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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 60, pp. 1--28.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/60\hfil Infinite systems of differential equations]
{Existence of solutions for infinite systems of differential equations
in spaces of tempered sequences}

\author[J. Bana\'s, M. Krajewska \hfil EJDE-2017/60\hfilneg]
{J\'ozef Bana\'s, Monika Krajewska}

\address{J\'ozef Bana\'s \newline
Department of Nonlinear Analysis,
Rzesz\'ow University of Technology,
al. Powsta\'nc\'ow  \newline Warszawy 8, 
35 - 959 Rzesz\'ow, Poland}
\email{jbanas@prz.edu.pl}

\address{Monika Krajewska \newline
Institute of Economics and Management,
State Higher School of Technology and Economics  in Jaros{\l}aw,
ul. Czarnieckiego 16, 37 - 500 Jaros{\l}aw, Poland}
\email{monika.krajewska@pwste.edu.pl}

\dedicatory{Communicated by Vicentiu Radulescu}

\thanks{Submitted January 3, 2017. Published February 27, 2017.}
\subjclass[2010]{34G20, 47H08}
\keywords{Infinite system of differential equations; tempered sequence;
\hfill\break\indent differential equation in Banach spaces;
measure of noncompactness}

\begin{abstract}
 The aim of this article is to study the existence of solutions
 for infinite systems of differential equations.
 We look for solutions in Banach tempered sequence spaces, using
 techniques associated with measures of noncompactness, and
 results from differential equations in abstract Banach spaces.
\end{abstract}

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


\section{Introduction}

The theory of differential equations in Banach spaces is nowadays almost
a closed branch of mathematical analysis.
Roughly speaking, after the publication of
\cite{BGoebel,Deimling,Deimling2,Herzog,Lemmert,monch,Oregan,szufla}
there have not appeared books or article presenting essential progress
in the theory in question.

One of the most important reason of such a situation is the generality
of problems raised in that theory. From this point of view we may consider
the mentioned theory as closed or almost closed.
Nevertheless, if we consider a particular case of differential equations
in Banach spaces created by infinite systems of differential equations,
the situation seems to be very far to be closed or even  to be
satisfactory developed. Up to now there appeared only a few papers devoted
to the study of the theory of infinite systems of differential equations.
The current state of that theory is presented in the recent monograph
\cite{Banas Mursaleen} (cf. also \cite{Banas Krajewska,Banas Lecko}).

On the other hand infinite systems of differential equations find numerous
applications in describing of several real world problems which can be
 encountered in the theory of neural nets, the theory of branching processes,
the theory of dissociation of polymers and a lot of others
(see for example \cite{Bellman,Deimling,Hille,Oguz,Zautykov}).
It is also worthwhile mentioning that infinite systems of differential
equations are applied to solve some problems investigated in mechanics
\cite{Persi1,Persi2,Zautykov2}.
Moreover, when we consider some problems of partial differential equations,
we can use the process of semidiscretization to transform those problems
into infinite systems of differential equations (cf. \cite{Deimling,Voigt,Walter}).

To establish some preliminary facts let us consider the infinite system
of ordinary differential equations which can be written in the  general form
\begin{equation}\label{rrozn1}
x'_n=f_n(t,x_1,x_2,\dots)
\end{equation}
for $t\in I=[0,T]$ and for $n=1,2,\dots$.

The Cauchy problem for system \eqref{rrozn1}
can be formulated as the initial value conditions
\begin{equation}\label{rrozn2}
x_n(0)=x_o ^n, \quad \text{for } n=1,2,\dots\,.
\end{equation}
Let us pay attention to be fact that any solution of \eqref{rrozn1}-\eqref{rrozn2}
has the form of a function sequence
\begin{equation}\label{rrozn3}
x(t)=(x_n(t))=(x_1(t),x_2(t),\dots)
\end{equation}
where $t$ runs over an interval $[0,T]$ (or $[0,T_1]\subset[0,T]$).
To avoid unnecessary ambiguities we will denote the interval of the
definition of solution \eqref{rrozn3} of \eqref{rrozn1}-\eqref{rrozn2} by
$I$ i.e., $I=[0,T]$.
Thus, for each fixed $t\in I$ the sequence ($x_n(t)$) presents certain
sequence of real numbers.
Therefore, we consider the solvability of problem \eqref{rrozn1}-\eqref{rrozn2}
in some sequence space $c_0$, $c$, $l_p$, $l_\infty$
(cf. \cite{Banas Mursaleen,Deimling}). Details concerning the mentioned
sequence spaces will be given later on.

Now we show that even in rather simple situations the mentioned
classical sequence spaces are not sufficient for the location of our investigations.

\begin{example} \label{examp1.1} \rm
To show the influence of the choice of initial values in a sequence space
in which are located solutions of a considered initial value problem
for an infinite system of differential equations,
let us consider the  linear diagonal infinite system of differential
equations
\begin{equation}\label{rrozn4}
x_n'=x_n
\end{equation}\\
with the initial conditions
\begin{equation}\label{rrozn5}
 x_n(0)=n, \quad \text{for } n=1,2,\dots\,.
\end{equation}
 We consider problem \eqref{rrozn4}--\eqref{rrozn5}
on an interval $I=[0,T]$.

It is easily seen that the solution of  \eqref{rrozn4}-\eqref{rrozn5} has the form
$$
x(t)=(x_n(t))=(ne^t)=(e^t,2e^t,3e^t,\dots).
$$
This means that $x(t)\notin l_\infty$ for each $t\in I$.
Thus the sequence space $l_\infty$ is not suitable to consider solvability
of problem \eqref{rrozn4}--\eqref{rrozn5} in this space. Obviously, such a
situation appears quite naturally since the initial point $(x_n^o)=(n)$
is not a member of $l_\infty$.
\end{example}

\begin{example} \label{examp1.2} \rm
 Let us consider the infinite system of differential equations
 \begin{equation}\label{rrozn6}
 x_n'=n\frac{\sqrt{|x_n|}}{\sqrt{|x_n|}+1}
 \end{equation}
for $n=1,2,\dots, $ together with initial conditions
 \begin{equation}\label{rrozn7}
 x_n(0)=0, \quad\text{for }   n=1,2,\dots\,.
 \end{equation}

Let us fix arbitrarily a natural number $n$. Then, we can easy calculate
that the solution of problem \eqref{rrozn6}--\eqref{rrozn7} has the form
$$
x_n(t)=\frac{n^2t^2}{2+nt+2\sqrt{1+nt}}
$$
for $t\in I $. Hence, we obtain the  estimate
\begin{equation} \label{rrozn8}
\begin{aligned}
x_n(t)
&\geq\frac{n^2t^2}{2+nt+2\sqrt{1+2nt+n^2t^2}} \\
&=\frac{n^2t^2}{2+nt+2(1+nt)}\geq\frac{n^2t^2}{4+4nt}
&=\frac{1}{4}\Big(nt-1+\frac{1}{nt+1}\Big)
\end{aligned}
\end{equation}
for $n=1,2,\dots$  and for $t\in I$.
\end{example}

Further, let us represent the solution of  \eqref{rrozn6}--\eqref{rrozn7}
in the form $x(t)=(x_n(t))=(x_1(t),x_2(t),\dots)$. Then, from estimate
\eqref{rrozn8} we infer that $x(t)\notin l_\infty$ for any $t>0$.
On the other hand let us notice that the right-hand sides of equations
\eqref{rrozn6} are not bounded. Indeed, we have
$$
\frac{n\sqrt{x}}{\sqrt{x}+1}\to n, \quad\text{as } x \to \infty.
$$

The above given examples suggest that we have to enlarge the spaces under
considerations  to ensure that solutions of infinite systems of differential
 equations starting from a point in such a space remain in the space in question
when $t$ runs over some interval $I$. It seems that a natural way to realize
the enlargement is to consider the so - called tempered sequence spaces.
Those spaces can be obtained from classical sequence spaces with help of a
tempering sequence. For example, if we take the classical space $l_\infty$ and
the tempering sequence $\beta_n=\frac{1}{n}$ $(n=1,2,\dots)$ then the new sequence
space $l_\infty^\beta$ with $\beta=(\beta_n)=(\frac{1}{n})$ is understood
as the space of all sequences $(x_n)$ such that the sequence
$(\beta_n x_n)=(\frac{1}{n}x_n)$ is bounded.
The details concerning tempered sequence spaces will be described later on.

It is worthwhile noticing that such an approach enables us to study
an essentially larger class of infinite systems of differential equations
in comparison with the classical setting.

In this article we discuss some classes of infinite systems of differential
 equations having solutions in the above mentioned tempered sequence spaces.
The results of the paper generalize several ones obtained up to now in
classical sequence spaces
(see \cite{Banas Krajewska,Banas Lecko,Banas Mursaleen,Deimling,Deimling2,
Mursaleen}).


\section{Auxiliary facts concerning the theory of measures of noncompactness}

This section is devoted to recall a few facts concerning the theory of measures
of noncompactness, which will be needed in our further considerations.
Those facts come mainly from monograph \cite{BGoebel} (cf. also \cite{Akhm,Ayer}).
To set the stage for our study we establish first the notation used in this article.

By the symbol $\mathbb{R}$
we will denote the set of real numbers, and by $\mathbb{N}$ the set of natural
numbers (positive integers). We write $\mathbb{R_+}$ to denote the interval
 $[0, \infty)$.
Further, assume that $E$ is a Banach space with the norm $ \| \cdot \| $ and
the zero element $\theta$. Denote by $B(x,r)$ the closed ball in $E$ centered
at $x$ and with radius $r$. We write $B_r$ to denote the ball $B(\theta, r)$.
If $X$ is a subset of $E$ then by $\bar{X}$, Conv$X$ we will denote the
closure and convex closure of $X$, respectively. Moreover, the symbols $X+Y$,
$\lambda X$, $(\lambda \in \mathbb{R})$ stand for standard algebraic
operations on sets $X$ and $Y$.

We denote by $\mathfrak{M}_{E}$
the family of all nonempty and bounded subset of the space $E$, and by
$\mathfrak{N}_{E}$ its subfamily consisting of all relatively compact sets.

In what follows we will accept the following axiomatic definition of the
 concept of a measure of noncompactness \cite{BGoebel}.

\begin{definition} \label{def2.1} \rm
A function $\mu : \mathfrak{M}_{E} \to \mathbb{R_+}$ is called a measure
of noncompactness if the following conditions are satisfied:
\begin{itemize}
 \item[(i)] The family $\ker \mu = \{ X \in \mathfrak{M}_{E}: \mu (X)=0\} $
is nonempty and $ \ker \mu \subset \mathfrak{N}_{E}$;

 \item[(ii)] $X\subset Y \Rightarrow \mu (X) \leq \mu (Y) $;

 \item[(iii)] $\mu (\bar{X})=\mu(X)$;

 \item[(iv)] $\mu (\operatorname{Conv}X)= \mu(X)$;

 \item[(v)] $\mu( \lambda X+(1-\lambda)Y)
\leq \lambda \mu (X)+(1-\lambda) \mu (Y)$ for $\lambda \in[0,1]$;

 \item[(vi)] if $(X_n) $ is a sequence of closed sets from
$\mathfrak{M}_{E}$ such that $ X_{n+1} \subset X_n$ for $n=1,2,\dots$ and
$\lim_{n \to \infty} \mu (X_n)=0$  then the set
$X_{\infty}=\cap_{n=1}^{\infty} X_n$ is nonempty.
\end{itemize}
\end{definition}


The family $\ker \mu$ from axiom (i) is said to be
\textit{the kernel of the measure $\mu$}.

Further, let us observe that from axiom (vi) it follows that
$\mu (X_\infty) \leq \mu (X_n)$ for $n=1,2,\dots$. This yields that
$\mu (X_\infty)=0$. Hence we conclude that the intersection set $X_\infty$
belongs to the kernel $\ker \mu$. This simple fact plays a very essential
role in applications.

In the sequel we will also consider measures of noncompactness having some
additional properties. Thus, a measure $\mu$ is referred to as \textit{sublinear}
if it satisfies the following two conditions:
\begin{itemize}
 \item [(vii)] $\mu (\lambda X)=| \lambda | \mu (X)$, $\lambda \in \mathbb{R}$;
 \item [(viii)] $\mu (X+Y) \leq \mu (X) + \mu (Y) $.
\end{itemize}
We say that a measure of noncompactness has \textit{maximum property} if
\begin{itemize}
 \item [(ix)] $\mu (X \cup Y)= \max \{ \mu (X), \mu (Y) \}$.
\end{itemize}
The measure $\mu $ is said to be \textit{full} if
\begin{itemize}
 \item [(x)] $\ker \mu =\mathfrak{N}_{E} $.
\end{itemize}

Finally, the measure of noncompactness $\mu $ is called \textit{regular}
 if it is sublinear, full and has maximum property.

The most convenient and simultaneously important regular measure of
noncompactness is the so - called \textit{Hausdorff measure $\chi $}
defined in the following way
$$
\chi (X)= \inf \{ \varepsilon >0: X \text{ has a finite $\varepsilon$-net in }
 E \}.$$
It can be shown that this measure has also some other interesting and useful
properties (cf. \cite{Akhm,Ayer,BGoebel,Banas Martinon}).

