O\\ ⋮& ⋮& \ddots & ⋮\\ O& O& \cdots & M_n\end{array}\right),M_1\in {\mathcal{X}}_1,M_2\in {\mathcal{X}}_2,\dots ,M_n\in {\mathcal{X}}_n.$One can use Theorems 4.3 and 4.4 for the direct sum of commutative groups if at least one of the groups satisfies the conditions of the theorems.

Since Theorem 5.1 can be used for any group which satisfies the conditions of Theorems 4.3 and 4.4, we can consider examples mentioned above. We highlight at least the group $\{I\}\oplus SO(2)$ for which one cannot apply Theorems 4.1 and 4.2 (consider $u=(1,0,0{)}^T$).

The main difference between Theorem 5.1 and Theorems 4.3 and 4.4 is given by the fact Theorem 5.1 covers perturbations of any system $\{A_k{\}}_{k\in {\mathbb{N}}_0}$, while only limit periodic and almost periodic systems are treated in Theorems 4.3 and 4.4. Nevertheless, Theorem 5.1 is new also for $\{A_k{\}}_{k\in {\mathbb{N}}_0}\in \mathcal{L}\mathcal{P}(\mathcal{X})$ and $\{A_k{\}}_{k\in {\mathbb{N}}_0}\in \mathcal{A}\mathcal{P}(\mathcal{X})$, which is commented in Example 5.2 below as well.

Theorem 5.1 has several new corollaries. Concerning the motivation in the previous section, we explicitly mention the following two new consequences.

Corollary 5.1

Let $\mathcal{X}$ be a commutative group having the property that there exists $\xi >0$ such that, for any $\delta >0$, there exist matrices $M_1,M_2,\dots ,M_l\in \mathcal{X}$ satisfying $M_i\in {\mathcal{O}}_{\delta }^{\varrho }(I),i\in \{1,2,\dots ,l\},\Vert M_l\cdots M_2\cdot M_1-I\Vert >\xi .$Then, for any $\epsilon >0$ and any $\{A_k{\}}_{k\in {\mathbb{N}}_0}\in \mathcal{L}\mathcal{P}(\mathcal{X})$, there exists $\{B_k{\}}_{k\in {\mathbb{N}}_0}\in {\mathcal{O}}_{\epsilon }^{\sigma }(\{A_k\})\cap \mathcal{L}\mathcal{P}(\mathcal{X})$ with the property that the fundamental matrix of $x_{k+1}=B_k\cdot x_k,k\in {\mathbb{N}}_0,$is not almost periodic.

Proof.

Since $\{A_k{\}}_{k\in {\mathbb{N}}_0}$ is limit periodic (consider Theorem 2.2 and Remark 2.3), there exists $K>\Vert I\Vert$ such that (41) $\Vert A_k\Vert <K,k\in {\mathbb{N}}_0.$(41) From [22, Lemma 4.4], it follows the uniform continuity of the multiplication of matrices on the set $(\mathcal{X}\cap {\mathcal{O}}_K^{\varrho }(O))\times (\mathcal{X}\cap {\mathcal{O}}_{\theta }^{\varrho }(I))$ for any $\theta >0$. Thus, for any $\epsilon >0$, there exists $\vartheta \in (0,1)$ with the property that (42) $A\cdot B\in {\mathcal{O}}_{\epsilon /2}^{\varrho }(A)\mathrm{if}A\in \mathcal{X}\cap {\mathcal{O}}_K^{\varrho }(O),B\in \mathcal{X}\cap {\mathcal{O}}_{\vartheta }^{\varrho }(I).$(42)

