\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 126, pp. 1--22.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/126\hfil Construction of almost periodic sequences]
{Construction of almost periodic sequences with given properties}

\author[M. Vesel\'y\hfil EJDE-2008/126\hfilneg]
{Michal Vesel\'y}

\address{Michal Vesel\'y \newline
Department of Mathematics and Statistics\\
Masaryk University\\
Jan\'{a}\v{c}kovo n\'{a}m. 2a \\
CZ-602 00 Brno, Czech Republic}
\email{michal.vesely@mail.muni.cz}

\thanks{Submitted October 9, 2007. Published September 10, 2008.}
\thanks{Supported by  grant 201/07/0145 of the Czech Grant Agency}
\subjclass[2000]{39A10, 42A75}
\keywords{keywords Almost periodic sequences;
\hfill\break\indent
almost periodic solutions to difference systems}

\begin{abstract}
 We define almost periodic sequences with values in a pseudometric
 space $\mathcal{X}$ and we modify the Bochner definition of
 almost periodicity so that it remains equivalent with the Bohr
 definition.
 Then, we present one (easily modifiable) method for constructing
 almost periodic sequences in $\mathcal{X}$.  Using this method,
 we find almost periodic homogeneous linear difference systems that
 do not have any non-trivial almost periodic solution.
 We treat this problem in a general setting where we suppose that
 entries of matrices in linear systems belong to a ring with a unit.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks


\section{Introduction}

First of all we mention the article \cite{ves796fa} by Fan which considers
almost periodic sequences of elements of a metric space and the
article \cite{ves794to} by Tornehave about almost periodic functions of
the real variable with values in a metric space.
In these papers, it is shown that many theorems that are valid for
complex valued sequences and functions
are no longer true. For continuous functions,
it was observed that the important property
is the local connection by arcs of the space
of values and also its completeness.
However, we will not use their results or other theorems and we
will define the notion of the almost periodicity of sequences
in pseudometric spaces without any conditions, i.e.,
the definition is similar to the classical definition of  Bohr,
the modulus being replaced by the distance.
We also refer to \cite{ves797z4} (or \cite{ves797z1}).
We add that
the concept of almost periodic functions of several variables with
respect to Hausdorff metrics can be found in \cite{vesfSU} which is
an extension of \cite{vesfDZ}.

In Banach spaces, a sequence $\{\varphi_k\}_{k \in \mathbb{Z}}$
is almost periodic if and only if any sequence of
translates of $\{\varphi_k\}$ has
a subsequence which converges and
its convergence is uniform with respect to $k$ in the sense
of the norm.
In 1933, the continuous case of the previous result was proved
by  Bochner in \cite{ves797bo}, where the fundamental
theorems of the theory of almost periodic functions with values in
a Banach space are proved too -- see, e.g.,
\cite{ves797am}, \cite[pp. 3--25]{ves795ap} or \cite{ves797ko},
where the theorems have been
redemonstrated by the methods of the functional analysis.
We add that the discrete version of this result can be proved similarly
as in \cite{ves797bo}. We also mention
directly the papers \cite{ves798wa} and \cite{ves794se}.

In pseudometric spaces, the above result is not generally true.
Nevertheless, by a modification of the Bochner proof of this result, we
will prove that a necessary and sufficient condition
for a sequence $\{\varphi_k\}_{k \in \mathbb{Z}}$
to be almost periodic is that
any sequence of translates of $\{\varphi_k\}$ has
a subsequence satisfying the Cauchy condition, uniformly with respect to $k$.

The paper is organized as follows. The next section presents
the definition of almost periodic sequences in a pseudo\-metric space,
the above
necessary and sufficient condition for the almost periodicity of a sequence
$\{\varphi_k\}_{k \in \mathbb{Z}}$,
and some basic properties of almost periodic sequences in pseudometric spaces
(see also, e.g., \cite{ves95MU}).

In Section 3, we show the way one can construct
almost periodic sequences in pseudometric spaces.
We present it in the below given theorems.
In Theorem \ref{thm3}, we consider almost periodic sequences for $k \in \mathbb{N}_0$;
in Theorem \ref{thm4} and Corolla\-ry~\ref{coro1}, sequences for $k \in \mathbb{Z}$
obtained from almost periodic sequences for $k \in \mathbb{N}_0$; and,
in Theorems \ref{thm5} and \ref{thm6}, sequences for $k \in \mathbb{Z}$.
We remark that our process is comprehensible and
easily modifiable. We add that methods of generating
almost periodic sequences are mentioned also in \cite[Section 4]{ves95MU}.

Then, in Section 4, we use results from the second and the third section
of this paper to obtain a theorem which will play important
role in the article \cite{ves792ve}, where
it is proved that the almost periodic homogeneous linear
difference systems which do not have any nonzero almost periodic solutions
form a dense subset of the set of all considered systems.
Using our method, one can get generalizations of statements
from \cite{ves798tk} and \cite{ves721ves},
where unitary (and orthogonal) systems are studied (see also \cite{ves798ves}).

We will analyse systems of the form
$$
x_{k+1} = A_k \cdot x_k, \quad  k \in \mathbb{Z} \quad
(\text{or }  k \in \mathbb{N}_0),
$$
where $\{A_k\}$ is almost periodic.
We want to prove that there exists a system of the above form
which does not have an almost periodic solution other than the trivial one.
(See Theorem \ref{thm7}.)
A closer examination of the methods used in constructions
reveals that the problem can be treated in possibly the most general
setting:
\begin{itemize}
\item[(1)] almost periodic sequences attain values
in a pseudometric space;
\item[(2)] the entries of almost periodic matrices are elements of an
infinite ring with a unit.
\end{itemize}

We note that many theorems about the existence of almost periodic solutions
of almost periodic difference systems of general forms are
published in \cite{ves9GOP, ves9HAM,ves794SO,ves797z1,ves797z3,ves797z4}
and several these existence theorems are proved there
in terms of discrete Lyapunov functions. Here, we can also refer to the
monograph \cite{ves792yo} and \cite[Theorems 3.6, 3.7, 3.8]{ves797z8}.
We add that the existence of an almost periodic homogeneous linear \emph{differential} system, which
has nontrivial bounded solutions and, at the same time, all the systems
from some neighbourhood
of it have no nontrivial almost periodic solutions, is proved
in \cite{ves704tk}.

As usual, $\mathbb{R}$ denotes the real line, $\mathbb{R}^{+}_0$ the set of all nonnegative
reals, $\mathbb C$ the complex plane, $\mathbb{Z}$ denotes
the set of integers, $\mathbb{N}$ the set of natural numbers, and
$\mathbb{N}_0$
the set of positive integers including the zero $0$.

Let $\mathcal{X} \ne \emptyset$ be an arbitrary set and let
$d : \mathcal{X} \times \mathcal{X} \to \mathbb{R}^+_0$ have these properties:
\begin{itemize}
 \item[(i)] $d(x, x) = 0$ for all $x \in \mathcal{X}$,
\item[(ii)] $d(x, y) = d(y, x)$ for all $x, y \in \mathcal{X}$,
\item[(iii)] $d(x, y) \le d(x,z) + d(z, y)$ for all $x, y, z \in \mathcal{X}$.
\end{itemize}
We say that $d$ is \emph{a pseudometric} on $\mathcal{X}$ and
$(\mathcal{X},d)$ \emph{a pseudometric space}.

For given $\varepsilon >0$, $x \in \mathcal{X}$,
in the same way as in metric spaces,
we define \emph{the} $\varepsilon$-\emph{neigh\-bourhood} of $x$
in $\mathcal{X}$ as the
set $\{y \in \mathcal{X}; \, d(x, y) < \varepsilon \}$. It will be
denoted by~$\mathcal{O}_\varepsilon  (x)$.

All sequences, which we will consider, will be subsets of $\mathcal{X}$.
The scalar (and vector) valued sequences will be denoted by the
lower-case letters,
the matrix valued sequences by the capital letters ($\mathcal{X}$ is a set
of matrices in this case), and each one of the scalar and matrix
valued sequences
by the symbols $\{\varphi_k\}$, $\{\psi_k\}$, $\{\chi_k\}$.

\section{Almost periodic sequences in pseudometric spaces}

Now we introduce a ``natural'' generalization of the almost periodicity.
We remark that our approach is very general and that
the theory of almost periodic sequences presented here does not
distinguish between
$x \in \mathcal{X}$ and $y \in \mathcal{X}$ if $d(x, y) = 0$.

\begin{definition} \label{def1}\rm
A sequence $\{\varphi_k\}$ is called \emph{almost periodic}
if, for any $\varepsilon > 0$,
there exists a positive integer $p (\varepsilon) $ such that any set
consisting of $p (\varepsilon) $ consecutive integers
(nonnegative integers if $k \in \mathbb{N}_0$)
contains at least one integer $l$ with the property that
$$
d ( \varphi_{k + l}, \varphi_k ) < \varepsilon, \quad  k \in \mathbb{Z}
\quad (\text{or } k \in \mathbb{N}_0).
$$
In the above definition, $l$ is called an
$\varepsilon$-\emph{translation number} of $\{\varphi_k\}$.
 \end{definition}

Consider again $\varepsilon> 0$. Henceforward, the set
of all ${\varepsilon}$-translation numbers
of a~sequence $\{\varphi_k\}$
will be denoted by $\mathfrak{T} (\{\varphi_k\}, \varepsilon)$.

\begin{remark} \label{rmk1} \rm
If $\mathcal{X}$ is a Banach space ($d(x,y)$ is given by $||\,x - y\,||$),
then a necessary and sufficient condition for a sequence
$\{\varphi_k\}_{k \in \mathbb{Z}}$
to be almost periodic is it to be \emph{normal}; i.e.,
$\{\varphi_k\}$ is almost periodic if and only if any sequence of
translates of $\{\varphi_k\}$
has a subsequence, uniformly convergent for $k \in \mathbb{Z}$ in the sense
of the norm.
This statement and the below given Theorem \ref{thm1} are not valid if
$\{\varphi_k\}$ is defined for $k \in \mathbb{N}_0$ and if we consider only
translates to the right -- see the example
$\mathcal{X} = \mathbb{R}$, $\varphi_0 = 1$, and $\varphi_k = 0$,
$k \in \mathbb{N}$.
But, if we consider translates to the left, then both of results
are valid for $k \in \mathbb{N}_0$ too.
 \end{remark}

It is seen that the above result is no longer valid
if the space of values fails to be complete.
Especially, in a pseudometric space
$(\mathcal{X}, d)$, it is not generally true that a sequence
$\{\varphi_k\}_{k \in \mathbb{Z}}$ is almost periodic
if and only if it is normal.
Nevertheless, applying the methods from any one of the proofs
of the results \cite[Theorem 1.10, p. 16]{ves711co},
\cite[Theorem 1.14, pp. 9--10]{ves798fi},
and \cite[Statement ($\zeta $), pp. 8--9]{ves795ap},
one can easily prove that every normal sequence
$\{\varphi_k\}_{k \in \mathbb{Z}}$ is almost periodic.
Further, we can prove the next theorem which we will need later. We add that
its proof is a~modification of the proof of
\cite[Theorem 1.26, pp. 45--46]{ves711co}.

\begin{theorem} \label{thm1}
Let $\{ \varphi_k \}_{k \in \mathbb{Z}} $ be given.
For an arbitrary sequence $\{ h_n\}_{n \in \mathbb{N}} \subseteq \mathbb{Z} $,
there exists a sub\-se\-quence
$\{\tilde{h}_n\}_{n \in \mathbb{N}} \subseteq \{ h_n\}_{n \in \mathbb{N}} $
with the Cauchy property with respect to~$\{ \varphi_k \}$, i.e., for any
$\varepsilon > 0$, there exists $ M \in \mathbb{N}$ for which the inequality
$$
d \big(\varphi_{k + \tilde{h}_{i}}, \varphi_{k + \tilde{h}_{j}} \big)
< \varepsilon
$$
holds for all $i, j, k \in \mathbb{Z}$, $i, j > M$, if and only if
$\{ \varphi_k \}_{k \in \mathbb{Z}}$ is almost periodic.
\end{theorem}

\begin{proof}
If any sequence of translates of $\{ \varphi_k \}$ has
a subsequence which has the Cauchy property, then $\{ \varphi_k \}$
is almost periodic. It can be proved similarly as
\cite[Theorem~1.10, p.~16]{ves711co}, where it is not used that 
$\mathcal{X}$ is complete.
To prove the opposite implication, we will assume that $\{ \varphi_k \}$
is almost periodic, and we will use the known method of the
diagonal extraction.