The usefulness  of the Hausdorff measure $\chi$ leads to the question if each
regular measure of noncompactness $\mu$ is equivalent to the Hausdorff
measure $\chi$. It was shown in \cite{Banas Martion2} that, in general,
the answer is negative. Nevertheless, we have the following theorem \cite{BGoebel}
which shows that each regular measure of noncompactness is one - sided
comparable with the Hausdorff measure.

\begin{theorem} \label{thm2.2}
 If $\mu$ is a regular measure then
 $$ \mu (X) \leq \mu (B_1) \chi (X)$$
 for any set $X\in \mathfrak{M}_{E}$.
\end{theorem}

In practice we use those measures of noncompactness which can be expressed
 with help of a convenient  formula associated with the structure of a
considered Banach space. It turns out that we know only a few Banach spaces
in which the Hausdorff measure of noncompactness can be expressed
(or, at least, estimated) in such a way  \cite{BGoebel}.
By these regards we mostly apply measures of noncompactness being not
regular but which are connected with sufficient conditions for relative
compactness in Banach spaces under considerations \cite{BGoebel,Banas Mursaleen}.

\section{Measures of noncompactness in classical sequence spaces}

Now we  work in the sequence spaces $c_0$, $c$, $l_p$ and $l_\infty$ being
the classical sequence spaces.
We recall briefly the definition of these spaces.

By the space $c_0$ we mean the set of all real (or complex) sequences $x=(x_n)$
converging to zero and normed by the classical supremum (or maximum) norm:
$$
\| x \| _{c_{0}} = \| (x_n) \| _{c_{0}}
=\sup \{| x_n | :n=1,2,\dots\}
=\max \{| x_n | : n=1,2,\dots\}.
$$
Obviously $c_0$ with this norm creates the Banach space.

Next, denote by $c$ the space of all sequences $x=(x_n)$ converging to
a (finite) limit, with the norm
$$
\| x \| _{c} = \| (x_n) \| _{c}=\sup\{| x_n | :n=1,2,\dots\}.
$$
The space $c$ with the norm $\| \cdot \| _{c}$ is a Banach space and $c_0$
is a closed subspace of $c$.

If we fix a number $p$, $p \geq1$, then by $l_p$ we denote the space consisting
of all sequences $x=(x_n)$ such that
$\sum_{n=1}^{\infty} | x_n | ^p < \infty$ . If we normed it by
$$
\| x \| _{l_{p}} = \| (x_n) \| _{l_{p}}
=\Big(\sum_{n=1}^{\infty}| x_n|^p\Big)^{1/p}
 $$
it becomes a Banach space.

Finally, by the symbol $l_\infty$ we denote the space of all bounded sequences
$x=(x_n)$ with the supremum norm
$$
\| x \| _{l_{\infty}} = \| (x_n) \| _{l_{\infty}}
=\sup  \{| x_n | :n=1,2,\dots\}.
 $$

Now, we present the known facts concerning the measures of noncompactness
in the above mentioned sequence spaces \cite{BGoebel,Banas Mursaleen}.
In the case of sequence spaces  $c_0$, $c$ and $l_p$ the situation
concerning measures of noncompactness seems to be thoroughly recognized.
Indeed, in the spaces  $c_0$ and $l_p$ we know formulas expressing the most
convenient measure of noncompactness i.e., the Hausdorff measure $\chi$
(cf. Section 2).
To present the mentioned formulas let us consider first the space $c_0$
and let us take an arbitrary nonempty and bounded subset of
$c_0$ i.e., take a set $X \in \mathfrak{M}_{c_0}$. Then we have \cite{BGoebel}
$$
\chi (X)=\lim_{n \to \infty} \Big\{ \sup_{(x_n)\in X}
\Big\{\sup \{| x_i|: i\geq n\} \Big\} \Big\} .
$$
Next, if we fix arbitrarily a number $p$, $p\geq1$, then for
$X \in \mathfrak{M}_{l_p}$ we have \cite{BGoebel,Banas Mursaleen}
$$
\chi (X)=\lim_{n \to \infty} \Big\{ \sup \Big\{
\Big(\sum_{k=n}^{\infty}| x_n|^p\Big)^{1/p}: x=(x_i) \in X \Big\} \Big\} .
$$

In the case of the sequence space $c$ the situation is a bit more complicated.
Namely, we do not know a formula for the Hausdorff measure $\chi$ in $c$ but
we know only a good estimate $\chi$.
Indeed, for  $X \in \mathfrak{M}_{c}$ let us define the quantity $\mu (X)$ by
the  formula
\begin{equation}\label{rrozn9}
\mu(X)=\lim_{n \to \infty} \Big\{ \sup_{(x_k)\in X}\Big\{\sup
\{| x_i- \lim_{k \to \infty} x_k |: i\geq n\} \Big\} \Big\}.
\end{equation}
Then we have the estimate
\begin{equation}\label{rozn1}
\frac{1}{2}\mu (X) \leq \chi (X) \leq \mu (X)
\end{equation}
and this estimate is sharp \cite{BGoebel}.

It can be shown that measure \eqref{rrozn9} is regular.
Nevertheless, let us pay attention to the fact that the measure $\mu$
has only theoretical meaning since the use of formula \eqref{rrozn9}
requires to know limits of sequences belonging to a set $X$.
Therefore, to obtain a more convenient formula we can use the classical
Cauchy condition associated with the limit of a sequence, since such an
approach does not require the use of the limit of a sequence.
Thus, for $X \in \mathfrak{M}_{c}$ we define the quantity
\begin{equation}\label{rozn2}
\mu_c(X)=\lim_{k \to \infty} \Big\{ \sup_{(x_i)\in X}\Big\{
\sup \{| x_n- x_m |: n,m\geq k\} \Big\} \Big\}.
\end{equation}

It is worthwhile mentioning that in a few papers and monographs
(see \cite{BGoebel,Banas Lecko,Banas Mursaleen}, for example) we can encounter
 results asserting that the measure $\mu_c$ defined by formula \eqref{rozn2}
is regular and equivalent to the Hausdorff measure $\chi$ in the space $c$.
On the other hand there are no proof of that fact.
Therefore, to bridge this gap we provide below the complete proof of the
following theorem.

\begin{theorem} \label{thm3.1}
 The quantity $\mu_c$ defined by formula \eqref{rozn2} is a regular measure of
 noncompactness in the space $c$. Moreover, the following inequalities are satisfied
 \begin{equation}\label{rozn3}
 \chi(X)\leq \mu _c (X) \leq 2 \chi (X)
 \end{equation}
 for $X \in \mathfrak{M}_{c}$
\end{theorem}

\begin{proof}
 At the beginning let us observe that keeping in mind formula \eqref{rozn2}
it is not hard to show the quantity $\mu_c$ satisfies axioms (ii)--(v) and
 (vii)--(ix) of the definition of a regular measure of noncompactness
(cf. Section 2 and Definition \ref{def2.1}).

 Next, fix arbitrarily a set $X \in \mathfrak{M}_{c}$ and choose a sequence
$x=(x_i) \in X$. Take a fixed natural number $k$. Then, for arbitrary
$n, m \geq k$ we have
$$
| x_n- x_m | \leq | x_n- \lim_{i \to \infty} x_i |
+ | x_m- \lim_{i \to \infty} x_i |.
$$
Hence we derive the estimate
 \begin{equation}\label{rozn4}
 \mu _c (X) \leq 2 \mu (X),
 \end{equation}
where $\mu$ is the measure of noncompactness defined by \eqref{rrozn9}.
Linking \eqref{rozn1} and \eqref{rozn4} we obtain
\begin{equation}\label{rozn5}
 \mu _c(X) \leq 4\chi (X)
\end{equation}
for $X \in \mathfrak{M}_{c}$.

 Now, let us denote $r=\mu _c(X)$. Fix $\varepsilon>0$ and find a natural
number $k_0$ such that
 \begin{equation}\label{rozw6}
 | x_n- x_m | \leq r+ \varepsilon
 \end{equation}
for each $x=(x_i) \in X$ and  $n,m \geq k_0$. Consider the set
$X_{k_o}=\{(x_1, x_2, \dots, x_{k_0}): x=(x_1, x_2, \dots, x_{k_0}, x_{k_0+1},\dots)
\in X \}$. Obviously $X_{k_0}$ is a bounded subset of the Euclidean space
$\mathbb{R}^{k_0}$. Thus there exists a finite $\varepsilon$-net of the set
$X_{k_0}$ formed by some $k_0$ - tuples
$\tilde{y_1}, \tilde{y_2}, \dots , \tilde{y}_m$, where
$\tilde{y}_p=(y_1^p, y_2^p, \dots, y_{k_0}^p)$ for $p=1,2,\dots,m$.

 Next, we consider the sequence $y_p$ $(p=1,2,\dots,m)$ defined as
 $$
y_p=(y_1^p, y_2^p, \dots, y_{k_0}^p, y_{k_0}^p, y_{k_0}^p, \dots).
$$
 We show that the set $\{y_1, y_2, \dots, y_m\}$ forms the $r+2 \varepsilon$-net
of the set $X$ in the space $c$. To this end take an arbitrary sequence
$x=(x_i)\in X$. Then, we can find a $k_0$-tuple
$\tilde{y}_p=(y_1^p, y_2^p, \dots, y_{k_0}^p)$ $(1\leq p \leq m)$ such that
\begin{equation}\label{rozw7}
 | x_i - y_i^p | \leq \varepsilon
 \end{equation}
 for $i=1,2,\dots,k_0$.
 Further, for $i \geq k_0$, we obtain
 $$
| x_i - y_i^p | \leq | x_i - x_{k_0}| +| x_{k_0} - y_i^p |
= |  x_i - x_{k_0} |+| x_{k_0} - y_{k_0}^p |.
$$
 Hence, from \eqref{rozw6} and \eqref{rozw7} we obtain
\begin{equation} \label{rozw8}
 | x_i - y_i^p | \leq r+ \varepsilon +\varepsilon=r+2\varepsilon.
\end{equation}
 Linking \eqref{rozw7} and \eqref{rozw8} we conclude that the set
 $\{y_1, y_2, \dots, y_m\}$ forms an $r+2 \varepsilon$-net of the set $X$
in the space $c$. Moreover, in view of the arbitrariness of $\varepsilon$ this yields
 $$ \chi(X)\leq r,$$
 which leads to the inequality
 \begin{equation}\label{rozw9}
 \chi(X)\leq \mu_c(X).
 \end{equation}
 Combining estimates \eqref{rozn5} and \eqref{rozw9} we derive the
following inequalities
 \begin{equation}\label{rozw10}
 \chi(X)\leq \mu_c(X)\leq 4\chi(X),
 \end{equation}
 which are satisfied for $X \in \mathfrak{M}_{c}$.

 Now, let us observe that from inequalities \eqref{rozw10} we obtain that
the quantity $\mu_c$ satisfies axioms (i) and (vi) of Definition \ref{def2.1}.
Thus, $\mu_c$ is a sublinear measure of noncompactness with maximum property
in the space $c$.
 Applying \eqref{rozw10} again we deduce that $\mu_c$ is a regular measure
equivalent to the Hausdorff measure $\chi$.

 In what follows let us observe that the estimate on the right hand side
of \eqref{rozw10} i.e., estimate \eqref{rozn5} can be improved.
Indeed, since $\mu_c$ is a regular measure of noncompactness then, in view
of Theorem \ref{thm2.2} we have
 \begin{equation}\label{rozw11}
 \mu_c(X)\leq \mu_c(B_1)\chi(X)
 \end{equation}
 for an arbitrary set $X \in \mathfrak{M}_{c}$ (the symbol $B_1$ stands
for the unit ball in $c$). On the other hand it is easy to calculate
that $\mu_c (B_1)=2$. Thus, from \eqref{rozw11} we obtain
\begin{equation}\label{rozw12}
 \mu_c(X)\leq 2\chi(X).
 \end{equation}
 Finally, combining estimates \eqref{rozw9} and \eqref{rozw12} we obtain
desired estimate \eqref{rozn3}. The proof is complete.
\end{proof}

In the sequel we shall deal with measures of noncompactness in the space $l_\infty$.
Firstly, let us notice that in this space we do not know a formula which expresses
the Hausdorff measure of noncompactness $\chi$. Even more, we do not know
formulas for regular measures in $l_\infty$ \cite{Akhm,BGoebel,Banas Mursaleen}.
Thus, in this case we can only obtain formulas for measures of noncompactness
defined in an axiomatic way (cf. Definition \ref{def2.1}). It is worthwhile mentioning
that there are known and used some convenient formulas for measures of
noncompactness in the space $l_\infty$ \cite{BGoebel,Banas Mursaleen}.
Unfortunately, in the literature there are no proofs of the correctness of
those formulas. More precisely, there are no proofs of the fact that the
formulas in question are measures on noncompactness in $l_\infty$.
Below we are going to fill  this gap.

To present the above mentioned formulas let us fix a set
$X \in \mathfrak{M}_{l_\infty}$. Next, we define the following three quantities:
\begin{gather}\label{rozw13}
 \mu^\infty_1(X)=\lim_{n\to \infty} \Big\{ \sup_{(x_i)\in X}
\big\{\sup \{ | x_i | : i \geq n\} \big\}\Big\}, \\
\label{rozw14}
 \mu^\infty_2(X)=\lim_{k\to \infty} \Big\{ \sup_{(x_i)\in X}
\big\{\sup  \{ | x_n-x_m | : n,m \geq k \} \big\}\Big\}, \\
\label{rozw15}
 \mu^\infty_3(X)=\limsup_{n\to \infty} \operatorname{diam} X_n,
\end{gather}
where $X_n=\{x_n: x=(x_i) \in X\}$ and
$\operatorname{diam}X_n=\sup\big \{| x_n - y_n | : x=(x_i), y=(y_i) \in X \big \}$.
Observe that the formula expressing the quantity $\mu^\infty_1$ coincides
 with the formula for the Hausdorff measure of noncompactness in the space $c_0$.
On the other hand, formula \eqref{rozw14} for the quantity $\mu^\infty_2$
coincides with formula \eqref{rozn2} for the measure of noncompactness $\mu_c$
in the sequence space $c$.

\begin{theorem} \label{thm3.2}
 The quantities $\mu _i^\infty (i=1,2,3)$ are sublinear measures of noncompactness
in the space $l_\infty$. In addition, the measures $\mu _1^\infty$ and
$\mu _2^\infty$ have maximum property. Moreover, for an arbitrary
set $X \in \mathfrak{M}_{l_\infty}$ the following inequalities hold
\begin{gather}\label{rozw16}
 \chi(X) \leq  \mu _2^\infty (X), \\
\label{rozw17}
 \chi(X) \leq  \mu _3^\infty (X), \\
\label{rozw18}
 \mu _2^\infty (X) \leq  2\mu _1^\infty (X), \\
\label{rozw19}
 \mu _3^\infty (X) \leq  2\mu _1^\infty (X).
\end{gather}
\end{theorem}

\begin{proof}
 The proof of  \eqref{rozw16} can be conducted in the same way
as the proof of  \eqref{rozw9}. Indeed, it follows easily from the
fact that $c$ is a subspace of the space $l_\infty$.