We use Theorem 5.1, where we put $\kappa =0$ for the considered $\{A_k{\}}_{k\in {\mathbb{N}}_0}\in \mathcal{L}\mathcal{P}(\mathcal{X})$. Then, (7(7) $\Vert M_l\cdots M_2\cdot M_1\cdot A_{\kappa -1}\cdots A_0-A_{\kappa -1}\cdots A_0\Vert >\xi .$(7) ) reduces to (see (6(6) $A_{-1}\cdots A_0:=I.$(6) )) $\Vert M_l\cdots M_2\cdot M_1-I\Vert >\xi .$Applying Theorem 5.1 for $\vartheta >0$ introduced in the previous paragraph, we have a limit periodic sequence $\{C_k{\}}_{k\in {\mathbb{N}}_0}\subseteq \mathcal{X}\cap {\mathcal{O}}_{\vartheta }^{\varrho }(I)$ such that the fundamental matrix of $x_{k+1}=A_k\cdot C_k\cdot x_k,k\in {\mathbb{N}}_0,$is not almost periodic. Therefore, we consider the homogeneous linear difference system given by $\{B_k{\}}_{k\in {\mathbb{N}}_0}:=\{A_k\cdot C_k{\}}_{k\in {\mathbb{N}}_0}$ whose fundamental matrix is not almost periodic. Evidently (see (5(5) $\sigma (\{A_k\},\{B_k\}):=\underset{k\in {\mathbb{N}}_0}{sup}\varrho (A_k,B_k),\{A_k{\}}_{k\in {\mathbb{N}}_0},\{B_k{\}}_{k\in {\mathbb{N}}_0}\in \mathcal{A}\mathcal{P}(\mathcal{X}).$(5) ), (41(41) $\Vert A_k\Vert <K,k\in {\mathbb{N}}_0.$(41) ), and (42(42) $A\cdot B\in {\mathcal{O}}_{\epsilon /2}^{\varrho }(A)\mathrm{if}A\in \mathcal{X}\cap {\mathcal{O}}_K^{\varrho }(O),B\in \mathcal{X}\cap {\mathcal{O}}_{\vartheta }^{\varrho }(I).$(42) )), we have that $\{B_k{\}}_{k\in {\mathbb{N}}_0}\in {\mathcal{O}}_{\epsilon }^{\sigma }(\{A_k\})$, because $\{C_k{\}}_{k\in {\mathbb{N}}_0}\subseteq {\mathcal{O}}_{\vartheta }^{\varrho }(I)\cap \mathcal{X}$. Moreover, (43) $C_k\in {\mathcal{O}}_{\vartheta }^{\varrho }(I)\subset {\mathcal{O}}_1^{\varrho }(I)\subset {\mathcal{O}}_{K+1}^{\varrho }(O),k\in {\mathbb{N}}_0.$(43)

Considering the limit periodicity of $\{A_k{\}}_{k\in {\mathbb{N}}_0}$ and $\{C_k{\}}_{k\in {\mathbb{N}}_0}$, there exist periodic sequences $\{A_k^p{\}}_{k\in {\mathbb{N}}_0}$ and $\{C_k^p{\}}_{k\in {\mathbb{N}}_0}$ for all $p\in \mathbb{N}$ such that $\Vert A_k^p-A_k\Vert <\frac{1}{p},\Vert C_k^p-C_k\Vert <\frac{1}{p},k\in {\mathbb{N}}_0,$which yield (see also (41(41) $\Vert A_k\Vert <K,k\in {\mathbb{N}}_0.$(41) ) and (43(43) $C_k\in {\mathcal{O}}_{\vartheta }^{\varrho }(I)\subset {\mathcal{O}}_1^{\varrho }(I)\subset {\mathcal{O}}_{K+1}^{\varrho }(O),k\in {\mathbb{N}}_0.$(43) )) $\begin{array}{rl}\Vert C_k^p\cdot A_k^p-C_k\cdot A_k\Vert & \le \Vert C_k^p\cdot A_k^p-C_k\cdot A_k^p\Vert +\Vert C_k\cdot A_k^p-C_k\cdot A_k\Vert \\ & \le \Vert C_k^p-C_k\Vert \cdot \Vert A_k^p\Vert +\Vert C_k\Vert \cdot \Vert A_k^p-A_k\Vert \le \frac{2(K+1)}{p},k\in {\mathbb{N}}_0.\end{array}$These estimations together with the periodicity of all sequences $\{C_k^p\cdot A_k^p{\}}_{k\in {\mathbb{N}}_0}$ prove that $\{B_k{\}}_{k\in {\mathbb{N}}_0}$ is limit periodic if $\{A_k{\}}_{k\in {\mathbb{N}}_0}$ and $\{C_k{\}}_{k\in {\mathbb{N}}_0}$ are limit periodic. The proof is complete.

Remark 5.2

It is evident that the statement of Corollary 5.1 does not change if one replaces the system $\{A_k{\}}_{k\in {\mathbb{N}}_0}\in \mathcal{L}\mathcal{P}(\mathcal{X})$ by a periodic one, i.e. if $\{A_k{\}}_{k\in {\mathbb{N}}_0}\in \mathcal{P}(\mathcal{X})$. Indeed, it follows directly from Definition 2.2.

Corollary 5.2

Let $\mathcal{X}$ be a commutative group having the property that there exists $\xi >0$ such that, for any $\delta >0$, there exist matrices $M_1,M_2,\dots ,M_l\in \mathcal{X}$ satisfying $M_i\in {\mathcal{O}}_{\delta }^{\varrho }(I),i\in \{1,2,\dots ,l\},\Vert M_l\cdots M_2\cdot M_1-I\Vert >\xi .$Then, for any $\epsilon >0$ and any $\{A_k{\}}_{k\in {\mathbb{N}}_0}\in \mathcal{A}\mathcal{P}(\mathcal{X})$, there exists $\{B_k{\}}_{k\in {\mathbb{N}}_0}\in {\mathcal{O}}_{\epsilon }^{\sigma }(\{A_k\})$ with the property that the fundamental matrix of $x_{k+1}=B_k\cdot x_k,k\in {\mathbb{N}}_0,$is not almost periodic.