Let $\{ h_n\}_{n \in \mathbb{N}} \subseteq \mathbb{Z} $
and $\vartheta > 0$ be arbitrary. By Definition \ref{def1}, there exists
a~po\-sitive integer $p$ such that, in any set
$\{h_n - p, h_n - p + 1, \dots, h_n \}$, there exists
a~${\vartheta}$-tran\-sla\-tion number $l_n$.
We know that $0 \le h_n - l_n \le p$ for all $n \in \mathbb{N}$.
We put $k_n := h_n - l_n$, $n \in \mathbb{N}$.
Clearly, $k_n = c =$ const. (a constant value from $\{0, 1, \dots, p\}$) for
infinitely many values of $n$.
Since
$$
d (\varphi_{k + h_n}, \varphi_{k + k_n})
=  d (\varphi_{(k + h_n - l_n) + l_n}, \varphi_{k + h_n - l_n})
 < \vartheta,\quad  k \in \mathbb{Z},
$$
there exists a subsequence $\{ {h}_n^1 \}$ of $\{ h_n \}$ and
an integer $c_1$ such that
\begin{equation}  \label{ves751a}
d (\varphi_{k + h_n^1}, \varphi_{k + c_1})  <  \vartheta, \quad
 k \in \mathbb{Z}, \; n \in \mathbb{N}.
\end{equation}

Consider now a sequence of positive numbers
$\vartheta_1 > \vartheta_2 > \dots > \vartheta_n > \dots$
converging to 0. We extract from the sequence
$\{ {\varphi_{k + h_n}} \}$
a subsequence $\{ {\varphi_{k + h_n^1}} \}$ which satisfies \eqref{ves751a}
for $\vartheta = \vartheta_1$. From this sequence we extract
a subsequence $\{ {\varphi_{k + h_n^2}} \}$
for which an
inequality analogous to \eqref{ves751a} is valid.
Of course, $c$ will not be the same, but will depend on the
subsequence. We proceed further in the same way. Next, we form
the sequence $\{ {\varphi_{k + h_n^n}} \}_{n \in \mathbb{N}}$.
Assume that $\varepsilon > 0$ is given and that we have
$2 \vartheta_m < \varepsilon$ for $m \in \mathbb{N}$.
As a result, for $i, j > m$, $i,j \in \mathbb{N}$, we obtain
$$
d \big(\varphi_{k + h_i^i}, \varphi_{k + h_j^j} \big)  \leq
d \big(\varphi_{k + h_i^i}, \varphi_{k + c_m} \big)
+ d \big(\varphi_{k + c_m}, \varphi_{k + h_j^j} \big) < \varepsilon, \quad  k \in \mathbb{Z},
 $$
where $c_m$ is the number corresponding to the sequence
$\{ {\varphi_{k + h_n^m}} \}_{n \in \mathbb{N}}$ and $\vartheta_m$.
\end{proof}

\begin{theorem} \label{thm2}
Let $\mathcal{X}_1, \mathcal{X}_2$ be arbitrary pseudometric spaces
and $\Phi : \mathcal{X}_1 \to \mathcal{X}_2$ be a~uni\-formly continuous map.
If $ \{\varphi_k\}$ is almost periodic, then
the se\-quence $\{\Phi \circ \varphi_k\}$ is almost periodic too.
\end{theorem}

\begin{proof}
Taking $\varepsilon > 0$ arbitrarily,
let $\delta (\varepsilon) > 0$ be the number corresponding
to ${\varepsilon}$ from the definition of the uniform continuity of $\Phi$.
Now, Theorem \ref{thm2} follows from the fact that the set of all
$\varepsilon$-translation
numbers of $\{\Phi \circ \varphi_k\}$ contains the set of
all $\delta (\varepsilon)$-tran\-slation
numbers of $\{\varphi_k\}$, i.e., from the inclusion
$$
\mathfrak{T} (\{\varphi_k\}, \delta (\varepsilon))
\subseteq \mathfrak{T} (\{\Phi \circ \varphi_k\}, \varepsilon).
$$
\end{proof}

We note that we can prove many theorems which
are valid in the classical case also for
pseudometric spaces. For example, we mention the following result:
\begin{itemize}
\item[(a)] For every
sequence of almost periodic sequences $\{\varphi_k^1\}, \dots,
\{\varphi_k^n\}, \dots $, the
sequence of
$ \lim _{n \to \infty } \varphi_k^n$
is almost periodic if the convergence is uniform with respect to $k$.
 \end{itemize}
Taking $n \in \mathbb{N}$ and using Theorem \ref{thm1}
(and Remark \ref{rmk1}) $n$-times,
one can easily prove also:
\begin{itemize}
 \item[(b)]
If $(\mathcal{X}_1, d_1), \dots, (\mathcal{X}_n, d_n)$
are pseudometric spaces and $\{\varphi_k^1\}, \dots , \{\varphi_k^n\}$
are arbitrary almost periodic sequences with values in
$\mathcal{X}_1, \dots, \mathcal{X}_n$,
respectively, then the sequence $\{\psi_k\}$,
with values in $ \mathcal{X}_1 \times \dots \times \mathcal{X}_n$
given by
$$
\psi_k := (\varphi_k^1 , \dots, \varphi_k^n)\quad
 \text{for all considered } k ,
$$
is also almost periodic.

\item[(c)] Let the sequences $\{ \varphi^1_{k} \},\dots, \{ \varphi^n_{k} \}$
$({k \in \mathbb{Z}}$ or ${k \in \mathbb{N}_0})$
be given. Then, the se\-quence~$\{ \psi_{k} \}$ which is defined by
$$
\psi_k := \varphi^{i + 1}_{j} \quad  \text{for all considered } k ,
$$
where $k = jn + i $, $j \in \mathbb{Z}$ ($j \in \mathbb{N}_0$),
$i \in \{0, \dots, n - 1\}$,
is almost periodic if and only if all sequences $\{\varphi_k^1\}, \dots , \{\varphi_k^n\}$ 
are almost periodic.
\end{itemize}
For the first result (in $\mathbb C$),
see \cite[Theorem 6.4, p. 139]{ves711co}.
We remark that one can use the above theorems to obtain more
general versions of Theorems \ref{thm3}, \ref{thm5}, \ref{thm6}.

\section{Construction of almost periodic sequences}

Now we prove several theorems which facilitate to find almost periodic
sequences having certain specific properties:

\begin{theorem} \label{thm3}
Let $ m \in \mathbb{N}_0 $, $\varphi_0, \dots, \varphi_m \in \mathcal{X}$,
and $j \in \mathbb{N}$ be arbitrarily given.
Let $\{r_{n}\}_{n \in \mathbb{N}}$ be an arbitrary
sequence of nonnegative real numbers such that
\begin{equation} \label{ves759sa}
\sum_{n=1}^\infty { r_{n}} < + \infty.
\end{equation}
Then, any sequence
$\{\varphi_{k}\}_{k \in \mathbb{N}_0} \subseteq \mathcal{X}$,
where $(n > 2, n \in \mathbb{N})$
\begin{gather*}
\varphi_{k} \in \mathcal{O}_{r_{1}} (\varphi_{k - (m + 1)}),
  \quad  k \in \{ m + 1, \dots, 2m + 1 \}, \\
\varphi_{k} \in \mathcal{O}_{r_{1}} (\varphi_{k - 2 (m + 1)}),
  \quad  k \in \{ 2(m + 1), \dots, 3(m + 1) - 1 \}, \\
\dots \\
\varphi_{k} \in \mathcal{O}_{r_{1}} (\varphi_{k - j (m + 1)}),
 \quad  k \in \{ j(m + 1), \dots, (j + 1)(m + 1) - 1 \}, \\
\varphi_{k} \in \mathcal{O}_{r_{2}} (\varphi_{k - (j + 1)(m + 1)}),
 \quad  k \in \{ (j + 1) (m + 1), \dots, 2(j + 1)(m + 1) - 1 \}, \\
\varphi_{k} \in \mathcal{O}_{r_{2}} (\varphi_{k - 2(j + 1)(m + 1)}),
 \quad  k \in \{ 2(j + 1) (m + 1), \dots, 3(j + 1)(m + 1) - 1 \}, \\
\dots \\
\varphi_{k} \in \mathcal{O}_{r_{2}} (\varphi_{k - j(j + 1) (m + 1)}),
  \quad  k \in \{ j(j + 1) (m + 1), \dots, (j + 1)^2 (m + 1) - 1 \}, \\
\dots\\
\begin{aligned}
\varphi_{k} \in \mathcal{O}_{r_{n}}
(\varphi_{k - (j + 1)^{n - 1}(m + 1)}),
 \quad  k \in \big\{ &(j + 1)^{n - 1} (m + 1), \\
 &\dots, 2(j + 1)^{n - 1} (m + 1) - 1 \big\},
\end{aligned}
\\
\begin{aligned}
\varphi_{k} \in \mathcal{O}_{r_{n}}
(\varphi_{k - 2(j + 1)^{n - 1} (m + 1)}),
 \quad  k \in \big\{& 2(j + 1)^{n - 1} (m + 1), \\
&\dots, 3(j + 1)^{n - 1}  (m + 1) - 1 \big\},
\end{aligned} \\
\dots \\
\begin{aligned}
\varphi_{k} \in \mathcal{O}_{r_{n}}
 (\varphi_{k - j(j + 1)^{n - 1} (m + 1)}),
 \quad  k \in \big\{ &j(j + 1)^{n - 1} (m + 1), \\
&\dots, (j + 1)^{n} (m + 1) - 1 \}, \dots
\end{aligned}
\end{gather*}
are arbitrary too, is almost periodic.
\end{theorem}