 To prove \eqref{rozw17} let us fix  $X \in \mathfrak{M}_{l_\infty}$ and
put $r=\mu _3^\infty (X)$. Next, take an arbitrary number $\varepsilon>0$.
Then, in view of definition \eqref{rozw15} we can find a natural number
$n_0$ such that diam$X_n \leq r+\varepsilon$ for $n\geq n_0$. Hence we infer
that for arbitrary elements $x=(x_i)$, $y=(y_i)$ of the set $X$ we have
\begin{equation}\label{rozw20}
 | x_n- y_n | \leq r+ \varepsilon
 \end{equation}
 for $n\geq n_0$.
 Further, we consider the set
$\bar{X}_{n_{0}}=\{(x_1,x_2,\dots,x_{n_0}): (x_i) \in X\}$.
This set is a relatively compact subset of the Euclidean space $\mathbb{R}^{n_0}$.
Thus, there exists a finite $\varepsilon$-net of the set $\bar{X}_{n_{0}}$
composed by $n_0$ - tuples $\tilde{y}_1=(y_1^1, y_2^1, \dots, y_{n_0}^1)$,
$\tilde{y_2}=(y_1^2, y_2^2, \dots, y_{n_0}^2)$,
$\tilde{y}_m=(y_1^m, y_2^m, \dots, y_{n_0}^m)$.
 Next, fix an arbitrary element
$y=(y_i)=(y_1, y_2, \dots, y_{n_0}, y_{n_0+1},\dots)$ 
of the set $X$ and consider the finite subset 
$Y=\{y_1,y_2,\dots,y_m\}$
of the space $l_\infty$ such that
 $$
y_i=(y_1^i, y_2^i, \dots, y_{n_0}^i, y_{n_0+1}, y_{n_0+2},\dots)
$$
 for $i=1,2,\dots,m$. We show that $Y$ forms a finite $r+\varepsilon$-net
of the set $X$.
 To this end take an arbitrary element $x=(x_i) \in X$ and consider the
$n_0$-tuple $\tilde{x}=(x_1, x_2, \dots, x_{n_0})$. Then we can find a
$n_0$-tuple $\tilde{y}_k \in \bar{X}_{n_0}$,
$\tilde{y}_k=(y_1^k, y_2^k, \dots, y_{n_0}^k)$ such that
\begin{equation}\label{rozw21}
 | x_i- y_i^k | \leq \varepsilon
 \end{equation}
for $i=1,2,\dots,n_0$.

 Now, we take the element
$y_k=(y_1^k, y_2^k, \dots, y_{n_0}^k, y_{n_0+1}, y_{n_0+2},\dots) \in Y$.
Then, in view of \eqref{rozw20} and \eqref{rozw21}, we have
 $$
| x_n- y_n | \leq r+\varepsilon
$$
for $n=1,2,\dots$. This means that $\| x-y_k \| _{l_\infty} \leq r+\varepsilon$.
Thus the set $Y$ forms a finite $r+\varepsilon$-net of the set $X$.
Hence we conclude that $\chi(X) \leq r+\varepsilon$. In view of the arbitrariness
of $\varepsilon$ this implies inequality \eqref{rozw17}.
 Further, let us observe that estimates \eqref{rozw18} and \eqref{rozw19}
are a simple consequence of the triangle inequality for absolute value.

 Next, from \eqref{rozw16} and \eqref{rozw18}
(or from \eqref{rozw17} and \eqref{rozw19}) we obtain the following estimate
 \begin{equation}\label{rozw22}
 \frac{1}{2}\chi (X) \leq \mu_1^\infty (X).
 \end{equation}

 Finally, taking into account inequalities \eqref{rozw16}, \eqref{rozw17}
and \eqref{rozw22} we conclude that the quantities $\mu_i^\infty$ $(i=1,2,3)$
satisfy axioms (i) and (vi) of Definition \ref{def2.1}. The fact that there are
satisfied other conditions (ii)--(v) and (vii), (viii) for all quantities
$\mu_i^\infty$ $(i=1,2,3)$ and condition (ix) for $\mu_1^\infty $ and
$\mu_2^\infty $ is easy to prove. This completes the proof.
\end{proof}


\section{Measures of noncompactness in spaces of tempered sequences}

As we saw in introduction, classical sequence spaces are not always
suitable to consider initial value problems for infinite systems of
differential equations. Therefore, in order to consider those initial
value problems we are frequently forced to treat the problems in
question in enlarged sequence spaces. Such sequence spaces can be obtained
if we consider the so - called tempered sequence spaces.

To define the mentioned spaces let us fix a real sequence $\beta =(\beta_n)$
such that $\beta_n$ is positive for $n=1,2,\dots$ and the sequence $(\beta_n)$
is nonincreasing. Such a sequence $\beta$ will be called the tempering sequence.
Next, consider the set $X$ consisting of all real (or complex) sequences
 $x=(x_n)$ such that $\beta _n x_n \to 0$ as $n \to \infty$. It is easily seen
that $X$ forms a linear space over the field of real (or complex) numbers.
We will denote this space by the symbol $c_0^{\beta}$.

It is easy to check that $c_0^{\beta}$ is a Banach space under the norm
$$
\| x\|_{ c_0^{\beta}}=\| (x_n)\|_{ c_0^{\beta}}=
 \sup \{\beta_n | x_n | : n=1,2,\dots \}
= \max \{\beta_n | x_n | : n=1,2,\dots \}.
$$

In a similar way we may consider the space $c^\beta$ consisting of real (complex)
sequences $(x_n)$ such that the sequence $(\beta _n x_n)$ converges to a
finite limit. Obviously $c^\beta$ forms a linear space and it becomes a
Banach space if we normed it by the supremum norm
$$
\| x\|_{ c^{\beta}}=\| (x_n)\|_{ c^{\beta}}
= \sup \{\beta_n | x_n | : n=1,2,\dots \}.
$$
In the same way we can consider the tempered sequence space
$l_ \infty ^ \beta$ of all sequences $(x_n)$ (real or complex) such that
the sequence $(\beta _n x_n)$ is bounded.
The space $l_ \infty ^ \beta$ is a Banach space under the norm
$$
\| x\|_{l_ \infty ^ \beta}=\| (x_n)\|_{ l_ \infty ^ \beta}
= \sup \{\beta_n | x_n | : n=1,2,\dots \}.
$$

Let us pay attention to the fact that taking $\beta _n =1$ for
$n=1,2,\dots$ we obtain spaces $c_0^{\beta}=c_0$, $c^\beta = c$ and
$l_ \infty ^ \beta=l_ \infty$. Similarly, if the sequence $(\beta _n)$ is
bounded from below by a positive constant $m$ i.e., if $\beta _n \geq m>0$
for $n=1,2,\dots$, then the norms in the tempered sequence spaces
$c_0^{\beta}$, $c^\beta$ and $l_ \infty ^ \beta$ are equivalent to the classical
supremum norm in each of the spaces $c_0$, $c$ and $l_ \infty$.
Thus,  to obtain an essential enlargement of the spaces $c_0$, $c$ and
$l_ \infty$  we should to assume that the tempering sequence $(\beta _n)$
converges to zero. In what follows we will impose such a requirement.

The most important fact for our further purposes is the assertion saying
that the pairs of the spaces $(c_0, c_0^{\beta})$, $(c, c^{\beta})$ and
$(l_ \infty, l_ \infty ^ \beta)$ are isometric. Indeed, consider for example
the spaces $l_ \infty$ and $ l_ \infty ^ \beta$.
Next, take the mapping $J:l_ \infty ^ \beta \to l_ \infty $ defined in the
following way
$$
J(x)=J((x_n))=(\beta_n x_n).
$$
Then, for arbitrarily fixed $x,y \in l_\infty ^\beta $ we have
\begin{align*}
\| J(x)-J(y) \| _{l_{\infty}}
&=\| J((x_n))-J((y_n)) \| _{l_{\infty}} \\
&=\|( \beta_n x_n) -(\beta_n y_n) \| _{l_{\infty}}\\
&=\sup  \{ |\beta _n x_n - \beta_n y_n | : n=1,2,\dots \}\\
&= \sup \{ \beta _n | x_n -y_n | : n=1,2,\dots \}
= \| x-y \| _ {l_\infty ^\beta}.
\end{align*}
This shows that the mapping $J$ is an isometry between the spaces
$l_ \infty ^ \beta$ and $l_ \infty$. Obviously, the same mapping establishes
the isometry between the spaces $c^ \beta$ and $c$ and the spaces $c_0^\beta$
and $c_0$, respectively.

The above assertions enable us to define measures of noncompactness in the
tempered sequence spaces $c_0^\beta, c^\beta$ and
$ l_ \infty ^ \beta$. In fact, the Hausdorff measure of noncompactness
$\chi(X)$ for $X \in \mathfrak{M}_{c_0^\beta}$ can be expressed in the
following way (cf. Section 3):
\begin{equation}\label{4.1}
 \chi(X)=\lim_{n\to \infty} \Big\{ \sup_{(x_i)\in X}
\big\{\sup  \big \{ \beta_i | x_i | : i \geq n\big \} \big\}\Big\}.
\end{equation}
Similarly, the analogue of the measure of noncompactness $\mu _c$ defined
by formula \eqref{rozn2} has the form
\begin{equation}\label{4.2}
\mu_{c^\beta}(X)=\lim_{k \to \infty} \Big\{ \sup_{(x_i)\in X}
\big\{\sup \{| \beta _n x_n- \beta_m x_m |: n,m\geq k\} \big\} \Big\},
\end{equation}
where $X \in \mathfrak{M}_{c^\beta}$.

Obviously, in view of the fact that the spaces $c$ and $c^\beta$ are
isometric (by the above mentioned isometry $J$), on the basis of Theorem \ref{thm3.1}
we have the estimates
$$
\chi(X) \leq \mu_ {c^\beta}(X) \leq 2\chi (X)
$$
for each $X \in \mathfrak{M}_{c^\beta}$, where $\chi$ denotes the Hausdorff
measure of noncompactness in the space $c^\beta$.

Now, let us take into account the tempered sequence space $l_\infty^\beta$.
Then, keeping in mind formulas \eqref{rozw13} - \eqref{rozw15} expressing
measures of noncompactness in the space $l_\infty$, we obtain the following
formulas for the counterparts of those measures in the space $l_\infty^\beta$:
\begin{gather}\label{4.3}
\mu_1^\beta(X)=\lim_{n\to \infty} \Big\{ \sup_{(x_i)\in X}
\big\{\sup  \big \{\beta_i |  x_i | : i \geq n\big \} \big\}\Big\}, \\
\label{4.4}
\mu_2^\beta(X)=\lim_{k \to \infty} \Big\{ \sup_{(x_i)\in X}
\big\{\sup \{| \beta _n x_n- \beta_m x_m |: n,m\geq k\} \big\} \Big\}, \\
\label{4.5}
\mu^\beta_3(X)=\limsup_{n\to \infty} \operatorname{diam}X_n^\beta,
\end{gather}
where $X \in \mathfrak{M}_{l_\infty^\beta}$. Moreover, $X^\beta_n$ in \eqref{4.5}
is understood in the following way
$$
X_n^\beta = \{x_n \beta_n : (x_i) \in X\}.
$$
Apart from this $\operatorname{diam}X^\beta_n=\sup \big \{\beta _n |  x_n - y_n |
: (x_i),(y_i) \in X \big \}$.

Further, taking into account Theorem \ref{thm3.2} we deduce the inequalities
\begin{gather}\label{4.6}
\chi(X) \leq  \mu _2^\beta (X), \\
\label{4.7}
\chi(X) \leq  \mu _3^\beta (X), \\
\label{4.8}
\mu _2^\beta (X) \leq  2\mu _1^\beta (X), \\
\label{4.9}
\mu _3^\beta (X) \leq  2\mu _1^\beta (X),
\end{gather}
where $X \in \mathfrak{M}_{l_\infty^\beta}$ and the symbol
$\chi $ denotes the Hausdorff measure of noncompactness in the
space $l_\infty^\beta$.

In view of inequalities \eqref{4.6}--\eqref{4.9} it is easily seen that
the kernel $\ker  \mu_1^\beta$ consists of all sets $X$ belonging to
the family $\mathfrak{M}_{l_\infty^\beta}$ such that the sequences
$(\beta_n x_n)$ tend to zero at infinity uniformly with respect to the set $X$
i.e., for any $\varepsilon>0$ there exists a natural number $n_0$ such that
$\beta _n | x_n | \leq \varepsilon $ for all $(x_i) \in X$ and for $n \geq n_0$.