Proof.

We can proceed as in the proof of Corollary 5.1 for (44) $\{B_k{\}}_{k\in {\mathbb{N}}_0}:=\{A_k\cdot C_k{\}}_{k\in {\mathbb{N}}_0}\in {\mathcal{O}}_{\epsilon }^{\sigma }(\{A_k\}),$(44) where the boundedness of $\{A_k{\}}_{k\in {\mathbb{N}}_0}$ (see (41(41) $\Vert A_k\Vert <K,k\in {\mathbb{N}}_0.$(41) )) remains valid. Note that the sequences $\{A_k{\}}_{k\in {\mathbb{N}}_0}$, $\{C_k{\}}_{k\in {\mathbb{N}}_0}$ are almost periodic ($\{C_k{\}}_{k\in {\mathbb{N}}_0}$ is even limit periodic, see Remark 2.3). Thus, the almost periodicity of the sequence $\{B_k{\}}_{k\in {\mathbb{N}}_0}$ (introduced in (44(44) $\{B_k{\}}_{k\in {\mathbb{N}}_0}:=\{A_k\cdot C_k{\}}_{k\in {\mathbb{N}}_0}\in {\mathcal{O}}_{\epsilon }^{\sigma }(\{A_k\}),$(44) )) follows from Theorem 2.1 (consider also Corollary 2.1), because any linear combination of almost periodic sequences is almost periodic and the product of two almost periodic sequences is almost periodic in the considered spaces.

Remark 5.3

Corollary 5.2 does not follow from Corollary 5.1. In [4], there is proved that there exist sequences which are almost periodic and, at the same time, which are not limit periodic. It suffices to consider the almost periodic sequence $\{A_k{\}}_{k\in {\mathbb{N}}_0}:=\{{\mathrm{e}}^{\mathrm{i}k}{\}}_{k\in {\mathbb{N}}_0}$ for the unitary group $\mathcal{X}$ in the scalar case. Then, the system determined by $\{A_k{\}}_{k\in {\mathbb{N}}_0}$ has a neighbourhood (in $\mathcal{A}\mathcal{P}(\mathcal{X})$) without any limit periodic system.

Remark 5.4

In fact, considering the statements of our results above and their proofs, one can see that it is not necessary to introduce the absolute value $|\cdot |$ on the whole field. It suffices to have it on a neighbourhood of $e_0$ and for all elements of matrices from the matrix group $\mathcal{X}$. For details, we refer to [23, Remark 15].

Example 5.2

To show that Corollaries 5.1 and 5.2 do not follow from Theorems 4.3 and 4.4, it suffices to consider a field F for which the unit ball $\{u\in F^m;\Vert u\Vert \le 1\}$ is not compact. For example, let us consider the field of all meromorphic functions defined on an arbitrarily given non-empty open connected set T, where meromorphic functions can be added, subtracted, multiplied, and the quotient can be formed applying the analytic continuation to eliminate removable singularities. For simplicity, we use Remark 5.4 and we put m = 1 (i.e. we consider the scalar case, when all groups are commutative). Let $F_0\subset F$ be the set of all bounded functions on T. For any $f\in F_0$, we introduce $|f|:=\underset{z\in T}{sup}|f(z)|,$where $|f(z)|$ is the absolute value of complex number $f(z)$. Note that $\Vert f\Vert =|f|$, $f\in F_0$.

Now, for any group $\mathcal{X}$ whose all elements are from $F_0$, one can apply Corollaries 5.1 and 5.2 if ${\mathcal{X}}_{\delta }:=\mathcal{X}\cap ({\mathcal{O}}_{\delta }^{\varrho }(I)\setminus \{I\})\equiv \mathcal{X}\cap ({\mathcal{O}}_{\delta }^{\varrho }((1))\setminus \{(1)\})\ne \mathrm{\varnothing }$for all $\delta >0$, where 1 denotes a constant function. Indeed, it suffices to consider only powers of matrices from ${\mathcal{X}}_{\delta }$. We remark that $\mathcal{X}$ does not need to be bounded (cf. [23, Example 4]).  

Acknowledgments

The authors would like to thank the referees.

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No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research presented in this paper was supported by the Czech Science Foundation [grant number GA20-11846S].