\begin{proof}
Consider an arbitrary $\varepsilon > 0$. We need to prove that
the set of all $\varepsilon$-tran\-s\-la\-tion numbers of $\{\varphi_{k}\}$
is relative dense in $\mathbb{N}_0$.
Using \eqref{ves759sa}, one can find $n(\varepsilon)$ for which
\begin{equation} \label{ves7ggsa}
\sum_{n=n(\varepsilon)}^\infty { r_{n}} < \frac{\varepsilon}{2}.
\end{equation}
We see that
\begin{equation}  \label{vesh38g}
\begin{gathered}
\varphi_{k + (j + 1)^{n(\varepsilon) - 1} (m + 1)} \in \mathcal{O}_{r_{n(\varepsilon)}} (\varphi_k), \\
\varphi_{k + 2(j + 1)^{n(\varepsilon) - 1} (m + 1)} \in \mathcal{O}_{r_{n(\varepsilon)}} (\varphi_k),  \\
\dots \\
\varphi_{k + j(j + 1)^{n(\varepsilon) - 1} (m + 1)} \in \mathcal{O}_{r_{n(\varepsilon)}} (\varphi_k)
\end{gathered}
\end{equation}
for
$$
0 \le k < (j + 1)^{n(\varepsilon) - 1} (m + 1).
$$
Next, from (iii) and \eqref{vesh38g} it follows
($i \in \{(j + 1)^n, \dots, (j + 1)^{n +1} - 1 \}$, $n \in \mathbb{N}$)
\begin{gather*}
\varphi_{k + (j + 1) (j + 1)^{n(\varepsilon) - 1} (m + 1)}
 \in \mathcal{O}_{r_{n(\varepsilon)}
 + r_{n(\varepsilon) + 1}} ( \varphi_{k}), \\
\dots \\
\varphi_{k + ((j + 1)^2 - 1)(j + 1)^{n(\varepsilon) - 1} (m + 1)}
\in \mathcal{O}_{r_{n(\varepsilon)} + r_{n(\varepsilon) + 1}}
( \varphi_{k} ), \\
 \dots \\
\varphi_{k + i(j + 1)^{n(\varepsilon) - 1} (m + 1)}
  \in \mathcal{O}_{r_{n(\varepsilon)} + r_{n(\varepsilon) + 1}
+ \dots + r_{n(\varepsilon) + n}} ( \varphi_{k} ), \\
\dots
\end{gather*}
for $ k \in \{0, \dots , (j + 1)^{n(\varepsilon) - 1} (m + 1) - 1\}$.
Therefore (consider \eqref{ves7ggsa}), we have
\begin{equation} \label{ves7792}
 \varphi_{k + l(j + 1)^{n(\varepsilon) - 1} (m + 1)}
  \in \mathcal{O}_{\frac{\varepsilon}{2}} (\varphi_k), \quad
 0 \le k < (j + 1)^{n(\varepsilon) - 1} (m + 1), \, l \in \mathbb{N}_0.
\end{equation}
We put
\begin{equation} \label{ves5tt0}
q(\varepsilon) := (j + 1)^{n(\varepsilon) - 1} (m + 1).
\end{equation}
Any $p \in \mathbb{N}_0$ can be expressed uniquely in the form
$$
p = k (p) +  l(p)  q(\varepsilon) \quad  \text{for some }
 k(p) \in \{0, \dots,  q(\varepsilon) - 1\}  \text{ and }
l(p) \in \mathbb{N}_0.
$$
Applying \eqref{ves7792}, we obtain
\begin{equation}  \label{ves6g31}
\begin{aligned}
d (\varphi_{p}, \varphi_{p + l q(\varepsilon)})
&=  d (\varphi_{k (p) +  l(p) q(\varepsilon)}, \varphi_{k (p)
 +  l(p)  q(\varepsilon) + l q(\varepsilon)}) \\
&\le  d (\varphi_{k (p) +  l(p)  q(\varepsilon)}, \varphi_{k (p)})
   + d (\varphi_{k (p)}, \varphi_{k (p) + (l + l(p)) q(\varepsilon)}
) \\
&< \frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon,
\end{aligned}
\end{equation}
where $p, l \in \mathbb{N}_0$ are arbitrary; i.e.,
$l q(\varepsilon)$
is an $\varepsilon$-translation number of $\{\varphi_{k}\}$ for all
$l \in \mathbb{N}_0$.
The fact that the set $\{l q(\varepsilon);  l \in \mathbb{N}_0 \}$
is relative dense in $\mathbb{N}_0$ proves the theorem.
 \end{proof}

\begin{remark} \label{rmk2} \rm
From the proof of Theorem \ref{thm3} (see \eqref{ves5tt0} and \eqref{ves6g31}),
for any $\varepsilon > 0$ and
any sequence $\{\varphi_{k}\}$ considered there,
we get the existence of $n(\varepsilon) \in \mathbb{N}$ such that
the set of all $\varepsilon$-translation numbers of $\{\varphi_{k}\}$
contains
$\{l (j + 1)^{n(\varepsilon)- 1} (m + 1); l \in \mathbb{N}\}$;
i.e., we have
\begin{equation} \label{ves720l}
T (\{\varphi_{k}\}, n(\varepsilon)) := \{l (j + 1)^{n(\varepsilon) - 1} (m + 1);
 \, l \in \mathbb{N}\} \subseteq \mathfrak{T} (\{\varphi_{k}\}, \varepsilon)
\end{equation}
for every $\varepsilon > 0$.
\end{remark}

\begin{theorem} \label{thm4}
Let $\{\varphi_{k}\}_{k \in \mathbb{N}_0}$ be an almost periodic sequence
and let the sequences $\{r_{n}\}_{n \in \mathbb{N}} \subset \mathbb{R}^+_0$
and $\{l_{n}\}_{n \in \mathbb{N}} \subseteq \mathbb{N}$ be such that
\begin{equation} \label{ves14w2}
r_{n} l_{n} \to 0 \quad  \text{as }  n \to \infty.
\end{equation}
If, for all $n$, there exists a set $ T (r_{n})$ of some $r_{n}$-translation numbers
of $\{\varphi_{k}\}$ which is relative dense in $\mathbb{N}_0$ and, for
every nonzero $l = l(r_{n}) \in T (r_{n})$,
there exists $i = i (l) \in \{1, \dots, l_{n}  + 1\}$ 
with the property that
\begin{equation} \label{ves3w2}
\varphi_{(i - 1)l + k} \in \mathcal{O}_{r_{n} l_{n}}
(\varphi_{i l - k}), \quad  k \in
\{0, \dots, l \},
\end{equation}
then the sequence $\{\psi_{k}\}_{k \in \mathbb{Z}}$, given by
the formula
\begin{equation} \label{ves9qj1}
\psi_{k} := \varphi_{k}  \text{ for } k \in \mathbb{N}_0 \quad \text{and}
 \quad  \psi_{k} := \varphi_{-k}  \text{ for } k \in \mathbb{Z}
\setminus \mathbb{N}_0,
\end{equation}
is almost periodic.

If, for all $n$, there exists a set $ \widetilde{T} (r_{n})$ of 
some $r_{n}$-translation numbers
of $\{\varphi_{k}\}$ which is relative dense in $\mathbb{N}_0$ and, for
every nonzero $m = m(r_{n}) \in \widetilde{T} (r_{n})$,
there exists $i = i (m) \in \{1, \dots, l_{n} + 1\}$ 
with the property that
\begin{equation} \label{ves7w2d}
\varphi_{(i - 1)m + k} \in \mathcal{O}_{r_{n} l_{n}}
(\varphi_{i m - k - 1}), \quad  k \in
\{0, \dots, m - 1 \},
\end{equation}
then the sequence $\{\chi_{k}\}_{k \in \mathbb{Z}}$, given by
the formula
\begin{equation} \label{ves7qjd}
\chi_{k} := \varphi_{k}  \text{ for } k \in \mathbb{N}_0 \quad \text{and}
 \quad  \chi_{k} := \varphi_{-(k + 1)}  \text{ for } k \in \mathbb{Z}
\setminus \mathbb{N}_0,
\end{equation}
is almost periodic.
\end{theorem}

\begin{proof} 
We will prove only the first part of Theorem \ref{thm4}.
The proof of the second case (the almost periodicity of $\{\chi_{k}\}$) 
is analogical.
Let $\varepsilon >0$ be arbitrarily small.
Consider $n \in \mathbb{N}$ satisfying (see \eqref{ves14w2})
\begin{equation} \label{vesaas1}
r_{n} l_{n} < \frac{\varepsilon}{3}.
\end{equation}
We will prove that the
set $\mathfrak{T} (\{\psi_{k}\}, \varepsilon)$ of all
${\varepsilon}$-translation numbers
of $\{\psi_{k}\}$ contains the numbers $\{ \pm l ; l \in T (r_{n}) \}$;
i.e., we will get the inequality
\begin{equation} \label{vesyt95}
d (\psi_{k}, \psi_{k \pm l}) < \varepsilon, \quad
 l \in T (r_{n}), \; k \in \mathbb{Z}
\end{equation}
which proves the theorem because $ \{ \pm l ;\,l \in T (r_{n}) \}$
is relative dense in $\mathbb{Z}$.

First of all we choose arbitrary $l \in T (r_{n})$.
From the theorem, we have $i= i (l)$.
Without loss of the generality, we can
consider only $+l$. (For $-l$, we can proceed similarly.)
Because of $ l_{n} \in \mathbb{N}$ and $l \in T (r_{n})$,
from \eqref{ves9qj1} and \eqref{vesaas1} it follows
\begin{equation} \label{vesyy7z}
d (\psi_{k}, \psi_{k + l}) < \frac{\varepsilon}{3}, \quad
 k \notin \{-l, \dots, -1\}, \; k \in \mathbb{Z}.
\end{equation}

Let $k \in \{-l, \dots, -1 \}$ be also arbitrarily chosen.
Evidently, we have 
$$
k + (1 -i)l \in \{- i l, \dots, - (i - 1)l - 1 \}
 $$
and
\begin{equation}  \label{ves4gt6}
\begin{aligned}
d (\psi_{k}, \psi_{k + l})
&\le d (\psi_{k}, \psi_{k + (1 -i) l})
 + d (\psi_{k + (1 -i) l}, \psi_{k + l}) \\
&= d (\varphi_{ -k}, \varphi_{ (i - 1) l - k })
 + d (\varphi_{(i - 1) l - k}, \varphi_{l + k}).
\end{aligned}
\end{equation}
The number $ (i - 1) l $ is an $\frac{\varepsilon}{3}$-translation
number of $\{\varphi_{k}\}$. Indeed, it follows from (iii),
\eqref{vesaas1}, and from $i \le l_{n} + 1$.
Therefore, we have
\begin{equation} \label{ves4111}
d (\varphi_{-k}, \varphi_{(i - 1) l - k}) < \frac{\varepsilon}{3}.
\end{equation}
Using \eqref{ves3w2} and \eqref{vesaas1}, we
get 
$$
d (\varphi_{(i - 1) l - k}, \varphi_{ i l + k}) < r_n l_n < \frac{\varepsilon}{3} .
$$
Thus, it holds
\begin{equation} \label{ves4112}
d (\varphi_{(i - 1) l - k}, \varphi_{l + k})
< \frac{2\varepsilon}{3}.
\end{equation}
Indeed, $(i-1) l $ is an
$\frac{\varepsilon}{3}$-translation number of $\{\varphi_{k}\}$
(consider again (iii),
\eqref{vesaas1}, and the inequality $i - 1 \le l_n $).