Similarly, the kernel $\ker  \mu_2^\beta$ consists of all sets
$X \in \mathfrak{M}_{l_\infty^\beta}$ such that the sequences $(\beta_n x_n)$
tend to finite limits uniformly on the set $X$. In other words, the sequences
$(\beta_n x_n)$ satisfy Cauchy condition uniformly with respect to $X$.

Finally, the kernel $\ker  \mu_3^\beta$ consists of all sets $X$ belonging
to the family $\mathfrak{M}_{l_\infty^\beta}$ such that the thickness of the
bundle formed by sequences $(\beta_n x_n)$, where $(x_i)\in X$, tends to zero
 at infinity.

Let us also observe that the measures of noncompactness $\mu _1^\beta $,
$\mu _2^\beta $, $\mu _3^\beta $ are not regular in the space $l_\infty^\beta$.

\section{Results from  differential equations in Banach spaces}

This section has an auxiliary character and contains a few results from the
theory of ordinary differential equations in Banach spaces
(cf. \cite{Banas Lecko,Banas Mursaleen,Deimling}). To present those results
let us assume that $E$ is a Banach space with a norm $\| \cdot \|$.
Let $x_0$ be a fixed element of $E$ i.e., $x_0 \in E$ and let $B(x_0, r)$
denotes a ball in $E$. We will consider the differential equation
\begin{equation}\label{5.1}
x'=f(t,x)
\end{equation}
with the initial condition
\begin{equation}\label{5.2}
x(0)=x_0.
\end{equation}
Here, we assume that $f=f(t,x)$ is a given function such that
$f: [0,T] \times B(x_0,r) \to E$. We will write $I=[0,T]$. Throughout this
section we will assume that $\mu$ is a measure of noncompactness in the space $E$.

Further, by the symbol $E_\mu$ we will denote the so - called
\textit{kernel set} of the measure of noncompactness $\mu$ \cite{Banas Mursaleen}
 which is defined in the following way
$$
E_\mu=\{x \in E: \{x\} \in \ker  \hspace{3pt} \mu \}.
$$
It can be shown that $E_\mu$ is a closed, convex subset of the space $E$.
Moreover, if $\mu$ is a sublinear measure then $E_\mu$ is a linear closed
subspace of $E$.
It is worthwhile mentioning that the concept of the kernel set plays
an important role in the theory of differential equations in Banach spaces.

Now, we recall a result concerning initial value problem \eqref{5.1}--\eqref{5.2}
 which is not very general but is useful for our purposes (cf. \cite{BGoebel}).

\begin{theorem} \label{thm5.1}
 Suppose the function f is uniformly continuous on $I \times B(x_0,r)$ and
$\| f(t,x) \| \leq A$, where $AT \leq r$. Further, let $\mu$ be a sublinear
measure of noncompactness in $E$ such that $ \{x_0\} \in\ker \mu$.
 We assume that for any nonempty set $X \subset B(x_0,r)$ and for almost all
$t\in I$ the following inequality holds
 \begin{equation}\label{5.3}
 \mu (f(t,X)) \leq p(t)\mu (X),
 \end{equation}
 where $p(t)$ is an integrable function on the interval $I$.
Then  \eqref{5.1}--\eqref{5.2} has at least one solution $x=x(t)$ on the
interval $I$ such that $x(t)\in E_\mu$ for $t\in I$.
\end{theorem}

The below given theorem is a slightly modified version of the result
contained in Theorem \ref{thm5.1}, which will be more convenient in our further
 considerations (cf. \cite{Banas Lecko,Banas Mursaleen}).

\begin{theorem} \label{thm5.2}
 Assume that $f$ is a function defined on $[0,T] \times E$ with values in $E$
such that
  \begin{equation}\label{5.4}
  \| f(t,x)\| \leq P+Q \| x \|
  \end{equation}
 for each $t\in [0,T]$ and $x\in E$, where $P$ and $Q$ are nonnegative constants.
Further, assume that $f$ is uniformly continuous on the set
$[0, T_1] \times B(x_0,r)$, where $QT_1 <1$ and $r=\frac{(P+Q)T_1 \| x_0 \|}{1-QT_1}$.
 Moreover, we assume that $f$ satisfies condition \eqref{5.3} with a sublinear
measure of noncompactness $\mu$ such that $x_0 \in E_\mu$.
Then, initial value problem \eqref{5.1}--\eqref{5.2} has a solution $x=x(t)$
on the interval $[0,T_1]$ such that $x(t) \in E_\mu$ for $t\in[0,T_1]$.
\end{theorem}

\begin{remark} \label{rmk5.3} \rm
 Observe that in the case when $\mu = \chi$
(the Hausdorff measure of noncompactness), the assumption on the uniform
continuity of the function $f$ can be replaced by the weaker one requiring
only the continuity \cite{monch}. The same assertion is also true if $\mu$
is a regular measure of noncompactness equivalent to the Hausdorff measure
\cite{Heinz,monch}.
\end{remark}

\section{Infinite systems of differential equations in the tempered sequence
space $c_o^\beta$}

The considerations of this section will be located in the Banach tempered
sequence space $c_o^\beta$ described in Section 4. Thus, we will assume
that $\beta=(\beta_n)$ is a sequence with positive terms which is nonincreasing.
The space $c_o^\beta$ consists of all sequences $(x_n)$ such that the sequence
$(\beta _n x_n)$ converges to zero. We will consider here only real sequences
$(x_n)$. The norm in the space $c_o^\beta$ is defined by the formula
$$
\| x\|_{c_ 0 ^ \beta}=\| (x_n)\|_{c_ 0 ^ \beta}
= \sup  \{\beta_n | x_n | : n=1,2,\dots \}.
$$
To simplify the notation we will use the symbol $\| \cdot \|$ instead of
$\| \cdot \| _{c_ 0 ^ \beta}$.

The object of our study in this section will be first semilinear lower diagonal
infinite systems of differential equations having the form
 \begin{equation}\label{6.1}
 x_n'= \sum_{i=1}^{k_n}a_{nn_i} (t) x_{n_i} +f_n (t,x_1,x_2,\dots)
 \end{equation}
with the initial value conditions
 \begin{equation}\label{6.2}
 x_n(0)=x_0^n, \quad\text{for } x=1,2,\dots\,.
 \end{equation}
We assume that for any fixed $n \in \mathbb{N} $ the sequence
$(n_1, n_2,\dots,n_{k_n})$ is such that $1\leq n_1 <n_2<\dots<n_{k_n} \leq n $.
Moreover, the sequence $(n_1)$ tends to infinity when $ n \to \infty $.
Apart from that we assume that there exists a natural number $K$ such that
 $k_n \leq K$ for all $n=1,2,\dots$\hspace{5pt}. In other words, this means
that any ``linear part'' of system \eqref{6.1} contains only finite number
of nonzero terms and the number of those terms does not exceed $K$.
In what follows infinite systems \eqref{6.1} satisfying the above requirement
will be called infinite systems of differential equations with
\textit{linear parts of constant width}.

Apart of the requirement concerning the constant width of linear parts
we will impose the following assumptions in our study of initial value
problem \eqref{6.1}--\eqref{6.2}:
\begin{itemize}
 \item[(i)] The function $a_{nn_i}=a_{nn_i}(t)$ is continuous on a fixed
interval $I=[0,T]$ for $n=1,2,\dots$ and for $i=1,2,\dots,k_n$;

 \item[(ii)] the functions $a_{nn_i}(t)$ are uniformly bounded on the interval
$I$ by a positive constant $A$ i.e., $| a_{nn_i}(t)| \leq A $ for
$t \in I$ and for $n=1,2\dots$ and for $i=1,2,\dots,k_n$;

 \item[(iii)] the sequence $(x_0^n)$ belongs to the space $c_0^\beta $;

 \item[(iv)] for each fixed $n$ the function $f_n(t,x_1,x_2,\dots)=f_n(t,x)$
acts from the set $I \times \mathbb{R}^\infty$ into $\mathbb{R}$.
Moreover, the function  $f_n :I \times c_0^\beta \to  \mathbb{R} $ is
continuous on $I \times c_0^\beta$;

 \item[(v)] there exists a sequence $(p_n)$ of nonnegative terms with the
property that $\beta _n p_n \to 0$ as $n \to \infty$ and such that
 $| f_n(t,x) | \leq  p_n$ for $t \in I$, $x \in c_0^\beta$ and for
$n=1,2\dots$\hspace{5pt}.
\end{itemize}
Now, we can formulate our existence result.

\begin{theorem} \label{thm6.1}
 Assume that the functions involved in system  \eqref{6.1}
having linear parts of constant width $K$, satisfy conditions (i)--(v).
Then initial value problem \eqref{6.1}--\eqref{6.2} has at least one solution
$x(t)=(x_n(t))=((x_1(t), x_2(t),\dots)$ in the sequence space $c_0^\beta$
on the interval $I$.
\end{theorem}

\begin{proof}
 For arbitrarily fixed $n\in \mathbb{N}$ let us denote
 $$
 g_n(t,x)=g_n(t, x_1, x_2,\dots)= \sum_{i=1}^{k_n}
a_{nn_i}(t) x_{n_i}+f_n(t, x_1, x_2,\dots),
$$
 where $t \in I$ and $x=(x_n) \in c_0^\beta $. Then, keeping in mind our
assumptions, we obtain
\begin{align*}
\beta_n | g_n(t, x_1, x_2,\dots)|
&\leq \beta_n \sum_{i=1}^{k_n} | a_{nn_i}(t)| |  x_{n_i}|
  + \beta _n | f_n(t, x_1, x_2,\dots)| \\
&\leq \beta_n A \sum_{i=1}^{k_n} | x_{n_i} | +\beta _n p_n
 = A\sum_{i=1}^{k_n} \beta _n | x_{n_i} | +\beta _n p_n \\
&\leq  A\sum_{i=1}^{k_n} \beta _{n_i} | x_{n_i} | +\beta _n p_n \\
&\leq AK \max \{\beta _{n_i} | x_{n_i} | : i=1,2,\dots,k_n\}+\beta _n p_n\\
&\leq AK \sup\{\beta_j | x_j | : j \geq n_1\}+\beta _n p_n.
\end{align*}
 Hence, replacing $n$ by $j$ and $j$ by $i$, we can write the above
 inequality in the form
  \begin{equation}\label{6.3}
  \beta_j | g_j(t, x_1, x_2,\dots)| \leq AK \sup\{\beta_i | x_i |
: i \geq j_1\}+ \beta _j p_j.
  \end{equation}
 Next, let us notice that from estimate \eqref{6.3} it follows that the
following inequality holds
\begin{equation}\label{6.4}
\begin{aligned}
\| g(t, x)\|&=\sup \{ \beta_j | g_j (t, x_1,x_2,\dots) | : j=1,2,\dots\}  \\
&\leq AK \sup_{j \in \mathbb{N}} \big\{\sup \{\beta_i | x_i | : i \geq j_1\}\big\}
+ \sup \{\beta _j p_j : j=1,2,\dots \}\\
&=AK \| x \| +P,
\end{aligned}
\end{equation}
where the operator $g=g(t,x)$ is defined on the set $I \times c_0^\beta$
in the following way
 $$
g(t,x) = (g_1(t,x), g_2 (t,x),\dots).
$$
 In view of estimate \eqref{6.4} we see that $g$ transforms the set
$I \times c_0^\beta$ into the space $c_0^\beta$.

 Now, we show that the operator $g$ is continuous on the set $I \times c_0^\beta$.
To this end we split the operator $g$ into two terms
 $$
g(t,x)=(Lx)(t)+f(t,x),
$$
 where the operators $L$ and $f$ are defined as follows:
 $$
 (Lx)(t)=((L_1 x)(t), (L_2x)(t),\dots)
$$
 where
$$
(L_nx)(t)=\sum_{i=1}^{k_n}a_{nn_i}(t)x_{n_i}
$$
 $(n=1,2,\dots)$, and
 $$
 f(t,x)=(f(t_1,x),f(t_2,x),\dots).
$$
 First we show that the operator $f$ is continuous on the set
 $I \times c_0^\beta$. To do this fix arbitrarily a number $\varepsilon>0$
and a point $x \in c_0^\beta$. According to assumption (v) we can choose
a natural number $n_0$ such that
  \begin{equation}\label{6.5}
  \beta _n p_n \leq \frac{\varepsilon}{2}
  \end{equation}
 for $n \geq n_0$. Next, in view of assumption (iv) we can find a number
$\delta _i$ $(i=1,2,\dots,n_0)$ such that for any $y \in c_0^\beta$ such that
 $\| x-y \| \leq \delta_i$ and for arbitrary $t \in I$ we have
$$
| f_i (t,x) - f_i (t,y) | \leq \frac{\varepsilon}{\beta_1}.
$$
 Let us take $\delta=\min\{\delta_1, \delta_2, \dots, \delta_{n_0}\}$.
Then, for arbitrary $y\in c_0^\beta$ such that  $\| x-y \| \leq \delta$
and for $t \in I$ we have
  \begin{equation}\label{6.6}
  | f_i (t,x) - f_i (t,y) | \leq \frac{\varepsilon}{\beta_1}
  \end{equation}
Combining \eqref{6.5} and \eqref{6.6}, for $y \in c_0^\beta$ with
$\| x-y \| \leq \delta$ and for $t \in I$, we obtain
\begin{align*}
\| f(t,x) - f(t,y)\|
&=\sup \{ \beta_n | f_n (t, x)-f_n(t,y) | : n=1,2,\dots\} \\
&=\max \Big \{ \max \big \{ \beta_n | f_n (t, x)-f_n(t,y) | :
  n=1,2,\dots, n_0\big \}, \\
&\qquad    \sup \big \{ \beta_n | f_n (t, x)-f_n(t,y) | : n>n_0\big \} \Big \} \\
&\leq  \max \Big \{ \max \big \{ \beta_1 | f_n (t, x)-f_n(t,y) | :
 n=1,2,\dots, n_0\big \}, \\
&\qquad \sup \{ \beta_n \big [ | f_n (t, x)|+ |f_n(t,y) |\big ] : n>n_0\} \Big \}\\
&\leq \max \Big \{ \beta_1 \Big (\frac{\varepsilon}{\beta _1}\Big), \sup \big
\{ 2 \beta_n p_n : n> n_0 \big\}\Big\}=\varepsilon.
\end{align*}
This shows that the operator $f$ is continuous at an arbitrary point
$(t,x) \in I \times c_0^\beta$.