Altogether, from \eqref{ves4gt6}, \eqref{ves4111}, and \eqref{ves4112},
we obtain
\begin{equation} \label{ves415n}
d (\psi_{k}, \psi_{k + l}) \, <\, \frac{\varepsilon}{3} + \frac{2\varepsilon}{3} = \varepsilon.
\end{equation}
Since the choice of $k$, $l$ was arbitrary (see \eqref{vesyy7z}),
\eqref{ves415n} gives \eqref{vesyt95}.
\end{proof}

\begin{corollary} \label{coro1}
Let $ m \in \mathbb{N}_0 $, $j \in \mathbb{N}$,
and the sequence $\{\varphi_{k}\}_{k \in \mathbb{N}_0}$ be from
Theorem~\ref{thm3}
and $M > 0$ be arbitrary.
If, for all $n > M$, $n \in \mathbb{N}$, there exists at
least one $i \in \{1, \dots, j \}$ satisfying
\begin{equation} \label{ves4122}
\varphi_{i (j + 1)^{n} (m + 1) + k}
= \varphi_{(i + 1)(j + 1)^{n} (m + 1) - k}, \quad
  k \in \{0, \dots, (j + 1)^{n} (m + 1) \},
\end{equation}
then the sequence $\{\psi_{k}\}_{k \in \mathbb{Z}}$ given by \eqref{ves9qj1}
is almost periodic.
If, for all $n > M$, $n \in \mathbb{N}$, there exists at
least one $i \in \{1, \dots, j \}$ satisfying
\begin{equation} \label{ves4123}
\varphi_{i (j + 1)^{n} (m + 1) + k}
= \varphi_{(i + 1)(j + 1)^{n} (m + 1) - k - 1}, \quad
  k \in \{0, \dots, (j + 1)^{n} (m + 1) - 1 \},
\end{equation}
then the sequence $\{\chi_{k}\}_{k \in \mathbb{Z}}$ given by \eqref{ves7qjd}
is almost periodic.
\end{corollary}

\begin{proof}
We put
$$
r_n := \frac{1}{n}, \quad l_n := 1, \quad
T (r_{n}) := T (\{\varphi_{k}\}, n(\frac{r_n}{2})) \quad
 \text{for all } n \in \mathbb{N},
$$
where $T (\{\varphi_{k}\}, n(\varepsilon))$ is defined
by \eqref{ves720l}. Because we can assume 
that $n(\frac{1}{2}) > M - 1$,
it suffices to consider Theorem \ref{thm4} and Remark \ref{rmk2} 
(from (iii), using \eqref{ves4122} and \eqref{ves4123}, we get 
\eqref{ves3w2} and \eqref{ves7w2d}, respectively).
\end{proof}

For $k \in \mathbb{Z}$, we can prove (analogously as Theorem \ref{thm3})
the following two theorems:

\begin{theorem} \label{thm5}
Let $ m \in \mathbb{N}_0 $, $\psi_0, \dots, \psi_m \in \mathcal{X}$,
$j \in \mathbb{N}$, and the sequences of non\-negative real
numbers $\{r_{n}^1\}_{n \in \mathbb{N}}, \dots , \{r_{n}^j\}_{n \in \mathbb{N}}$
be arbitrarily given so that
\begin{equation} \label{vess11a}
\sum_{n=1}^\infty { r_{n}^i} < + \infty, \quad  i \in \{ 1, \dots, j\}
\end{equation}
holds.
Then, every sequence
$\{\psi_{k}\}_{k \in \mathbb{Z}} \subseteq \mathcal{X}$
for which it is true
\begin{gather*}
\psi_{k} \in \mathcal{O}_{r_{1}^1} (\psi_{k - (m + 1)}),
 \quad k \in \{ m + 1 , \dots, 2 (m +1) - 1 \}; \\
 \dots \\
\psi_{k} \in \mathcal{O}_{r_{1}^j} (\psi_{k - j(m + 1)}),
 \quad  k \in \{ j(m + 1), \dots, (j+1)(m + 1) - 1 \};\\
\psi_{k} \in \mathcal{O}_{r_{2}^1} (\psi_{k + (j+1)(m + 1)}),
 \quad  k \in \{ - ( j+1) (m + 1), \dots, -1 \}; \\
\dots \\
\begin{aligned}
&\psi_{k} \in \mathcal{O}_{r_{2}^j} (\psi_{k + j(j+1) (m + 1)}),\\
& k \in \{- j(j +1)(m + 1), \dots, - (j-1)(j +1)(m + 1) - 1 \} ;
\end{aligned}
\\
\begin{aligned}
&\psi_{k} \in \mathcal{O}_{r_{3}^1} (\psi_{k - (j+1)^2 (m + 1)}),\\
& k \in \{(j+1) (m + 1), \dots, (j+1)(m+1) + (j+1)^2 (m + 1) -1 \} ;
\end{aligned}
\\
\dots \\
\begin{aligned}
& \psi_{k} \in \mathcal{O}_{r_{3}^{j}} (\psi_{k - j(j +1)^2 (m + 1)}
 ), \\
& k \in \{(j +1) (m + 1) + (j - 1)(j+1)^2 (m + 1), \dots,\\
&\quad  (j +1) (m + 1) + j(j +1)^2 (m + 1) - 1 \};
\end{aligned}
\\
\begin{aligned}
& \psi_{k} \in \mathcal{O}_{r_{4}^1} (\psi_{k + (j+1)^3 (m + 1)}),\\
&k \in \{ -(j +1)^3 (m + 1) - j (j+1) (m+1), \dots, - j (j+1) (m+1) - 1 \};
\end{aligned}
\\
 \dots \\
\begin{aligned}
& \psi_{k} \in \mathcal{O}_{r_{4}^{j}} (\psi_{k + j(j +1)^3 (m + 1)}),\\
&k \in \{ -j(j +1)^3 (m + 1) - j(j +1)(m + 1), \dots,\\
&\quad  - (j - 1)(j +1)^3 (m + 1) - j(j +1)(m + 1) - 1  \};
\end{aligned}
\\
 \dots \\
\begin{aligned}
&\psi_{k} \in \mathcal{O}_{r_{2n}^1} (\psi_{k + (j + 1)^{2n - 1} (m + 1)}
), \\
& k \in \{ -((j + 1)^{2n - 1} + \dots + j(j +1)^3 + j(j +1))(m + 1),  \\
&\quad \dots, -((j + 1)^{2n - 3} + \dots + j(j +1)^3 + j(j +1))(m + 1) - 1 \};
\end{aligned}
\\
\dots\\
\begin{aligned}
&\psi_{k} \in \mathcal{O}_{r_{2n}^j} (\psi_{k + j(j + 1)^{2n - 1}
 (m + 1)}), \\
& k \in \{-(j(j + 1)^{2n - 1} + \dots + j(j +1)^3 + j(j +1))(m + 1), \\
&\quad  \dots, -((j-1)(j + 1)^{2n - 1} + \dots + j(j +1)^3 + j(j +1))
 (m + 1) - 1\};
\end{aligned}
\\
\begin{aligned}
& \psi_{k} \in \mathcal{O}_{r_{2n + 1}^1} (\psi_{k - (j + 1)^{2n}
(m + 1)}), \\
&k \in \{ (j +1) (m + 1) + j(j +1)^2 (m + 1) + \dots + j(j + 1)^{2n -2}(m + 1), \\
&\quad  \dots , (j +1) (m + 1) + j(j +1)^2 (m + 1) +\\
&\quad  \dots + j(j + 1)^{2n -2}
 (m + 1) + (j + 1)^{2n} (m + 1) - 1 \};
\end{aligned}
\\
\dots \\
\begin{aligned}
& \psi_{k} \in \mathcal{O}_{r_{2n + 1}^j} (\psi_{k - j(j + 1)^{2n}
(m + 1)}),\\
&k \in \{(j +1) (m + 1) + j(j +1)^2 (m + 1) + \\
&\quad \dots  + j(j + 1)^{2n -2}
(m + 1) + (j-1)(j + 1)^{2n} (m + 1), \\
&\quad \dots (j +1) (m + 1) + j(j +1)^2 (m + 1) + \dots + j(j + 1)^{2n}
(m + 1) - 1 \};
\end{aligned}
\\
\dots
\end{gather*}
is almost periodic.
\end{theorem}

\begin{proof} We put $ r_n := \max_{1 \leq i \leq j} r_n^i$,
$n\in \mathbb{N}$.
Then \eqref{vess11a} implies
$ \sum_{n=1}^\infty { r_{n}} < + \infty$.
Let the number $n(\varepsilon)\in \mathbb{N}$ satisfy the condition
\eqref{ves7ggsa}. 
We can show that, for arbi\-tra\-ri\-ly given $\varepsilon>0$,
in any set $Z \subseteq \mathbb{Z}$
consisting of
$$
(j+1)^{n(\varepsilon) - 1} (m + 1) \text{ consecutive integers},
$$
there exists an $\varepsilon$-translation number of $\{\psi_{k}\}$.
From it follows the theorem.
\end{proof}

\begin{theorem} \label{thm6}
 Let $\varphi_0, \dots, \varphi_m \in \mathcal{X}$ be given,
$\{r_{i}\}_{i \in \mathbb{N}} \subset \mathbb{R}^{+}_0$,
$\{j_{i}\}_{i \in \mathbb{N}} \subseteq \mathbb{N}$,
and $n \in \mathbb{N}_0$ be arbitrary such that $ m +n$ is even and
\begin{equation} \label{vest81a}
\sum_{i=1}^\infty { r_{i} j_i} < + \infty.
\end{equation}
For any $\varphi_{m+1}, \dots, \varphi_{m+n}$, if we set
\begin{gather*}
\psi_k := \varphi_{k + \frac{ m +n}{2} }, \quad
 k \in \{- \frac{m +n}{2}, \dots, \frac{m +n}{2}  \}, \\
 M := \frac{m +n}{2}, \quad   N := m +n
\end{gather*}
and we choose arbitrarily
\begin{gather*}
\psi_k \in \mathcal{O}_{r_{1}} (\psi_{k + N +1}), \quad
 k \in \{-N -M - 1 , \dots, -M - 1 \}, \\
 \dots  \\
\psi_k \in \mathcal{O}_{r_{1}} (\psi_{k + N + 1}) , \quad
 k \in \{- j_1 N - M - 1, \dots,- (j_1 - 1)N - M - 1 \}, \\
\psi_k \in \mathcal{O}_{r_{1}} (\psi_{k - N - 1}) , \quad
 k \in \{M + 1, \dots, N + M + 1 \}, \\
 \dots \\
\psi_k \in \mathcal{O}_{r_{1}} (\psi_{k - N - 1}) , \quad
 k \in \{(j_1 - 1) N + M + 1, \dots, j_1 N + M + 1 \}, \\
 \dots  \\
\psi_k \in \mathcal{O}_{r_{i}} (\psi_{k + p_i}), \quad
 k \in \{- p_{i} - p_{i-1} - \dots - p_1 , \dots, - p_{i-1} - \dots - p_1 \},\\
 \dots  \\
\begin{aligned}
&\psi_k \in \mathcal{O}_{r_{i}} (\psi_{k + p_i}),\\
& k \in \{ - j_i p_i - p_{i-1} - \dots - p_1, \dots, - (j_{i} - 1) p_{i}
- p_{i-1} - \dots - p_1 \},
\end{aligned}
\\
\psi_k \in \mathcal{O}_{r_{i}} (\psi_{k - p_i}) , \quad
 k \in \{ p_{i-1} + \dots + p_1, \dots, p_i + p_{i-1} + \dots + p_1 \}, \\
\dots \\
\psi_k \in \mathcal{O}_{r_{i}} (\psi_{k - p_i}) , \quad
 k \in \{(j_{i} - 1)p_i + p_{i-1} + \dots + p_1, \dots, j_ip_i + p_{i-1}
+ \dots + p_1 \}, \\
 \dots
\end{gather*}
where
\begin{gather*}
p_1:= (j_1 N + M + 1) + 1, \quad  p_2 := 2(j_1 N + M + 1) + 1, \\
p_3 := (2 j_2 + 1) p_2,  \dots ,
p_i := (2j_{i-1} + 1)p_{i-1}, \dots,
\end{gather*}
then the resulting sequence $\{\psi_k\}_{k \in \mathbb{Z}}$ is almost
periodic.
\end{theorem}

\begin{proof}
Consider arbitrary $\varepsilon > 0$ and
a positive integer $n(\varepsilon) \ge 2$
for which (see \eqref{vest81a})
$ \sum_{i=n(\varepsilon)}^\infty { r_{i}\, j_i} < \frac{\varepsilon}{4}$.
One can show that
$ \{ l p_{n(\varepsilon)} ; l \in \mathbb{Z} \} \subseteq \mathfrak{T}
 (\{\psi_k\}, \varepsilon)$
which completes the proof.
\end{proof}

\section{An application}

Let $m \in \mathbb{N}$ be arbitrarily given.
We will analyse almost periodic systems of $m$ homogeneous linear
difference equations of the form
\begin{equation} \label{ves69fr}
x_{k+1} = A_k \cdot x_k, \quad  k \in \mathbb{Z} \;
(\text{or }  k \in \mathbb{N}_0),
\end{equation}
where $\{A_k\}$ is almost periodic.
Let $\mathfrak X$ denote \emph{the set of all systems} \eqref{ves69fr}.
Our aim is to study the existence of a system
$\widehat{\mathfrak{S}} \in \mathfrak X$ which
does not have any nontrivial almost periodic solutions. We are going to
treat this problem in a~very general setting and
this motivates our requirements on the set of values of matrices $A_k$.

We need the set of entries of $A_k$ to be a subset of a~set~$R$
with two operations and unit elements such that $R$ with them
is a ring because the multiplication of matri\-ces~$A_k$
has to be associative (consider the
natural expression of solutions of \eqref{ves69fr}).
We also need the set of all considered $A_k$ to form a set $X$
which has the below given properties \eqref{ves621l},
and we need that there exists at least one of the below mentioned
functions $F_1, \, F_2 : [-1, 1] \to X$
-- see \eqref{ves6pi1}, \eqref{ves6pi2}, respectively.
The conditions~\eqref{ves6pi1} are
common, natural, and simple. However, the main theorem of this article
(the existence of the above sys\-tem $\widehat{\mathfrak{S}} \in \mathfrak X$)
is true for many subsets of the set of all unitary or orthogonal matrices
which contain of matrices that have the eigen\-value~$\lambda = 1$.
Thus, we will consider the existence of $F_2$.

We remark that it is possible to obtain results about the nonexistence of
nontrivial almost periodic solutions using different methods than those presented
in our paper.
For example, if the zero solution of a system ${\mathfrak{S}}$
of the form \eqref{ves69fr} is asymptotically (or even exponentially) stable,
then it is obviously that we can
choose $\widehat{\mathfrak{S}} := {\mathfrak{S}}$.
See \cite{ves790ki} and more general \cite{ves790do}, \cite{ves790IG}, and \cite{ves797z8},
where the method of Lyapunov function(al)s is used.

Let $R = (R, \oplus, \odot)$ be an infinite ring with a unit
and a zero denoted as $e_1$ and $e_0$, respectively.
The symbol $\mathcal{M} (R, m)$ will denote the set of all $m \times m$
matrices with elements from $R$. If we
consider the $i$-th column of $U \in \mathcal{M} (R, m)$, then we write $ U_i $;
and $R^m$ if we consider the set of all $m \times 1$ vectors with entries attaining
values from $R$.
As usual, we define the multiplication $\cdot$ of matrices from $\mathcal{M} (R, m)$
(and $U \cdot v$, $U \in \mathcal{M} (R, m)$, $v \in R^m$) by $\oplus$ and $\odot$.
Let $d$ be a pseudometric on $R$ and suppose that
\begin{equation} \label{vesrv13}
\text{the operations $\oplus$  and $\odot$
 are continuous with respect to $d$.}
\end{equation}
It gives the pseudometrics in
$R^m$ and $\mathcal{M} (R, m)$ because $\mathcal{M} (R, m)$
can be expressed as $R^{m \times m}$;
i.e., $d$ in $R^m$ and $\mathcal{M} (R, m)$ is the sum of $m$
and $m^2$ nonnegative numbers given by $d$ in $R$, respectively.
For simplicity, we will also denote these pseudo\-metrics as $d$.

The vector $v \in R^m$
is called \emph{nonzero} (or \emph{nontrivial}) if
$d\big(v, (e_0, \dots, e_0)^T \big) > 0$.
We say that a nonzero vector $(r_1, \dots, r_m)^T$,
where $r_1, \dots, r_m \in R$,
is an $e_1$-\emph{ei\-gen\-vector} of $U \in \mathcal{M} (R, m)$ if
$$
 d \Bigg( U \cdot \begin{pmatrix} r_1 \\ \vdots  \\  r_m \end{pmatrix},
\begin{pmatrix} r_1 \\ \vdots  \\ r_m \end{pmatrix}
\Bigg) = 0,
$$
and that $V \in \mathcal{M} (R, m)$ is \emph{regular for a nonzero vector}
$(r_1, \dots, r_m)^T \in R^m$ if
\begin{equation} \label{ves6uj3}
d \Bigg( V \cdot \begin{pmatrix} r_1 \\ \vdots \\ r_m \end{pmatrix},
 \begin{pmatrix} e_0 \\ \vdots  \\ e_0 \end{pmatrix} \Bigg) > 0.
\end{equation}
Next, we set
$$
\mathcal{I} := \begin{pmatrix} e_1 & e_0 &  \hdots & e_0 \\ e_0 & e_1 &
\hdots & e_0 \\ \vdots & \vdots & \ddots & \vdots  \\
e_0 & e_0 & \hdots & e_1
\end{pmatrix}
\in \mathcal{M} (R, m).
$$
If, for given $U \in \mathcal{M} (R, m)$ and
$X \subseteq \mathcal{M} (R, m)$, there exists
the unique ma\-trix $ V \in X$ (we put $V= W$ if $d(V, W) = 0$) 
for which
$$
U \cdot V = V \cdot U = \mathcal{I},
$$
then we define $U^{-1} := V$ and we say that $V$ is
 \emph{the inverse matrix} of $U$ in $X$.

For any function $H : [a, b] \to X$ ($a \le 0 < b$, $a, b \in \mathbb{R}$)
and $s \in \mathbb{R}$, we extend its domain of definition as follows
\begin{equation}  \label{vesutg}
H(s) := \begin{cases}
H(\sigma) \cdot (H(b))^l &\text{for } s \ge 0, \\
(H(a))^l \cdot H(\sigma) & \text{for } s < 0 \text{ if } a < 0,
\end{cases}
\end{equation}
where $s = l b + \sigma$ for $l \in \mathbb{N}_0$,
$\sigma \in [0, b)$
or $s = l a + \sigma $ for $l \in \mathbb{N}_0$,
$\sigma \in (a, 0]$.
Hereafter, we will restrict coefficients $A_k$ in \eqref{ves69fr}
to be elements of an infinite set $X \subseteq \mathcal{M} (R, m) $
with the following properties:
\begin{equation} \label{ves621l}
\mathcal{I} \in X; \quad  U, V \in X \, \Longrightarrow \,
U \cdot V \in X, \; U^{{-}1} \text{ exists in } X;
\end{equation}
and either
\begin{quote}
there exists a continuous function $F_1 :[-1, 1] \to X$ satisfying
\begin{equation}  \label{ves6pi1}
F_1 (0) = \mathcal{I}; \quad F_1(t) = F_1^{-1}(-t), \quad
 t\in [0, 1];
\end{equation}
and the matrix $ F_1 (1)$  has no $e_1$-eigenvector
\end{quote}
or
\begin{quote}
there exist continuous
$F_2 : [-1, 1] \to X$,  $t_1, \dots , t_q \in (0, 1]$,
$\delta > 0$  such that
\begin{equation}  \label{ves6pi2}
 F_2 (0) = \mathcal{I};\quad F_2 \Big( \sum_{i=1}^p {s_i} \Big)
= \prod_{i=1}^{p} {F_2 ({s_i})}, \quad 
s_1, \dots, s_p \in [-1, 1];
\end{equation}
and, for any $v \in R^m$,  one can find
$j \in \{1, \dots, q\}$  for which $v$ is not
an $e_1$-eigenvector of $F_2(t)$,
$t \in (\max  \{0, t_j - \delta \},
\min \{ t_j + \delta, 1 \})$.
\end{quote}
We mention that, for $U_1, \dots, U_p \in X$ ($p \in \mathbb{N}$),
we define
$$
 \prod_{i=1}^{p} U_i := U_1 \cdot U_2 \cdots U_p, 
 \quad \prod_{i=p}^{1} U_i := U_p \cdot U_{p - 1} \cdots U_1.
$$
For the above function $H$, we also use the conventional notation
$$
(H(s))^0 := \mathcal{I}, \quad H^{-1} (s) := \big( H(s) \big)^{-1}\quad
\text{for all considered } s \in \mathbb{R}.
$$
We remark that, because of \eqref{vesrv13},
the requirement for the existence
of $\delta > 0$ (in \eqref{ves6pi2}) can be dropped.
Now we comment our assumptions on $R$ and $X$:
We note that $R$ does not need to be commutative, and thus
the set of all solutions of \eqref{ves69fr}
is not generally a modulus over $R$ with the scalar multiplication given by
$$
r \begin{pmatrix} x^1_k \\ \vdots \\ x^m_k \end{pmatrix}
:= \begin{pmatrix} r \odot x^1_k \\ \vdots \\
r \odot x^m_k \end{pmatrix},
$$
where $\{ ( x^1_k, \dots, x^m_k )^T \}$
is a solution of \eqref{ves69fr}, $r \in R$, $k \in \mathbb{Z}$
($k \in \mathbb{N}_0$).