 Next, we show that the operator $L$ is continuous on the set $I \times c_0^\beta$.
Similarly as before, fix arbitrarily $x\in c_0^\beta$, $t\in I$ and a number
$\varepsilon>0$. Then, for $y \in c_0^\beta$ with $\| x-y \| \leq \varepsilon$
and for an arbitrary fixed natural number $n$, in view of imposed assumptions
we obtain
\begin{align*}
&\beta_n | (L_n x)(t)-(L_n y)(t))| \\
 & = \beta_n   \Big | \sum_{i=1}^{k_n} a_{nn_i}(t)x_{n_i} - \sum_{i=1}^{k_n}
 a_{nn_i}(t) y_{n_i} \Big | \\
&\leq \beta_n  \sum_{i=1}^{k_n} | a_{nn_i}(t) | | x_{n_i} - y_{n_i} | \\
&\leq  A\sum_{i=1}^{k_n} \beta _{n} | x_{n_i}- y_{n_i}|
  \leq A\sum_{i=1}^{k_n} \beta _i | x_{n_i}- y_{n_i}| \\
&\leq AK \max \{\beta _{i} | x_{n_i}-y_{n_i} | : i=1,2,\dots,k_n\} \\
&\leq AK \sup\{\beta_j | x_j-y_j | : j \geq n_1\} \\
&\leq AK \sup\{\beta_j | x_j-y_j | : j=1, 2, \dots \}
   =AK \| x-y \| \leq AK\varepsilon.
\end{align*}
 Hence we deduce that the operator $L$ is continuous on the set
$I \times c_0^\beta$. Consequently, as we announced before, we conclude that
the operator $g$ is continuous on the set $I \times c_0^\beta$.

 In what follows let us take a number $T_1$ such that $T_1 <T$ and $AKT_1<1$.
According to assumptions of our theorem take the number
$r=\frac{(P+AK)T_1 \| x_0 \| }{1-AKT_1}$ and consider the ball $B(x_0,r)$.
Next, choose an arbitrary subset $X$ of the ball $B(x_0,r)$.
Then, for $x\in X$ and $t\in [0,T_1]$, in view of estimate \eqref{6.3},
for an arbitrary fixed natural number $n$, we obtain:
\begin{align*}
&\sup \{\beta_j | g_j(t,x_1, x_2,\dots)|: j \geq n \} \\
&\leq \sup \big \{ AK \sup \{ \beta_i | x_i | : i \geq j_1 \} :
 j\geq n \big \}+ \sup \big \{ \beta_j p_j : j\geq n \big \} \\
&\leq AK \sup \big \{ \sup \{ \beta_i | x_i | : i \geq n_1 \},
 \sup \{ \beta_i| x_i | : i \geq (n+1)_1 \}, \\
&\quad   \sup \{ \beta_i | x_i | : i \geq (n+2)_1 \}, \dots \big \}
 + \sup \big \{ \beta_j p_j : j\geq n \big \}.
\end{align*}
This yields the estimate
\begin{align*}
&\sup_{x\in X} \big \{ \sup \{ \beta_j | g_j(t,x_1, x_2,\dots)|:
 j \geq n \} \big \}\\
&\leq AK \sup_{x\in X} \big\{ \sup \{ \sup \{\beta_i | x_i |
 : i \geq j_i \}: j \geq n \} +\sup \{ \beta_j p_j : j \geq n \} \big\}.
\end{align*}
  Passing with $n \to \infty $ in the above estimate and taking into account
that $j_1 \to \infty$ as $j \to \infty$, we obtain
 $$
\chi (g(t,X)) \leq AK\chi (X),
$$
 where $\chi$ denotes the Hausdorff measure of noncompactness in the space
$c_0^\beta$ expressed with help of formula \eqref{4.1}.
Finally, in view of the above established facts and Theorem \ref{thm5.2}, we complete
the proof.
\end{proof}

The following  example illustrates the result in Theorem \ref{thm6.1}.

\begin{example} \label{examp6.2} \rm
 Consider the  infinite system of differential equations
\begin{equation}\label{6.7}
 \begin{gathered}
   x_1'=x_1+\frac{\sqrt{| x_1 |}}{\sqrt{| x_1 |}+1},\\
   x_2'=x_1+x_2+2\frac{\sqrt{| x_2 |}}{\sqrt{| x_2 |}+1},\\
   x_3'=x_2+x_3+3\frac{\sqrt{| x_3 |}}{\sqrt{| x_3|}+1},\\
   \dots\\
   x_n'=  x_{n-1}+x_n+n\frac{\sqrt{| x_n |}}{\sqrt{| x_n|}+1},\\
   \dots .
 \end{gathered}
 \end{equation}
 with initial conditions
 \begin{equation}\label{6.8}
 x_n(0)=n\quad\text{for } n=1,2,\dots\,.
 \end{equation}
\end{example}

Observe that \eqref{6.7} is a semilinear lower diagonal infinite system of
differential equations with linear parts of constant width $K=2$.
Moreover, it is easily seen that system \eqref{6.7} is a particular case of
system \eqref{6.2} if we take $a_{nn_i}(t)=1$ for $t\in I$, where we put
$I=[0,T_1]$, where $T_1>0$ is a number chosen according to assumptions of
Theorem \ref{thm5.2}. Additionally, $n=1,2\dots$ and $i=1,2$ for $n\geq2$.
Hence we see that there is satisfied assumption (i) of Theorem \ref{thm6.1}.

Further, we have that $| a_{nn_i}(t)| \leq1 $ for $t \in I$ and
$n=1,2,\dots$, $i=1,2$. This means that functions $ a_{nn_i}(t)$
satisfy assumption (ii).

In what follows let us take the sequence $\beta_n=\frac{1}{n^2}$ for
$n=1,2\dots$. Obviously we have that $x_0=(x_0^n)=(n)\in c_0^\beta$, where
$\beta=(\beta_n)=(\frac{1}{n^2})$. Thus there is satisfied assumption (iii).
 From the form of system \eqref{6.7} we see that we can take
$$
f_n(t,x_1,x_2,\dots)=n\frac{\sqrt{| x_n |}}{\sqrt{| x_n |}+1}
$$
for $n=1,2\dots$. Obviously, the function $f_n=f_n(t,x)$ is continuous on
the set $I \times c_0^\beta$. Moreover, we have
$$
| f_n(t,x_1,x_2,\dots) | \leq n, \quad\text{for } n=1,2\dots\,.
$$
Thus we conclude that the functions $f_n$
satisfy assumptions (iv) and (v) with $p_n=n$ for $n=1,2\dots$.

Finally, on the basis of Theorem \ref{thm6.1} we deduce that there exists at least
 one solution $x(t)=(x_n(t))$ of initial value problem
 \eqref{6.7}--\eqref{6.8} defined on some interval $I=[0, T_1]$ such that
for each $t \in I$ the sequence $(x_n(t))$ belongs to the space $c_0^\beta$
with $\beta =(\frac{1}{n^2})$. This means that $x_n(t)=o(n^2)$ as $n \to \infty$,
for any fixed $t \in [0,T_1]$.

In the sequel we will also consider the semilinear lower diagonal infinite
system of differential equations of the form \eqref{6.1} i.e.,
\begin{equation}\label{6.9}
x_n'=\sum_{i=1}^{k_n}a_{nn_i}(t)x_{n_i}+f_n(t,x_1,x_2,\dots)
\end{equation}
with initial value conditions
\begin{equation}\label{6.10}
x_n(0)=x_n^0, \quad \text{for } n=1,2,\dots\,.
\end{equation}

Now, we dispense with the assumption requiring that system \eqref{6.9}
has linear parts of constant width. We replace this assumption, as well
as assumption (ii), by the following hypotheses:
\begin{itemize}
 \item [(ii')] The sequence $(n_1)$ tends to $\infty$ as $n\to \infty $;

 \item[(ii'')] the sequence $ \Big(  \sum_{i=1}^{k_n}| a_{nn_i}(t)| \Big)$
is uniformly bounded on the interval $I=[0,T_1]$ i.e., there exists a constant
$A>0$ such that
 $$
\sum_{i=1}^{k_n}| a_{nn_i}(t)| \leq A
$$
 for each $t\in I$ and for $n=1,2,\dots$.
\end{itemize}
Then we have the following result.

\begin{theorem} \label{thm6.3}
 Assume that  {\rm (i), (ii'), (ii''), (iii)--(v) }
of Theorem \ref{thm6.1}  are satisfied. Then initial value problem
\eqref{6.9}--\eqref{6.10} has at least one solution $x(t)=(x_n(t))$
in the sequence space $c_0^\beta$ defined on the interval $I=[0,T_1]$,
 where $T_1$ is a number chosen according to Theorem \ref{thm5.2}.
\end{theorem}

\begin{proof}
 Similarly, as in the proof of Theorem \ref{thm6.1}, for a fixed $n\in \mathbb{N}$
let us denote
\begin{gather*}
g_n(t,x)=\sum_{i=1}^{k_n}a_{nn_i}(t)x_{n_i}+f_n(t,x),\\
(L_nx)(t)=\sum_{i=1}^{k_n}a_{nn_i}(t)x_{n_i},
\end{gather*}
 where $t\in I$ and $x=(x_n)=(x_1,x_2,\dots)\in c_0^\beta$.
 Next, let us put
\begin{gather*}
g(t,x)=(g_1(t,x),g_2(t,x),\dots),\\
(Lx)(t)=((L_1x)(t), (L_2x)(t),\dots), \\
f(t,x)=(f_1(t,x),f_2(t,x),\dots).
\end{gather*}
 Now, in view of our assumptions, we obtain:
\begin{equation} \label{6.11}
\begin{aligned}
\beta_n | g_n(t,x_1,x_2,\dots)|
&\leq \beta _n \sum_{i=1}^{k_n} | a_{nn_i}(t)| | x_{n_i} |
 + \beta _n | f_n(t,x_1, x_2, \dots) | \\
&\leq \sum_{i=1}^{k_n} | a_{nn_i}(t)| \beta _{n_i} | x_{n_i} | + \beta _n p_n \\
&\leq \sum_{i=1}^{k_n} | a_{nn_i}| \max \Big \{ \beta _{n_i} | x_{n_i} |
 : i=1,2,\dots,k_n \Big \} + \beta _n p_n \\
&\leq A \sup \{ \beta _j | x_j |: j \geq n_1 \} + \beta _n p_n.
 \end{aligned}
\end{equation}
 Further, from the above estimate we obtain
 \begin{equation}\label{6.12}
 \| g(t,x) \| = \sup \{\beta _n | g_n(t,x_1,x_2,\dots)| \} \leq A \| x\| +P,
 \end{equation}
where $P=\sup \{\beta _n p_n: n=1,2,\dots\}$. Obviously $P < \infty$ since
 $\beta _n p_n \to 0$ as $n \to \infty$.
 Next, in virtue of estimate \eqref{6.12} we have that the operator $g$
transforms the set $I \times c_0^\beta$ into $c_0^\beta$.

 In what follows observe that because of  a suitable part of the proof
of Theorem \ref{thm6.1},
we conclude that $f$ is continuous on the set $I \times c_0^\beta$.
Thus, to show the continuity of the operator $g$ on the set $I \times c_0^\beta$
it is sufficient to show that the operator $L$ is continuous on this set.
To this end fix arbitrarily $x\in c_0^\beta, t\in I$ and a number $\varepsilon>0$.
Then, for $y\in c_0^\beta$ with $\| x-y \| \leq \varepsilon$ and for a fixed
$n \in \mathbb{N}$, we obtain:
\begin{align*}
\beta_n | (L_nx)(t)-(L_ny)(t)|
&=\beta_n | \sum_{i=1}^{k_n}a_{nn_i}(t)x_{n_i}
 - \sum_{i=1}^{k_n}a_{nn_i}(t)y_{n_i} | \\
&\leq \beta _n \sum_{i=1}^{k_n}| a_{nn_i}(t)| | x_{n_i}-y_{n_i}|
  \leq \sum_{i=1}^{k_n}| a_{nn_i}(t)| \beta _{n_i} | x_{n_i}-y_{n_i}| \\
&\leq  \sum_{i=1}^{k_n}| a_{nn_i}(t)| \sup \{ \beta _{n_i} | x_{n_i}-y_{n_i}|
 : i=1,2,\dots, k_n\} \\
&\leq A \sup \{ \beta _j | x_{j}-y_{j}|: j=1,2,\dots\}=A \| x-y \|
 \leq A\varepsilon.
\end{align*}
 Hence we obtain that $\| (Lx)(t)-(Ly)(t)\| \leq A\varepsilon$ which means
that the operator $L$ is continuous on the set $I \times c_0^\beta$.
Consequently we obtain the continuity of the operator $g$ on $I \times c_0^\beta$.