We need a sequence $\{a_k\}_{k \in \mathbb{N}_0}$ of real numbers,
which has special properties (mentioned in the below given
Lemmas \ref{lem1}--\ref{lem4}), to prove the main theorem of this paper.
We define the sequence $\{a_k\}_{k \in \mathbb{N}_0}$
by the recurrent formula
\begin{equation} \label{ves6p13}
a_0 := 1, \quad  a_1:= 0, \quad  a_{2^n + k} := a_{k} - \frac{1}{2^n}, \quad 
k = 0, \dots, 2^n - 1, \; n \in \mathbb{N}.
\end{equation}
For this sequence, we have the following results:

\begin{lemma} \label{lem1}
The sequence $\{a_k\}$ is almost periodic.
\end{lemma}

The above lemma follows from Theorem \ref{thm3} where we set $\varphi_k = a_k$
($k \in \mathbb{N}$) and
$$
 \mathcal{X}= \mathbb{R}, \quad  m=0, \quad  j = 1, \quad
 \varphi_0 = 1, \quad  r_n = \frac{4}{2^{n}}, \; n \in \mathbb{N}.
$$

\begin{lemma} \label{lem2}
The following holds
\begin{equation} \label{ves6j34}
a_{2^{n+2} -1 - i} = - a_{2^{n+1} +i}
\end{equation}
for any $n \in \mathbb{N}_0$ and $i \in \{0, \dots, 2^n - 1 \}$;
 i.e., $i \in \{0, \dots, 2^{n+1} -1\}$.
\end{lemma}

Before proving this statement, observe that \eqref{ves6j34} is equivalent to
$$
\sum_{k=2^{n+1} +i}^{2^{n+2} -1 - i} {a_k} = 0, \quad
 n \in \mathbb{N}_0, \; i \in \{0, \dots, 2^n-1 \};
$$
i.e., to
$$
\sum_{k=0}^{2^{n+1} - 1 + i} {a_k}
= \sum_{k=0}^{2^{n+2} -1 - i} {a_k}, \quad
 n \in \mathbb{N}_0, \; i \in \{0, \dots, 2^n - 1\}.
$$

\begin{proof}[Proof of Lemma \ref{lem2}]
 Obviously, \eqref{ves6j34} is true for $n \in \{0,1\}$
because
$$
a_2 = - a_3 = \frac{1}{2}, \quad a_4 = - a_7 = \frac{3}{4}, \quad  a_5 = - a_6 = - \frac{1}{4};
$$
i.e.,
$$
\sum_{k=0}^{1} {a_k} = \sum_{k=0}^{3} {a_k} = \sum_{k=0}^{7} {a_k} = 1, \quad  
\sum_{k=0}^{4} {a_k} =  \sum_{k=0}^{6} {a_k} = \frac{7}{4}.
$$
Suppose that \eqref{ves6j34} is true also for $2, \dots, n - 1$.
We choose $i \in \{0, \dots, 2^n - 1\}$ arbitrarily.
(We have $2^{n+2} -1 - i \ge 2^{n+1} + 2^{n}$.)
 From \eqref{ves6p13} and the induction hypothesis it follows
$$
a_{2^{n+2} -1 - i} + a_{2^{n+1} + 2^{n} + i}
= - \frac{1}{2^{n}}, \quad   a_{2^{n+1} + i} - a_{2^{n+1}
+ 2^{n} + i} =  \frac{1}{2^{n}}.
$$
Summing the above equalities, we get \eqref{ves6j34}.
\end{proof}

\begin{lemma} \label{lem3}
 We have
\begin{equation} \label{ves030z}
\sum_{k=0}^{n} {a_k} \ge 1, \quad  n \in \mathbb{N}_0.
\end{equation} \end{lemma}

\begin{proof}
Evidently, $ a_0= {a_0 + a_1} = 1$. It means that
\eqref{ves030z} is true for $n = 0$ and $n = 1 = 2^1 - 1$.
Let it be valid for arbitrarily given $2^p - 1$
and all $n < 2^p - 1$, i.e., let
$$
\sum_{k=0}^{n} {a_k} \ge 1, \quad  n \le 2^p - 1, \; n \in \mathbb{N}_0.
$$
Considering the definition of $\{a_k\}$,
we obtain
$$
\sum_{k=0}^{2^{p} + j - 1} {a_k} = \sum_{k=0}^{2^{p} - 1} {a_k}
+ \sum_{k=2^p}^{2^{p} + j - 1} {a_k} \ge 1
+ \sum_{k=0}^{j - 1} {a_k} - j \frac{1}{2^p} \ge 1 + 1 - 1 = 1
$$
for any $j \in \{1, \dots, 2^p \}$. Lemma \ref{lem3} now follows
by the induction.
\end{proof}

\begin{lemma} \label{lem4}
We have
\begin{gather} \label{ves6ju8}
\sum_{k=0}^{2^n - 1} {a_k} = 1, \\
 \label{ves5ju3}
\sum_{k=0}^{2^{n+i} + 2^{n} - 1} {a_k} = 2 - \frac{1}{2^i},
\end{gather}
where $n \in \mathbb{N}_0$, $i \in \mathbb{N}$.
\end{lemma}

\begin{proof} It is possible to prove this result
by means of Lemma \ref{lem2},
but we prove it directly using \eqref{ves6p13}
and the induction principle.
We have
$$
a_0 =1,\quad  a_0 + a_1 = 1, \quad  a_0 + a_1 + a_2 + a_3 = 1  .
$$
If we assume that
$$
 \sum_{k=0}^{2^{n - 1} - 1} {a_k} = 1,
$$
then we get (see \eqref{ves6p13})
\begin{align*}
\sum_{k=0}^{2^{n} - 1} {a_k}
&= \sum_{k=0}^{2^{n - 1} - 1} {a_k}
 + \sum_{k=2^{n - 1}}^{2^{n} - 1} {a_k} \\
&= \sum_{k=0}^{2^{n - 1} - 1} {a_k}
 + \sum_{k=0}^{2^{n - 1} - 1} {\Big( a_k - \frac{1}{2^{n -1}} \Big)}\\
&= 2\sum_{k=0}^{2^{n - 1} - 1} {a_k} - 1 = 1.
\end{align*}
Therefore, \eqref{ves6ju8} is proved.
Analogously, applying \eqref{ves6p13} and \eqref{ves6ju8}, one can compute
\begin{align*}
\sum_{k=0}^{2^{n+i} + 2^{n} - 1} {a_k}
&= \sum_{k=0}^{2^{n+i} - 1} {a_k}
+ \sum_{k=2^{n+i}}^{2^{n+i} + 2^{n} - 1} {a_k} \\
&=1 + \sum_{k=0}^{2^{n} - 1} \Big( {a_k} - \frac{1}{2^{n+i}}\Big)\\
&= 1 + \big(1 - \frac{1}{2^i} \big)
\end{align*}
what gives \eqref{ves5ju3}.
 \end{proof}

Applying the matrix valued functions $F_1$, $F_2$, we obtain the next lemma.

\begin{lemma} \label{lem5}
For each $j \in \{1,2\}$,
any $n \in \mathbb{N}_0$, and $i \in \{0, \dots, 2^n - 1 \}$,
it holds
$$
F_j (a_{2^{n+2} -1 - i}) = F_j^{-1} (a_{2^{n+1} +i})
$$
and, consequently,
$$
\prod_{k=2^{n+1} +i}^{2^{n+2} -1 - i} { F_j (a_{k })} = 
\prod_{k=2^{n+2} -1 - i}^{2^{n+1} +i} { F_j (a_{k })} = \mathcal{I}.
$$
\end{lemma}

\begin{proof}
Clearly, this is a corollary of Lemma \ref{lem2}. Consider \eqref{ves6pi1},
\eqref{ves6pi2}, and the fact that the multiplication of matrices
(in $\mathcal{M} (R, m)$)
is associative.
\end{proof}

Immediately, from Lemma \ref{lem4} (see \eqref{ves6pi2}), we have the
following formulas for the func\-tion $F_2$:

\begin{lemma} \label{lem6}
The equalities
$$
 \prod_{k=0}^{2^n - 1} {F_2 ({a_k})} = F_2 (1), \quad
 \prod_{k=0}^{2^{n+i} + 2^{n} - 1} {F_2 ( {a_k})}
= F_2 \big( 2 - \frac{1}{2^i}\big)
$$
hold for all $n \in \mathbb{N}_0$ and $i \in \mathbb{N}$.
\end{lemma}

Now we can prove the main statement of our paper.

\begin{theorem} \label{thm7}
There exists a system of the form \eqref{ves69fr} that
does not possess a~non\-zero almost periodic solution.
\end{theorem}

\begin{proof}
First we suppose that the coefficients $A_k$ belong to $X$
such that there exists a func\-tion $F_1$ from \eqref{ves6pi1}.
Using Theorem \ref{thm2}, we get
the almost periodicity of the sequence
$\{F_1 \circ a_k\}_{k \in \mathbb{N}_0} $,
where $\{a_k\} $ is given by \eqref{ves6p13}. We want to show
that all nonzero
solutions of the system $\mathfrak{S}_1 \in \mathfrak X$ determined by
$\{F_1 \circ a_k\}$
are not almost periodic.

By contradiction, suppose that there exist $c_1, \dots, c_m \in R$
such that the vector valued sequence
\begin{equation} \label{ves0nhh}
\{f_k\} := \Big\{ P_k \cdot \begin{pmatrix} c_1 \\ \vdots \\ c_m \end{pmatrix}
\Big\}, \quad  k \in \mathbb{N}_0,
\end{equation}
where $\{P_k\}_{k \in \mathbb{N}_0}$ is the principal
fundamental matrix of $\mathfrak{S}_1$, is nontrivial and almost periodic;
i.e., suppose that $\mathfrak{S}_1$ has a nontrivial almost periodic
solution $ \{f_k\}$.
Since $\{f_k\}$ is almost periodic, $(c_1, \dots,c_m)^T$ is nonzero, and,
because of $a_0=1$, it is valid
\begin{equation} \label{ves0t37}
f_i = U_i \cdot F_1(1) \cdot \begin{pmatrix} c_1 \\ \vdots \\ c_m
\end{pmatrix} \quad  \text{for any } i \in \mathbb{N}
\text{ and some } U_i \in X,
\end{equation}
we know that (see \eqref{ves6uj3})
\begin{equation} \label{ves0nte}
F_1(1) \text{ is regular for } c:= (c_1, \dots, c_m)^T.
\end{equation}

Considering \eqref{ves6p13},
the uniform continuity of 
$F_1$ and the continuity of
the multiplication of matrices (see \eqref{vesrv13}), 
(iii), Lemma \ref{lem5}, and \eqref{ves0nhh},
from the first part of Theorem \ref{thm4} (see the proof of Corolla\-ry~\ref{coro1}
and again Lemma \ref{lem5}), 
one can obtain that the sequence $\{g_k\}_{k \in \mathbb{Z}}$, where
\begin{equation} \label{ves01t2}
g_{k} := f_{k}, \quad  k \in \mathbb{N}_0, \quad
g_{k} := f_{-k}, \quad  k \in \mathbb{Z} \setminus \mathbb{N}_0,
\end{equation}
is almost periodic too. Now we use Theorem \ref{thm1}
for $\{\varphi_k\} \equiv \{g_k\}$ and
$\{h_n\}_{n \in \mathbb{N}} \equiv \{2^n\}_{n \in \mathbb{N}}$ (we can also 
consider directly $\{\varphi_k\} \equiv \{f_k\}$ and use Remark \ref{rmk1}).
This theorem implies that, for any $\varepsilon > 0$,
there exists an infinite set
$N(\varepsilon) \subseteq \mathbb{N}$ such that
the inequality
\begin{equation} \label{ves01tt}
d (g_{k + 2^{n_1}}, g_{k + 2^{n_2}}) < \varepsilon, \quad
k \in \mathbb{Z}
\end{equation}
holds for all $n_1, n_2 \in N(\varepsilon)$.

Using \eqref{ves6pi1} (twice), we get
$d (c, F_1 (1) \cdot c) > 0$,
and consequently (consider \eqref{ves0nte})
\begin{equation} \label{ves7u66}
\vartheta := d (F_1 (1) \cdot c, F_1 (1) \cdot F_1 (1)
 \cdot c) > 0.
\end{equation}
 From Lemma \ref{lem5} (for $i = 0$), \eqref{ves0nhh}, and \eqref{ves01t2}
(see also \eqref{ves0t37}), we have
\begin{equation} \label{ves71hk}
g_0 = c, \quad  g_1 = F_1 (1) \cdot c, \quad  \dots, \quad
 g_{2^n} = F_1 (1) \cdot c,
\end{equation}
where $n \in \mathbb{N}$ is arbitrary, and hence,
considering \eqref{ves6p13}, it holds
\begin{equation} \label{ves71hp}
d (g_{2^{i} + 2^{n}}, F_1 (1) \cdot F_1 (1) \cdot c) \to 0 \quad
 \text{as }   n \to \infty
\end{equation}
and for every $i \in \mathbb{N}$ because $F_1$ is uniformly continuous
and the multiplication of matrices
is continuous.
We also have
\begin{equation} \label{ves01hb}
d (g_{2^{n_2} + 2^{n_1}}, F_1 (1) \cdot c) < \frac{\vartheta}{2}
\end{equation}
for all $n_1, n_2 \in N (\frac{\vartheta}{2})$. Indeed,
put $k = 2^{n_2}$ in \eqref{ves01tt}
and consider \eqref{ves71hk} for $n = n_2 + 1$. If we choose
$n_1 \in N (\frac{\vartheta}{2})$
and put $i = n_1$ in \eqref{ves71hp}, then
there exists $n_0 \in \mathbb{N}$
such that, for any $n \ge n_0$, it holds
$$
d (g_{2^{n_1} + 2^{n}}, F_1 (1) \cdot F_1 (1) \cdot c)
< \frac{\vartheta}{2}.
$$
Thus, for arbitrarily given $n_2 \ge n_0$,
$n_2 \in N (\frac{\vartheta}{2})$,
we get
\begin{equation} \label{ves01ha}
d (g_{2^{n_2} + 2^{n_1}}, F_1 (1)\cdot F_1 (1) \cdot c)
< \frac{\vartheta}{2} .
\end{equation}
Finally, applying \eqref{ves7u66}, (iii), \eqref{ves01hb}, and \eqref{ves01ha},
 we have
\begin{align*}
\vartheta  \le  d (F_1 (1) \cdot c, g_{2^{n_2} + 2^{n_1}})
+ d (g_{2^{n_2} + 2^{n_1}}, F_1 (1) \cdot F_1 (1) \cdot c)
< \vartheta.
\end{align*}
This contradiction gives the proof when
we consider \eqref{ves6pi1} for ${k \in \mathbb{N}_0}$.

Let ${k \in \mathbb{Z}}$. Then, we can consider the system
$\widetilde{\mathfrak{S}}_1$
determined by the sequence
\begin{equation} \label{ves0109}
B_k := F_1 (a_k), \quad  {k \in \mathbb{N}_0},  \quad
B_{k} := F_1 ( - a_{-k - 1}), \quad  k \in \mathbb{Z} \setminus \mathbb{N}_0.
\end{equation}
Since the sequence $\{|\, a_k\,|\}_{k \in \mathbb{N}_0} $
is almost periodic (see Theorem \ref{thm2}) and has the form 
of $\{\varphi_k\}_{k \in \mathbb{N}_0}$ from Theorem \ref{thm3}
and since it is valid (see \eqref{ves6j34})
$$ |\,a_{2^{n+2} -1 - i}\,| = |\,a_{2^{n+1} +i}\,|, \quad 
n \in \mathbb{N}_0, \; i \in \{0, \dots, 2^n - 1 \},$$
the fact that $\{B_{k}\}$ is almost periodic follows 
from the second part of Corollary~\ref{coro1}, 
from (c) (mentioned in Section~2), and Theorem \ref{thm2}.
Next, the process is the same as for
${k \in \mathbb{N}_0}$.
Let $\{P_k\}_{k \in \mathbb{Z}}$ be the principal
fundamental matrix of $\widetilde{\mathfrak{S}}_1$ and
$g_k := f_k$, $k \in \mathbb{Z}$.
Also now we have \eqref{ves01tt}, and consequently we get the same
contradiction.

Let the coefficients $A_k$ belong to $X$
such that there exists a func\-tion $F_2$ from \eqref{ves6pi2}.
Consider the numbers $t_1, \dots , t_q \in (0, 1]$
and $\delta > 0$ from \eqref{ves6pi2}.
Without loss of the generality, we can assume
\begin{equation} \label{ves0x.9}
\delta < t_1 < \dots < t_q \quad  \text{and} \quad  t_q < 1 - \delta.
\end{equation}
Indeed, if $t_j=1$,
then we can put $t_j := 1 - \frac{\delta}{2}$ and redefine $\delta$.
We repeat that any vector $v \in R^m$ determines some
$j \in \{1, \dots, q\}$ (see again \eqref{ves6pi2}) such that
$v$ is not an $e_1$-eigenvector
of $F_2 (t)$ for $t \in ( t_j - \delta, t_j + \delta)$.

From Theorem \ref{thm2} it follows that
the sequence $\{F_2 \circ a_k\}_{k \in \mathbb{N}_0} $ is almost periodic. Thus,
it determines the system of the form \eqref{ves69fr}. We will denote
it as $\mathfrak{S}_2$.
Suppose that $\mathfrak{S}_2$ has a nontrivial almost periodic
solution $ \{x_k\}_{k \in \mathbb{N}_0}$. For the principal fundamental
matrix $\{P_k\}$ of the system $\mathfrak{S}_2$, we
have
$$
x_k = P_k \cdot x_{0}, \quad  k \in \mathbb{N}_0,
$$
where the vector $x_0$ is nonzero.
Using this fact and
taking into account Lemma \ref{lem3} and \eqref{ves6pi2}, we obtain
\begin{equation} \label{ves0t40}
x_n = F_2(t) \cdot F_2^{i}(1) \cdot x_0 \quad \text{for some }
i \in \mathbb{N}, \; t \in [ 0,  1),
\end{equation}
and for arbitrary $n \in \mathbb{N}$.
 From Lemma \ref{lem6}, we also get
\begin{equation} \label{ves0t41}
x_{2^n} = F_2 (1) \cdot x_0  \quad
\text{for all } n \in \mathbb{N}_0
\end{equation}
and
\begin{equation} \label{ves0t42}
x_{2^{n +i} + 2^n} = F_2 \Big(1 - \frac{1}{2^i} \Big) \cdot F_2 (1) \cdot x_0
\quad  \text{for all } n \in \mathbb{N}_0, \; i \in \mathbb{N}.
\end{equation}

Analogously as for $\{f_k\}$,
one can extend $\{x_k\}_{k \in \mathbb{N}_0}$ by the formula
$$
x_{k} := x_{-k}, \quad  k \in \mathbb{Z} \setminus \mathbb{N}_0
$$
for all $k \in \mathbb{Z}$ so that the sequence $\{x_k\}_{k \in \mathbb{Z}}$
is almost periodic too.
Now we apply Theorem \ref{thm1} for the sequences $\{x_k\}_{k \in \mathbb{Z}}$ and
$\{2^n\}_{n \in \mathbb{N}}$.
For any $\varepsilon > 0$, there exists an infinite set
$M(\varepsilon) \subseteq \mathbb{N}$
such that, for any $n_1, n_2 \in M(\varepsilon)$, we have
\begin{equation} \label{ves01tl}
d (x_{k + 2^{n_1}}, x_{k + 2^{n_2}}) < \varepsilon, \quad
 k \in \mathbb{Z} .
\end{equation}
Since $F_2$ is uniformly continuous 
and the multiplication of matrices is con\-ti\-nuous,
for arbitrary $i \in \mathbb{N}_0$ and $\varepsilon>0$,
we have from \eqref{ves6p13} and
\eqref{ves0t41} that
\begin{equation} \label{ves71tt}
d (x_{2^{i} + 2^{n}}, F_2 (1) \cdot F_2 (1) \cdot x_0)
< \varepsilon \quad \text{for sufficiently large } n \in \mathbb{N}.
\end{equation}

Because of the almost periodicity of $\{x_k\}$ and \eqref{ves0t40},
the matrix $F_2 (1)$ has to be
regular for $x_0$.
Let $\varepsilon>0$ be arbitrarily small
and $n_1 \in M(\varepsilon)$ arbitrarily large.
From \eqref{ves01tl} and \eqref{ves71tt}, where we
choose $k = 2^{n_1 - j}$
and $i= n_1 - j$ for $j \in \{0, \dots, n_1\}$,
it follows that, for given $n_1$,
there exists sufficiently large
$n_2 \in M(\varepsilon)$ for which
\begin{equation}  \label{ves7144}
\begin{aligned}
&d (x_{2^{n_1 - j} + 2^{n_1}}, F_2 (1) \cdot F_2 (1) \cdot x_0) \\
&\le d (x_{2^{n_1 - j} + 2^{n_1}}, x_{2^{n_1 - j} + 2^{n_2}})
+ d (x_{2^{n_1 - j} + 2^{n_2}}, F_2 (1) \cdot F_2 (1) \cdot x_0)
< 2\varepsilon.
\end{aligned}
\end{equation}
Since $\varepsilon$ (in \eqref{ves7144}) is arbitrarily small and,
choosing $j = 0$, we have
$$
d (x_{2^{n_1 +1}}, F_2 (1) \cdot F_2 (1) \cdot x_0) 
< 2\varepsilon,
$$
we know (see \eqref{ves0t41}) that $ F_2 (1) \cdot x_0$ is
an $e_1$-eigenvector of $F_2 (1)$, i.e., we get
\begin{equation} \label{ves7155}
d (F_2 (1) \cdot x_0, F_2 (1) \cdot F_2 (1) \cdot x_0) = 0.
\end{equation}
If we choose $j=1$, then we obtain (consider \eqref{ves0t42})
$$
d (F_2(\frac{1}{2}) \cdot F_2(1) \cdot x_0, F_2 (1) \cdot F_2 (1)
\cdot x_0) = 0.
$$
Analogously, for any $j$ (the number $n_1$ is arbitrarily large),
we get
$$ 
d (F_2\big(1- \frac{1}{2^j} \big) \cdot F_2(1) \cdot x_0, F_2 (1)
\cdot F_2 (1) \cdot x_0) = 0.
$$
Thus,
\begin{equation} \label{ves2.10}
d (F_2\big(2- \frac{1}{2^j} \big) \cdot x_0, 
F_2(1) \cdot F_2(1) \cdot x_0) = 0,
\quad  j \in \mathbb{N}.
\end{equation}
Hence, we have
\begin{equation} \label{ves7710}
d (F_2\big(2- \frac{1}{2^{j}} \big) \cdot x_0,
F_2\Big(2- \frac{1}{2^{j-1}} \Big) \cdot x_0) = 0, \quad
j \in \mathbb{N}.
\end{equation}
Because of
$$
F_2\big(2- \frac{1}{2^{j}} \big)
= F_2\big(\frac{1}{2^{j}} + 2- \frac{1}{2^{j -1}} \big)
= F_2\big(\frac{1}{2^{j}} \big) \cdot F_2\big(2- \frac{1}{2^{j -1}} \big)
$$
and (see \eqref{ves7155} and \eqref{ves2.10})
$$
d \Big(F_2\big(2- \frac{1}{2^{j -1}} \big) \cdot x_0, F_2(1) \cdot x_0 \Big)  = 0,
$$
from \eqref{ves7710} it follows
$$
d \Big( F_2\big(\frac{1}{2^{j}} \big) \cdot F_2(1) \cdot x_0, F_2(1)
\cdot x_0 \Big) = 0 \quad  \text{for all } j \in \mathbb{N},
$$
i.e., $F_2(1) \cdot x_0$ is an $e_1$-eigenvector of
$F_2(2^{-j})$ for all $j \in \mathbb{N}$.