 Now, let us choose a number $T_1$, $T_1<T$ such that $AT_1<1$. Next, take the
number $r=(P+A)T_1 \| x_o \| / (1-AT_1)$ and assume that $X$ is a nonempty subset
 of the ball $B(x_0,r)$. Then, arguing similarly as in the proof of
Theorem \ref{thm6.1}
and utilizing estimate \eqref{6.11} we obtain
 $$
\chi (g(t,X)) \leq A \chi (X),
$$
 where $\chi$ is the Hausdorff measure of noncompactness in the space
$c_0^\beta$  described by formula \eqref{4.1}.
 Hence, applying Theorem \ref{thm5.2} we complete the proof.
\end{proof}

Next we provide an example showing the applicability of Theorem \ref{thm6.3}.

\begin{example} \label{examp6.4} \rm
 We consider the lower diagonal infinite system of differential equations.
To expose this system in a transparent way we will assume that $n$ is an
even natural number, say $n=2k$. Then, we can present the announced system as follows:
 \begin{gather}
 x_1'=x_1+ \frac{x_1}{1+ x_1^2}, \nonumber \\
 x_2'=x_1+tx_2+2 \frac{x_1+x_2}{1+x_1^2+x_2^2}, \nonumber\\
 x_3'=\frac{t^2}{2!}x_3+3\frac{x_2+x_3}{1+x_2^2+x_3^2}, \nonumber\\
 x_4'=\frac{t^2}{2!}x_3+\frac{t^3}{3!}x_4+4\frac{x_3+x_4}{1+x_3^2+x_4^2}, \nonumber\\
 \dots    \label{6.13}\\
 x_{n-1}'(=x_{2k-1}')
=\frac{t^k}{k!} x_{k+1}+ \dots +\frac{t^{2k-2}}{(2k-2)!} x_{2k-1}
  +(2k-1)\frac{x_{2k-2}+x_{2k-1}}{1+x_{2k-2}^2+x_{2k-1}^2}, \nonumber\\
 x_n'(=x_{2k}')=\frac{t^k}{k!} x_{k+1}+ \dots +\frac{t^{2k-2}}{(2k-2)!} x_{2k-1}
 +\frac{t^{2k-1}}{(2k-1)!}x_{2k}
  +2k\frac{x_{2k-1}+x_{2k}}{1+x_{2k-1}^2+x_{2k}^2}, \nonumber\\
 \dots \,. \nonumber
 \end{gather}
\end{example}
We also assume that the following initial conditions are satisfied
\begin{equation}\label{6.14}
x_n(0)=n^2
\end{equation}
for $n=1,2,\dots$\hspace{5pt}.

Let us observe that initial value problem \eqref{6.13}--\eqref{6.14}
is a particular case of problem \eqref{6.9}--\eqref{6.10}.
To justify this assertion we show that the components involved in
\eqref{6.13}--\eqref{6.14} satisfy assumptions of Theorem \ref{thm6.3}.
First of all let us observe that functions $a_{nn_i}(t)$ appearing in
infinite system \eqref{6.13} have the form
$$
a_{nn_i}(t)=\frac{t^{n_i-1}}{(n_i-1)!}
$$
for $n_i=\frac{n}{2}+1, \frac{n}{2}+2, \dots,n $ (if $n$ is even) or
$n_i=[\frac{n}{2}]+2, [\frac{n}{2}]+3,\dots, n $ (if $n$ is odd).
 Obviously the functions $a_{nn_i}(t)$ are continuous on each interval
of the form $[0, T]$. Thus, there is satisfied assumption (i).

Since $n_1=\frac{n}{2}+1$ for $n$ even or $n_1=[\frac{n}{2}]+2$ for $n$ odd,
we see that assumption (ii') is satisfied. To check assumption (ii'')
observe that we have
$$
\sum_{i=1}^{k_n} | a_{nn_i}(t) | =\sum_{i=1}^n | a_{nn_i}(t) |
\leq 1+t+ \frac{t^2}{2!}+\dots+\frac{t^n}{n!}\leq e^t
$$
for $t \in [0,T]$. Hence we have that assumption (ii'') is satisfied with $A = e^T$.

Further, take the tempering sequence of the form $\beta=(\beta_n)=(\frac{1}{n^3})$.
Then the sequence $(x_0^n)=(n^2)$ is a member of the tempered sequence space
$c_0^\beta$ , so assumption (iii) is satisfied.
Similarly, it is not hard to verify that the functions $f_n$, where
$$
f_n(t,x)=f_n(t,x_1,x_2,\dots)=n\frac{x_{n-1}+x_n}{1+x_{n-1}^2 +x_n^2}
$$
$(n=2, 3 \dots)$ are continuous on the set $I\times c_0^\beta$.
 Moreover, for each fixed $n$ we obtain
$$
| f_n(t,x)| \leq n \frac{| x_{n-1} | +| x_n |}{1+x_{n-1}^2 +x_n^2} \leq n.
$$
Thus, we can put $p_n=n$ in assumption (v). Obviously we have that
$\beta_n p_n =\frac{1}{n^2}\to 0$ as $n \to \infty$. Thus assumption (v)
is satisfied.

Hence, in view of Theorem \ref{thm6.3}, initial value problem \eqref{6.13}--\eqref{6.14}
has at least one solution $x(t)=(x_n(t))$ belonging to the sequence space
$c_0^\beta$ and defined for $t \in I=[0,T_1]$, where $T_1$ satisfies the
inequality $T_1A=T_1e^{T_1}<1$. We can calculate that $T_1\leq 0.568\dots$.

\begin{remark} \label{rmk6.5} \rm
 Observe that in Example \ref{examp6.4} instead of 
$\beta=(\beta_n)=(1/n^3)$
we can take the tempering sequence of the form 
$\beta_n = 1/n^{2+\delta}$,
where $\delta$ is an arbitrary positive number.
Similarly, in Example \ref{examp6.2} we can take the tempering sequence of
the form $\beta_n = 1/n^{1+\delta}$,
where $\delta >0$ is an arbitrary number and $n=1,2\dots$.
\end{remark}

\section{Infinite perturbed diagonal systems}

In this section we study the existence of solutions of a perturbed diagonal
infinite system of differential equations in the sequence space $c^\beta$.
Consider the infinite perturbed diagonal systems
of differential equations of the form
\begin{equation}\label{7.1}
x_n'=a_n(t)x_n+g_n(t,x_1,x_2,\dots)
\end{equation}
with the initial conditions
\begin{equation}\label{7.2}
x_n(0)=x_n^0,
\end{equation}
for $n=1,2,\dots$ and $t\in I=[0,T]$.
Problem \eqref{7.1}--\eqref{7.2} will be investigated in the sequence space
$c^\beta$, where $\beta=(\beta_n)$ is a tempering sequence i.e., the sequence
$(\beta_n)$ is nonincreasing and has positive terms.

Infinite systems of differential equations \eqref{7.1}--\eqref{7.2} contain,
as particular cases, the systems considered in the theory of neural sets
(cf. \cite[pp. 86-87]{Deimling},  and \cite{Oguz}).
Let us also mention that system \eqref{7.1}--\eqref{7.2} was studied in
\cite{Banas Lecko}. The existence result concerning initial value
problem \eqref{7.1}--\eqref{7.2} which we are going to present here,
will generalize essentially results obtained in the above quoted papers
\cite{Banas Lecko,Oguz} and the monograph \cite{Deimling}.
In our considerations we will utilize the measure of noncompactness
$\mu_2^\beta$ in the space $c^\beta$ defined by formula \eqref{4.2}.

Initial value problem \eqref{7.1}--\eqref{7.2} will be studied under the
following assumptions.
\begin{itemize}
 \item [(i)] $x_0=(x_n^0) \in c^\beta $;
 \item[(ii)] the mapping $g=(g_1, g_2, \dots)$ acts from the set
$I \times c^\beta $ into $c^\beta$ and is continuous on $I \times c^\beta$;
 \item[(iii)] There exists a sequence $(p_n)$ with $\beta _n p_n \to 0$ as
$n \to \infty$ such that
 $$
| g_n(t, x_1, x_2, \dots) | \leq p_n
$$
 for $t \in I$, $x=(x_n) \in c^\beta$ and for $n=1,2,\dots$.

 \item[(iv)] The functions $a_n(t)$ are continuous on $I$ and the sequence
$(a_n(t))$ converges uniformly on $I$ (to a function $a=a(t)$).
\end{itemize}

Notice that in view of the imposed assumptions the sequence $(a_n(t))$ is
equibounded on $I$. This implies that the constant
$$
A= \sup \{a_n(t): t\in I, n=1,2,\dots\}
$$
is finite.

Now, we can formulate our result.

\begin{theorem} \label{thm7.1}
 Let assumptions (i)--(iv) be satisfied. If $AT<1$ then initial value
problem \eqref{7.1}--\eqref{7.2} has a solution $x(t)=(x_n(t))$ on the
interval $I$ such that $x(t) \in c^\beta$ for each $t \in I$.
\end{theorem}

\begin{proof}
 At the beginning, for $t\in I$ and $x=(x_n) \in c^\beta$ let us denote
\begin{gather*}
f_n(t,x)=a_n(t)x_n+g_n(t,x),
f(t,x)=(f_1(t,x), f_2(t,x),\dots),
\end{gather*}
 where $n$ is an arbitrarily fixed natural number. Further, fix arbitrary
natural numbers $m, n$. Without loss of generality we can assume that $m<n$.
Then, we obtain
\begin{equation} \label{7.3}
\begin{aligned}
&| \beta _n f_n(t,x)-\beta_m f_m (t,x)| \\
&\leq | \beta_n a_n(t) x_n -\beta_m a_m (t) x_m |
 +| \beta _n g_n (t,x) - \beta _m g_m (t,x)| \\
&\leq | \beta_n a_n(t) x_n -\beta_m a_n (t) x_m |
 + | \beta_m a_n(t) x_m -\beta_m a_m (t) x_m | \\
&\quad + \beta_n |   g_n (t,x)|   + \beta _m |  g_m (t,x)| \\
&\leq | a_n(t)|  | \beta_n x_n - \beta_m x_m| + \beta_m | x_m | | a_n(t) - a_m(t) |
+  \beta_n p_n+ \beta_m p_m.
\end{aligned}
\end{equation}
 In view of the imposed assumptions we deduce that $(\beta_k x_k)$ is a Cauchy
sequence. The same statement is also valid for the function sequence $(a_k(t))$.

 Moreover, we have that $\beta_n p_n \to 0 $ as $n \to \infty$.
Taking into account the above established facts, from estimate \eqref{7.3}
we deduce that $(\beta_n f_n (t, x))$ is a Cauchy sequence.
This yields that $ (f_n (t, x)) \subset c^\beta $.

 Next, observe that for arbitrary $n \in \mathbb{N}, t\in I$ and for a fixed
$x \in c^\beta$, we have
 \begin{equation} \label{7.4}
| \beta_n f_n(t,x) |
 \leq | \beta _n a_n (t)x_n | +| \beta _n g_n (t,x) |
\leq | a_n(t)| \beta_n | x_n| + \beta _n p_n \leq A \| x \| +P,
 \end{equation}
 where $P=\sup\{\beta _n p_n: n=1,2\dots\}$ and the symbol $ \| \cdot \| $
denotes the norm in the space $c^\beta$ (cf. Section 4). Obviously $P< \infty$.
From estimate \eqref{7.4} we deduce the following one
 \begin{equation}\label{7.5}
 \| f(t,x) \| \leq A \| x \| +P.
 \end{equation}

 Now, we consider the mapping $f(t, x)$ on the set $I \times B(x_0,r)$, where $r$
is taken according to Theorem \ref{thm5.2}, i.e.,
 $$
r=\frac{(A+P)T\| x_0 \|}{1-AT}.
$$
 To prove the continuity of the mapping $f(t,x)$ let us fix arbitrarily
$t \in I$ and $x\in B(x_0,r)$. Next, choose arbitrary $s \in I$ and
$y \in B(x_0,r)$. Then, in view of the imposed assumptions, we obtain
\begin{align*}
&\| f(t,x) - f(s,y) \| \\
&= \sup \Big \{ | \beta_n f_n (t, x) - \beta _n f_n (s,y) | : n=1,2,\dots \Big \} \\
&\leq \sup \Big \{ \beta_n | a_n (t) x_n - a_n (s) y _n | : n=1,2,\dots \Big \}\\
&\quad + \sup \Big \{ \beta_n | g_n (t, x) - g_n (s, y) | : n=1,2,\dots \Big \} \\
&\leq \sup \Big \{ \beta_n\Big[ | a_n (t) x_n - a_n (s) x_n |
 +| a_n (s) x_n - a_n (s) y_n | \Big] : n=1,2,\dots \Big \} \\
&\quad + \sup \Big \{ \beta_n | g_n (t, x) - g_n (s, y) | : n=1,2,\dots \Big \} \\
&\leq \sup \Big \{ \beta_n | x_n | | a_n (t) - a_n (s) | : n=1,2,\dots \Big \}\\
&\quad + \sup \Big \{|  a_n (s)| \beta_n | x_n-y_n | : n=1,2,\dots \Big \} \\
&\quad  + \sup \Big \{ \beta_n | g_n (t, x) - g_n (s, y) | : n=1,2,\dots \Big \} \\
&\leq (\| x_0 \| +r) \sup \Big \{ | a_n (t) - a_n (s) | : n=1,2,\dots \Big \} \\
&\quad +  A \| x - y \| + \| g(t, x) - g(s, y) \|.
\end{align*}
 Hence, keeping in mind the fact that the sequence $(a_n(t))$ is equicontinuous
on the interval $I$ and the mapping $g$ is continuous at the point $(t, x)$
we conclude that the mapping $f$ is continuous at $(t, x)$.
In view of the arbitrariness of $t$ and $x$ this yields that $f$ is continuous
on the set $I\times B(x_0,r)$.