Since any number $t \in \left[0, 1\right]$ can be expressed in the form
$$
 \sum_{i=1}^\infty {\frac{a_i}{2^i}}, \quad
\text{where  } a_i \in \{0, 1\},
$$
for considered $\delta>0$, there exists $n \in \mathbb{N}$
such that, for every $t \in \left[0, 1\right]$, there exist
$a_1, \dots, a_n \in \{0, 1\}$ satisfying
$$
\big|t - \sum_{i=1}^n {\frac{a_i}{2^i}} \big|
 < \delta.
$$
Thus, $F_2(1) \cdot x_0$ is an $e_1$-eigenvector of $F_2 (t_j + s_j)$
for some $s_j \in (-\delta, \delta)$ and any $j \in \{1, \dots, q\}$ 
which cannot be true.
This contradiction shows that
$\{x_k\}_{k \in \mathbb{N}_0}$
is not almost pe\-riodic.

If one considers the system $\widetilde{\mathfrak{S}}_2$ obtained from
${\mathfrak{S}_2}$ as in \eqref{ves0109} (after replacing
$\mathfrak{S}_1$ by $\mathfrak{S}_2$), then, analogously as
for $F_1$ and ${k \in \mathbb{N}_0}$, one can prove that
$\widetilde{\mathfrak{S}}_2 \in \mathfrak{X}$ and that any its
non\-trivial solution $\{x_k\}_{k \in \mathbb{Z}}$ is not
almost periodic.
\end{proof}

\begin{remark} \label{rmk3} \rm
 Let a nonzero $F_1 (1) \cdot v \in R^m$
not be an $e_1$-eigenvector of the matrix $F_1 (1)$ from \eqref{ves6pi1};
 i.e., the condition \eqref{ves6pi1} be weakened in this way.
Then, from the first part of the proof of Theorem \ref{thm7},
we obtain that the sequence $\{f_k\}$, given by \eqref{ves0nhh},
is not almost periodic for $(c_1, \dots, c_m)^T = v$.
It means that there exists
a~system $\mathfrak{S}^1 \in \mathfrak X$ with
the principal fundamental matrix $\{P_k^1\}$ such that the sequence
$\{ P_k^1 \cdot v \}_{k \in \mathbb{N}_0}$ or
$\{ P^1_k \cdot v \}_{k \in \mathbb{Z}}$
is not almost periodic.
Analogously, it is seen: If one requires in \eqref{ves6pi2} only that, for
a nonzero ve\-ctor $v \in R^m$, there exists $t \in (0, 1]$
for which $F_2 (1) \cdot v$ is not an $e_1$-eigenvector of $F_2(t)$, then there exists
a system $\mathfrak{S}^2 \in \mathfrak X$ satisfying that the sequence
$\{ P^2_k \cdot v \}_{k \in \mathbb{Z}}$
(or $\{ P^2_k \cdot v \}_{k \in \mathbb{N}_0}$),
where $\{P_k^2\}_{k \in \mathbb{Z}}$ (or $\{ P^2_k\}_{k \in \mathbb{N}_0}$)
is the principal fundamental matrix of $\mathfrak{S}^2$,
is not almost periodic.
\end{remark}

The condition
\begin{equation} \label{ves6pi3}
F_2 \Big( \sum_{i=1}^p {s_i} \Big)
= \prod_{i=1}^{p} {F_2 ({s_i})}, \quad
s_1, \dots, s_p \in [-1, 1], \; p \in \mathbb{N}
\end{equation}
in \eqref{ves6pi2} is ``strong''. For example, from it follows that the
multiplication of matrices from the set $\{F_2 (t);\,t \in \mathbb{R} \}$
is commutative. At the same time, we say that, for many subsets 
of unitary or orthogonal
matrices, it is not a limitation and that the method in the proof
of Theorem \ref{thm7} can be simplified
in many cases. We will show it in two important special cases.

\begin{example} \label{exa1} \rm
If, for any nontrivial vector $v \in R^m$, there exists $\varepsilon (v) >0$
with the property that
$$
F_2 (t) \cdot v \notin \mathcal{O}_{\varepsilon(v)} (v) \quad
\text{ for all } t \ge 1 \; \text{(see \eqref{vesutg})},
$$
then the fact, that the systems ${\mathfrak{S}_2}$ and
$\widetilde{\mathfrak{S}}_2$
from the proof of Theorem \ref{thm7} do not have nontrivial almost
periodic solutions, follows directly
from Lemma~\ref{lem3} and~\eqref{ves6pi3}.
Indeed, the set $\mathfrak{T} (\{x_k\}, \varepsilon (x_0)) \setminus \{0\}$ is empty
for any nonzero so\-lu\-tion $\{x_k\}$.
\end{example}

\begin{example} \label{exa2} \rm
Let the function $F_2$, in addition to \eqref{ves6pi2}, satisfy
\begin{equation} \label{vesf02i}
F_2 (s) = F_2 (0) = \mathcal{I}
\end{equation}
for some positive irrational number $s$, \eqref{ves0x.9} hold,
and $p \in \mathbb{N}$ be arbitrary.
Then, the sy\-stem ${\mathfrak{S}}$
determined by the sequence
$$
\{A_k\} := \{F_2(k/p) \},
 $$
where $k \in \mathbb{N}_0$ or $k \in \mathbb{Z}$, has
no nontrivial almost periodic solutions.

The function $F_2 (t/p)$, $t\in \mathbb{R}$
is continuous and periodic with a period ${p}s$
(see \eqref{ves6pi3}, \eqref{vesf02i}).
Using the compactness of the interval $[0, ps]$, 
\eqref{ves6pi3}, and Theorem \ref{thm1}, we get that
$\{F_2 (k/p) \}_{k \in \mathbb{Z}}$
is almost periodic. The almost periodicity of
$\{F_2 (k/p) \}_{k \in \mathbb{N}_0}$ is now
obvious.

Suppose, by contradiction, that $\{x_k\} \equiv \{P_k \cdot x_0 \} $
is a nontrivial almost periodic solution of ${\mathfrak{S}}$.
We mention that
there exists $\delta>0$ satisfying that,
for any nonzero $v \in R^m$, one can find $j \in \mathbb{N}$
such that there exists a~positive num\-ber $\vartheta(v)$ for which
\begin{equation} \label{vestf2j}
\vartheta(v) \le d (F_2 \big( \frac{j}{p} + t\big) \cdot v, v),
\quad t \in (-\delta, \delta ),
\end{equation}
because
\begin{equation} \label{vesf03j}
\{F_2 ( k/p);  k \in \mathbb{N} \}  \text{ is dense in }
\{F_2 (t); t \in \mathbb{R} \}
\end{equation}
which is proved (for a continuous periodic function $F_2$
satisfying \eqref{ves6pi3}
with the smallest period $s > 0$ that is an irrational number)
in detail, e.g., in \cite[pp.~44--46]{ves798ves}.
Evidently, \eqref{vesf03j} gives that
\begin{equation} \label{veskkf}
\{F_2 ( k/p); k \in N \}  \text{ is dense in }
 \{F_2 (t); t \in \mathbb{R} \}
\end{equation}
for any set $N$ what is relative dense in $\mathbb{N}$.

Because the multiplication of matrices is continuous, there exists $\varepsilon>0$
which satisfies that every vector $u$ with the property $d(u, x_0)< \varepsilon$
determines the same $j$ in \eqref{vestf2j} as $x_0$ and one can find
\begin{equation} \label{vesg08j}
\vartheta(u) \ge \frac{\vartheta(x_0)}{2} .
\end{equation}
 From \eqref{ves6pi3}, we see that
\begin{equation} \label{vesg08w}
x_k = F_2 \Big( \sum_{i=0}^{k-1} \frac{i}{p} \Big)
\cdot x_0,\quad  k \in \mathbb{N}.
\end{equation}
Let $l$ be an arbitrary positive $\frac{\vartheta(x_0)}{2}$-translation number of
$\{x_k\}$, thus,
let
\begin{equation} \label{vesg0t2}
d(x_{k+l}, x_k) < \frac{\vartheta(x_0)}{2} \quad  \text{for all }
 k \in \mathbb{N},
\end{equation}
and let $N$ be the set of all positive $\varepsilon$-translation numbers
of $\{x_k\}$.
Since
$$
\sum_{i=0}^{k + l -1} \frac{i}{p} = \sum_{i=0}^{k -1} \frac{i}{p}
+ \frac{kl}{p} + \frac{l(l-1)}{2p}, \quad  k \in \mathbb{N},
$$
for all $k \in N$, we have (see again \eqref{ves6pi3})
\begin{equation} \label{vesd097}
d(x_{k+l}, x_k) = d \Big( F_2 \Big( \frac{kl}{p} + \frac{l(l-1)}{2p} \Big)
 \cdot F_2 \Big( \sum_{i=0}^{k -1} \frac{i}{p} \Big) \cdot x_0, F_2
\Big( \sum_{i=0}^{k -1} \frac{i}{p} \Big) \cdot x_0 \Big).
\end{equation}
 From \eqref{veskkf}, if we replace $\frac{1}{p}$ by $\frac{l}{p}$, we
get the choice of $k \in N$ such that
\begin{equation} \label{vesff09}
\big| \frac{j}{p} - \frac{k l}{p} - \frac{l(l-1)}{2p} \big|
< \delta \quad (\text{mod}\, s)
\end{equation}
for $j$ in \eqref{vestf2j} determined by $x_0$.
From \eqref{vestf2j}, \eqref{vesg08j} (consider the definition of $\varepsilon$),
\eqref{vesg08w}, \eqref{vesd097}, and \eqref{vesff09}, we have
$$
d(x_{k+l}, x_{k}) \ge \frac{\vartheta(x_0)}{2}
$$
for at least one $k \in \mathbb{N}$.
But, at the same time, we have \eqref{vesg0t2}.
This contradiction gives that $\{x_k\}$ cannot be almost periodic.
 See also the proof
of the first part of \cite[Proposition 2, p. 593]{ves798tk}, where
almost periodic unitary systems are studied.
\end{example}

At the end, we remark that the last considered system ${\mathfrak{S}}$
(in Example \ref{exa2}) has no physical interpretations
in any technical applications if we consider directly the sequence $\{{k}/p\}$; 
in contrast to ${\mathfrak{S}_2}$
and $\widetilde{\mathfrak{S}}_2$ (the sequence $\{a_{k}\}$).
In applications, the following can be utilized:
Let $\{u_{k}\}_{k \in \mathbb{Z}}$ (or $\{u_{k}\}_{k \in \mathbb{N}_0}$) be
a sequence of arbitrary values and let
the below considered function $\varphi$ be defined on the set
$\{u_{k}; k \in \mathbb{Z}\}$ or $\{u_{k};  k \in \mathbb{N}_0\}$.
If we extend the definition of the discrete almost periodicity
so that $\varphi$ is \emph{almost periodic} if, for every $\varepsilon > 0$,
one can find
$p (\varepsilon) \in \mathbb{N}_0$ with the pro\-per\-ty that any
set, in the form $\{k_0, \dots, k_0 + p (\varepsilon)\} $,
$k_0 \in \mathbb{Z}$
(or $k_0 \in \mathbb{N}_0$), contains a num\-ber $l$
satisfying the inequality
$$
d (\varphi (u_{k + l}), \varphi (u_k)) < \varepsilon
$$
for all $k \in \mathbb{Z}$ (or $k \in \mathbb{N}_0$),
then all results (mentioned in this article) about almost
periodic sequences are still valid.

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\end{document}