 Now, let us take a nonempty subset $X$ of the ball $B(x_0,r)$.
Fix $t\in I$ and $x=(x_n) \in X$. Then, in view of \eqref{7.3}, for
arbitrarily fixed natural numbers $m,n$ we obtain
\begin{align*}
&| \beta_n f_n (t, x) - \beta _m f_m (t,x) | \\
&\leq | a_n (t)| | \beta_n x_n -\beta_m x_m |
 +\| x \| | a_n(t)- a_m(t) | +\beta _n p_n +\beta_m p_m.
\end{align*}
 Hence, taking into account the imposed assumptions, we derive the estimate
 \begin{equation}\label{7.6}
 \mu^\beta _2 (f(t,X)) \leq a(t)\mu_2^\beta (X),
 \end{equation}
 where (as we mentioned above) $\mu_2^\beta$ is the measure of noncompactness
 in the space $c^\beta$ defined by formula \eqref{4.2}. Finally,
linking estimates \eqref{7.5} and \eqref{7.6}, in view of Theorem \ref{thm5.2} we
conclude that problem \eqref{7.1}--\eqref{7.2} has at least one solution in
the space $c^\beta$. The proof is complete.
\end{proof}

Now we given an example illustrating our considerations.

\begin{example} \label{examp7.2} \rm
 Consider the  perturbed diagonal infinite system of differential equations
 \begin{equation} \label{7.7}
 x_n'=\big (n\sin\frac{t}{n}\big)x_n+\arctan(x_n+x_{n+1})
 \end{equation}
 with the initial conditions of the form
 \begin{equation}\label{7.8}
 x_n(0)=n+1
 \end{equation}
 for $n=1,2,\dots$ and for $t \in I=[0,T]$, where $T$ is a fixed positive
number such that $T\leq \frac{\pi}{2}$. The value of $T$ will be estimated
precisely later.

 Observe that initial value problem  \eqref{7.7}--\eqref{7.8}
is a special case of problem \eqref{7.1} - \eqref{7.2} if we put
$a_n(t)=n\sin\frac{t}{n}$, $g_n(t,x_1,x_2,\dots)=\arctan(x_n+x_{n+1})$
and if we accept the tempering sequence $\beta=(\beta _n)=(\frac{1}{n})$.
We show briefly that in such a case infinite system \eqref{7.7} with initial
conditions \eqref{7.8} satisfies assumptions of Theorem \ref{thm7.1}.
To this end observe that the function sequence $(a_n(t))$ consists
of functions continuous on the interval $I$ and it is uniformly convergent
on $I$ to the function $a(t)=t,\hspace{5pt} t\in I$. Thus the sequence
$(a_n(t))$ satisfies assumption (iv).

 Further, we have
 $$
| g_n(t,x_1,x_2,\dots)| =| \arctan(x_n+x_{n+1})| \leq \frac{\pi}{2}
$$
 for $n=1,2\dots$. Thus, taking $p_n=\frac{\pi}{2}$ we see that assumption
(iii) is satisfied. Similarly we verify assumption (i).

 To check assumption (ii) let us fix arbitrarily $x,y \in c^\beta$, $x=(x_k)$,
$y=(y_k)$. Then, for a fixed $n \in \mathbb{N}$ we obtain:
\begin{equation}\label{7.9}
\begin{aligned}
&\beta _n  | g_n(t,x_1,x_2,\dots) - g_n(t,y_1,y_2,\dots)| \\
&=\frac{1}{n}| \arctan(x_n+x_{n+1}) - \arctan(y_n+y_{n+1})| \\
&\leq \frac{1}{n}| x_n+x_{n+1} - y_n - y_{n+1}|
 \leq \frac{1}{n}| x_n- y_n| + \frac{1}{n}| x_{n+1} - y_{n+1}| \\
&\leq \frac{n+1}{n} \Big( \frac{1}{n+1}| x_n- y_n|
 + \frac{1}{n+1}| x_{n+1} - y_{n+1}|\Big) \\
&\leq 2\Big(\frac{1}{n}| x_n- y_n| + \frac{1}{n+1}| x_{n+1} - y_{n+1}|\Big).
\end{aligned}
 \end{equation}
 Next, in view of \eqref{7.9}, for arbitrarily fixed $t,s \in I$ and
$x,y \in c^\beta$, we obtain
\begin{align*}
&\| g(t,x)-g(s,y)\|\\
& = \sup \Big\{ \frac{1}{n} | g_n(t,x) - g_n(s,y)| :n=1,2,\dots\Big\} \\
&\leq \sup \Big\{2\big(\frac{1}{n}| x_n-y_n|+\frac{1}{n+1} | x_{n+1}-y_{n+1}| \big )
 :n=1,2\dots\Big\} \\
&\leq 2\sup \Big\{\frac{1}{n}| x_n-y_n| :n=1,2\dots\Big\}
 +2\sup \Big\{\frac{1}{n+1} | x_{n+1}-y_{n+1}| :n=1,2\dots\Big\} \\
&\leq 4 \| x-y \|,
\end{align*}
 where the symbol $ \| \cdot \| $ denotes the norm in the space $c^\beta$.
Thus we showed that the mapping $g$ is continuous on the set $I\times c^\beta$
(even Lipschitz continuous). This means that the mapping $g$ satisfies assumption
(ii).

 Finally, let us observe that using standard methods of mathematical analysis,
we obtain
\begin{align*}
A&=\sup \big\{ | a_n(t)|: t\in [0,T], n=1,2,\dots\big \} \\
 &=\sup\big\{n\sin\frac{t}{n}: t\in [0,T], n=1,2,\dots\big\}
\leq \frac{\pi}{2}.
\end{align*}
 Thus, if we take $T<\frac{2}{\pi}$, then applying Theorem \ref{thm7.1}
we deduce that initial value problem \eqref{7.7}--\eqref{7.8}
has at least one solution $x(t)=(x_n(t))$ such that $(x_n(t)) \in c^\beta$
for any $t\in[0,T]$.
\end{example}

\section{Infinite systems of differential equations in the sequence space
$l^\beta_\infty$}


In this section we will work in the space $l^\beta_\infty$ described in
details in Section 4. We will assume here that the tempering sequence
$\beta=(\beta_n)$ consists of positive terms and is nonincreasing.
We will utilize the measure of noncompactness $\mu^\beta_3$ defined on the
family $\mathfrak{M}_{l^\beta_\infty}$ by formula \eqref{4.5}.
For simplicity, that measure will be denoted by $\mu$. Recall, that
for $X\in \mathfrak{M}_{l^\beta_\infty}$ we put
$$
\mu(X)=\limsup_{n\to \infty }\operatorname{diam} X_n^\beta,
$$
where $X_n^\beta=\{\beta_n x_n: x=(x_i) \in X\}$. Equivalently,
this formula can be written in a more convenient way
\begin{equation}\label{8.1}
\mu(X)=\limsup_{n\to \infty }\operatorname{diam} \beta_n X_n,
\end{equation}
where $X_n=\{x_n: x=(x_i) \in X\}$. We refer to Section 4 for the
properties of the measure $\mu$.

In what follows we will investigate the following perturbed semilinear
lower diagonal infinite system of differential equations
\begin{equation}\label{8.2}
x_n'=\sum_{j=k_n}^na_{nj}(t)x_j+g_n(t,x_1,x_2,\dots)
\end{equation}
with the initial conditions
\begin{equation}\label{8.3}
x_n(0)=x_n^0
\end{equation}
for $n=1,2\dots$ and $t\in I=[0,T]$.\\
Throughout this section we will assume that the sequence $(k_n)$ appearing
in \eqref{8.2} is such that $1 \leq k_n \leq n$ for $n=1,2,\dots$
and $k_n \to \infty$ as $n \to \infty$.

It is worthwhile mentioning that infinite systems of differential equations
having form \eqref{8.2} were up to now considered very seldom
 (cf. \cite{Banas Krajewska,Banas Mursaleen}).

For further purposes we denote by $f=f(t,x)$ the mapping defined on the set
$I \times l^\beta_\infty$ in the following way
$$
f(t,x)=(f_1(t,x),f_2(t,x),\dots),
$$
where
$$
f_n(t,x)=f_n(t,x_1,x_2,\dots)=\sum_{j=k_n}^na_{nj}(t)x_j+g_n(t,x_1,x_2,\dots)
$$
for $n=1,2,\dots$\hspace{5pt}. Moreover, we will also define the mapping $g(t,x)$
 on the set $I \times l^\beta_\infty$ by putting
$$
g(t,x)=(g_1(t,x), g_2(t,x),\dots).
$$

Now, we formulate assumptions under which problem \eqref{8.2}--\eqref{8.3}
will be studied. Namely, we will impose the following hypotheses.
\begin{itemize}
 \item [(i)] $x_0=(x_n^0) \in l^\beta_\infty$;

 \item[(ii)] the mapping $g$ acts from the set $I \times l^\beta_\infty$
into $l^\beta_\infty$ and is uniformly continuous on $I \times l^\beta_\infty$;

 \item[(iii)] there exists a sequence $(p_n)$ with $\beta_np_n \to 0$ as
$n \to \infty$ and such that
 $$ | g_n(t,x_1, x_2, \dots)| \leq p_n
$$
 for $t \in I, x=(x_n) \in l_\infty^\beta$ and $n=1,2\dots$;

 \item[(iv)] The functions $a_{nj} : I \to \mathbb{R}$
$(j=k_n, k_n+1,\dots, n$, $n=1,2\dots)$ are continuous and nondecreasing on $I$.
 Moreover, we assume that the function sequence $(A_n(t))$ is equicontinuous
on the interval $I$ and the sequence $(\bar{A_n}(t))$ is uniformly bounded on $I$,
 where
 $$
A_n(t)= \sum_{j=k_n}^na_{nj}(t), \quad
 \bar{A_n}(t)= \sum_{j=k_n}^n | a_{nj}(t) |
$$
 for $n=1,2,\dots$.
\end{itemize}

Keeping in mind assumption (iv), for further purposes we can define the constant
$$
A=\sup \{ \bar{A_n}(t) : t \in I, n=1,2,\dots\}.
$$
In view of assumptions (iv) we have that $A<\infty$.

Now, we can formulate that following result concerning initial value
problem \eqref{8.2}--\eqref{8.3}.

\begin{theorem} \label{thm8.1}
 Assume that conditions {\rm(i)--(iv)} are satisfied and $AT<1$.
Then initial value problem \eqref{8.2}--\eqref{8.3} has at least one
solution $x(t)=(x_k(t))$ on the interval $I=[0, T]$ such that
$x(t) \in l_\infty ^\beta $ for $t \in I$.
\end{theorem}

\begin{proof}
 Let us take an arbitrary element $x=(x_k)\in l_\infty ^\beta$.
Next, fix $t\in I$ and $n \in \mathbb{N}$. Then, in view of the imposed
assumptions we obtain
 \begin{equation} \label{8.4}
\begin{aligned}
| \beta_n f_n(t,x)|
&\leq \beta _n \sum_{j=k_n}^n | a_{nj}(t) | | x_j| +\beta _n | g_n (t,x) |\\
&\leq \sum_{j=k_n}^n | a_{nj}(t) | \beta _j | x_j| +\beta_n p_n \\
&\leq \sum_{j=k_n}^n | a_{nj}(t) | \max \{ \beta _j| x_j|: j=k_n, k_n+1, \dots, n\}
+  \beta_n p_n \\
&\leq \bar{A_n}(t)\sup \{ \beta _j | x_j | : j=1,2,\dots\}
+ \beta_n p_n \leq A \| x\| +P,
\end{aligned}
 \end{equation}
where  $\| \cdot \|$ stands for the norm in the space $l^\beta_\infty$ and
$P=\sup\{\beta _n p_n: n=1,2\dots\}. $ Obviously $P<\infty$ in view of
assumption (iii). Hence we infer that the mapping $f$ transforms the set
$I \times l^\beta_\infty$ into $l^\beta_\infty$.

Next we consider the mapping $f(t,x)$ on the set $I \times B(x_0,r)$,
where $r$ is taken according to Theorem \ref{thm5.2} i.e., $r=\frac{(A+P)T\| x_0 \| }{1-AT}$.
First, we show that $f$ is uniformly continuous on the set $I \times B(x_0,r)$.
In view of assumption (ii) it is sufficient to show that the linear operator
$L$ defined by
 $$
(Lx)(t)=((L_1x)(t), (L_2x)(t),\dots),
$$
 where
 $$
(L_nx)(t)=\sum_{j=k_n}^na_{nj}(t)x_j
$$
 for $n=1,2,\dots$, is continuous on the set $I \times l^\beta_\infty$.
To this end fix arbitrarily $x, y \in l^\beta_\infty$, $t, s \in I$ and
$n \in \mathbb{N}$. Without loss of generality we may assume that $s<t$.
Then, keeping in mind our assumptions, we obtain
\begin{align*}
&\beta_n | (L_nx)(t)-(L_ny)(s)| \\
&= \beta _n \Big | \sum_{j=k_n}^n a_{nj}(t) x_j
 - \sum_{j=k_n}^n a_{nj}(s) y_j \Big | \\
&\leq \beta _n \Big| \sum_{j=k_n}^na_{nj}(t) x_j
  - \sum_{j=k_n}^n a_{nj}(t) y_j\Big |
  + \beta _n \Big| \sum_{j=k_n}^n a_{nj}(t)y_j
  - \sum_{j=k_n}^n a_{nj} (s) y_j\Big |\\
&\leq \beta _n \sum_{j=k_n}^n | a_{nj}(t) | | x_j - y_j |
 + \beta _n \sum_{j=k_n}^n | a_{nj}(t) - a_{nj} (s) | | y_j |\\
&\leq \sum_{j=k_n}^n | a_{nj} (t)| \beta _j | x_j - y_j |
 + \sum_{j=k_n}^n \big( a_{nj}(t) - a_{nj} (s) \big) \beta _j  | y_j | \\
&\leq \bar{A_n}(t) \sup \big\{ \beta _j | x_j - y_j |: j=1,2,\dots \big\} \\
&\quad + \sum_{j=k_n}^n \big( a_{nj}(t) - a_{nj} (s) \big)
 \sup \{ \beta _j  | y_j |: j=1,2,\dots\} \\
&\leq A \| x-y \| + \Big( \sum_{j=k_n}^n  a_{nj}(t)
 - \sum_{j=k_n}^n  a_{nj}(s)\Big) \| y \| \\
&\leq A \| x-y \| + \big(A_n(t) - A_n(s)\big) \| y \|.
\end{align*}
Hence, we derive the  estimate
$$
\| (L x)(t)-(L y)(s)\|
\leq A\| x-y \| + \sup \big\{A_n(t) - A_n(s) : n=1,2\dots\big\} \| y \|.
$$
From this estimate and assumption (iv) we conclude that the operator
 $L$ is continuous on the set $I \times l^\beta_\infty$.
Obviously $L$ is uniformly continuous as linear.

Next, let us take a nonempty subset $X$ of the ball $B(x_0,r)$.
Fix arbitrarily $x,y \in X$ and $t\in I$. Then, with help of a similar
reasoning as above, for a fixed $n$ natural, we obtain
\begin{align*}
&\beta _n | f_n(t,x) -f_n(t,y)| \\
&\leq \beta _n \Big| \sum_{j=k_n}^na_{nj}(t) x_j
 - \sum_{j=k_n}^n a_{nj}(t) y_j\Big | + \beta _n | g_n(t,x)-g_n(t,y)| \\
&\leq \beta _n \Big| \sum_{j=k_n}^na_{nj}(t) ( x_j - y_j)\Big |
 + \beta _n | g_n(t,x)| + \beta _n | g_n(t,y)| \\
&\leq \beta _n  \sum_{j=k_n}^n| a_{nj}(t)| | x_j - y_j)| + 2 \beta _n p_n \\
&\leq  \sum_{j=k_n}^n| a_{nj}(t)| \beta _j  | x_j - y_j)| + 2 \beta _n p_n \\
&\leq \bar{A_n}(t) \sup \big\{ \beta _j | x_j - y_j |: j=k_n, k_n+1,\dots \big\}
  +2 \beta _n p_n  \\
&\leq \bar{A_n}(t) \sup \big\{ \beta _j \operatorname{diam} X_j: j=k_n, k_n+1,
\dots \big\} +2 \beta _n p_n  .
\end{align*}
Hence, we derive the inequality
$$
\operatorname{diam} \beta _n f_n(t,X)
\leq \bar{A_n}(t) \sup \big\{ \beta _j \operatorname{diam} X_j:
 j\geq k_n \big\} +2 \beta _n p_n  .
$$
From the above estimate, in view of assumptions (iii) and (iv) we have
\begin{equation}\label{8.5}
\mu(f(t,X)) \leq A\mu (X),
\end{equation}
where $\mu$ is the measure of noncompactness defined by \eqref{8.1}.
Finally, combining estimates \eqref{8.4} and \eqref{8.5}, on the basis
of Theorem \ref{thm5.2}, we complete the proof.
\end{proof}

\begin{remark} \label{rmk8.2} \rm
 Observe that instead of the requirement imposed in assumption (iv) that the
functions $a_{nj}$ ($j=k_n, k_n+1,\dots,n$; $n=1,2,\dots$) are nondecreasing
on $I$, we can assume that those functions are nonincreasing on $I$.
\end{remark}

The next example shows the applicability of the result  in Theorem \ref{thm8.1}.

\begin{example} \label{examp8.3} \rm
 Consider the semilinear lower diagonal perturbed infinite system of
differential equations
 \begin{equation}\label{8.6}
 x_n'=\sum_{j=k_n}^nt^{n+j}x_j +\sin (x_n+x_{n+1}+x_{n+2})
 \end{equation}
 with the initial conditions
 \begin{equation}\label{8.7}
 x_n(0)=n,
 \end{equation}
 for $n=1,2\dots$ and for $t\in I=[0,T],$ where $T<1$.
 Moreover, we assume that $(k_n)$ is a nondecreasing sequence of natural
numbers such that $1 \leq k_n \leq n$ and $k_n \to \infty$ as $n\to \infty$.

 Observe that  \eqref{8.6}--\eqref{8.7} is a special case of
\eqref{8.2}--\eqref{8.3}, where $a_{nj}(t)=t^{n+j}$ for $j=k_n, k_n+1,\dots,n$
and for $ n=1,2,\dots$. Apart from this, the function $g_n$ has the form
 $$
g_n(t,x_1,x_2,\dots)=\sin( x_n+x_{n+1}+x_{n+2})
$$
 for $ n=1,2,\dots$.
 It is easily seen that infinite system \eqref{8.6} with initial conditions
\eqref{8.7} satisfies assumptions of Theorem \ref{thm8.1} if we take the tempering
sequence $ (\beta _n) $ of the form $\beta _n =\frac{1}{n}$ for $ n=1,2,\dots$.
 Indeed, we have obviously that $(x_0^n)=(n) \in l^\beta _\infty$.
This means that assumption (i) is satisfied.

 Now, take an arbitrary element $x=(x_k) \in l^\beta _\infty$ and a number $t\in I$.
 Then, for a fixed natural number $n$ we obtain
\begin{align*}
\beta _n | g_n(t,x_1,x_2,\dots)|
&= \frac{1}{n} | \sin( x_n+x_{n+1}+x_{n+2})| \\
&\leq \frac{1}{n} \Big( | x_n| + | x_{n+1}| +| x_{n+2}| \Big) \\
&=\frac{n+2}{n} \Big( \frac{1}{n+2} | x_n| +\frac{1}{n+2} | x_{n+1}|
 + \frac{1}{n+2} | x_{n+2}| \Big) \\
&\leq 3\Big( \frac{1}{n} | x_n| +\frac{1}{n+1} | x_{n+1}|
 + \frac{1}{n+2} | x_{n+2}| \Big) \leq 3 \| x \|,
\end{align*}
where the symbol $\| \cdot \|$ denotes the norm in the space $l^\beta_\infty$.
Hence we obtain
 $$
\| g(t,x) \| \leq 3 \| x \|
$$
 which shows that $g$ acts from the set $I \times l^\beta_\infty$ into
$l^\beta_\infty$.

 Further, if we fix arbitrarily $n \in \mathbb{N}$,
$x=(x_1, x_2, \dots) \in l^\beta_\infty, y=(y_1, y_2, \dots) \in l^\beta_\infty$ and
$t,s \in I$, then we obtain
\begin{align*}
&\beta _n | g_n(t,x_1,x_2,\dots) - g_n(s,y_1,y_2,\dots)| \\
&=\frac{1}{n} | \sin( x_n+x_{n+1}+x_{n+2})- \sin( y_n+y_{n+1}+y_{n+2})| \\
&\leq \frac{1}{n} \Big( | x_n - y_n | + | x_{n+1}-y_{n+1}|
 + | x_{n+2}-y_{n+2}| \Big) \\
&=\frac{n+2}{n} \Big( \frac{1}{n+2} | x_n- y_n |
 +\frac{1}{n+2} | x_{n+1} - y_{n+1} | + \frac{1}{n+2} | x_{n+2} - y_{n+2} | \Big) \\
&\leq 3\Big( \frac{1}{n} | x_n - y_n | +\frac{1}{n+1} | x_{n+1} - y_{n+1} |
 + \frac{1}{n+2} | x_{n+2} - y_{n+2} | \Big) \\
&\leq  3 \| x-y \|.
\end{align*}
 Hence we derive the  estimate
 $$
\| g(t,x) - g(s,y) \| \leq  3 \| x-y \|
$$
 which shows that the mapping $g$ is uniformly continuous on the set
 $I \times l^\beta_\infty$.
 Thus the mapping $g$ satisfies assumption (ii) of Theorem \ref{thm8.1}.

 Now, we have
  $$
| g_n(t,x_1,x_2,\dots)| =  | \sin( x_n+x_{n+1}+x_{n+2})|\leq 1
$$
  which shows that there is satisfied assumption (iii) with $p_n =1$ for,
$n=1,2\dots$ \hspace{5pt}.
To show that there is satisfied assumption (iv) let us notice that we have
  $$
A_n (t)= \bar {A_n}(t)= t^{n+k_n} \frac{1-t^{n-k_n+1}}{1-t}
$$
  for $t\in I=[0,T]$ and for $n=1,2,\dots$ \hspace{5pt}.
Using the standard methods of analysis it is not hard to show that the sequence
$(A_n(t)$) is equicontinuous on the interval $I$. Moreover, we have the
 estimate
  $$
A_n (t)= \bar{A_n}(t)\leq A \leq \frac{1}{1-T}
$$
  for any $t \in I$.
Summing up we see that initial value problem \eqref{8.6}--\eqref{8.7}
satisfies the assumptions in Theorem \ref{thm8.1}. Therefore
the infinite system \eqref{8.6} with initial value conditions \eqref{8.7}
has at least one solution in the space $l^\beta_\infty$.
\end{example}

\begin{thebibliography}{9}

 \bibitem{Akhm} R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina,
B. N. Sadovskii;
\emph{Measures of Noncompactness and Condensing Operators}, Operator Theory:
 Advances and Applications 55, Birkh\"auser, Basel 1992.

 \bibitem{Ayer} J. M. Ayerbe Tolendo, T. Dominguez Benavides, G. Lopez Acedo;
\emph{Measures of Noncompactness in Metric Fixed Point Theory,} Birkh\"auser,
Basel 1997.

 \bibitem{BGoebel} J. Bana\'s, K. Goebel;
\emph{Measures of Noncompactness in Banach Spaces},
Lect. Notes in Pure and Appl. Math. 60, Marcel Dekker, New York, 1980.

 \bibitem{Banas Krajewska} J. Bana\'s, M. Krajewska;
\emph{On solutions of semilinear infinite systems of differential equations},
Dyn. Syst. Appl. 22 (2013) 301--316.

 \bibitem{Banas Lecko} J. Bana\'s, M. Lecko;
\emph{Solvability of infinite systems of differential equations in Banach
sequence spaces}, J. Comput. Appl. Math. 137 (2001) 363--375.

 \bibitem{Banas Martinon} J. Bana\'s, A. Martinon;
\emph{Some properties of the Hausdorff distance in metric spaces},
Bull. Austral. Math. Soc. 42 (1990) 511--516.

 \bibitem{Banas Martion2} J. Bana\'s, A. Martinon;
\emph{Measures of noncompactness in Banach sequence spaces},
Math. Slovaca 42 (1992) 497--503.

 \bibitem{Banas Mursaleen} J. Bana\'s, M. Mursaleen;
\emph{Sequence Spaces and Measures of Noncompactness with Applications
to Differential and Integral Equations}, Springer, New Delhi 2014.

 \bibitem{Bellman} R. Bellman;
\emph{Methods in Nonlinear Analysis II}, Academic Press, New York 1973.

 \bibitem{Deimling} K. Deimling;
\emph{Ordinary Differential Equations in Banach Spaces}, Lect. Notes in Math.
596, Springer, Berlin 1977.

 \bibitem{Deimling2} K. Deimling;
\emph{Nonlinear Functional Analysis}, Springer, Berlin 1985.

 \bibitem{Heinz} H. P. Heinz;
\emph{On the behaviour of measures of noncompactness with respect to
differentiation and integration of vector valued functions},
Nonlin. Anal. 7 (1983) 1351--1371.

 \bibitem{Herzog} G. Herzog;
\emph{On Lipschitz conditions for ordinary differential equations
in Fr\'echet spaces}, Czech. Math. J. 48 (1998) 95--103.

 \bibitem{Hille} E. Hille;
\emph{Pathology of infinite systems of linear first order differential
equations with constant coefficients}, Ann. Mat. Pura. Appl. 55 (1961) 135--144.

 \bibitem{Lemmert} R. Lemmert;
\emph{On ordinary differential equations in locally convex spaces},
Nonlin. Anal. 10 (1986) 1385--1390.

 \bibitem{monch} H. M\"onch, G.H. von Harten;
\emph{On the Cauchy problem for ordinary differential equations in Banach spaces},
Arch. Math. 39 (1982) 153--160.

 \bibitem{Mursaleen} M. Mursaleen, A. Alotaibi;
\emph{Infinite systems of differential equations in some BK spaces},
Abstr. Appl. Anal., vol. 2012, Article ID 863484, 20 pages.

 \bibitem{Oguz} M. N. Oguzt\"oreli;
\emph{On the neural equations of Cowan and Stein}, Utilitas Math. 2 (1972)
 305-315.

 \bibitem{Oregan} D. O'Regan, M. Meehan;
\emph{Existence Theory for Nonlinear Integral and Integrodifferential Equations},
Mathematics and its Applications 445, Kluwer Academic Publishers, Dordrecht 1998.

 \bibitem{Persi1} K. P. Persidskii;
\emph{Countable systems of differential equations and stability of their solutions},
Izv. Akad. Nauk. Kazach. SSR 7 (1959) 52--71.

 \bibitem{Persi2} K. P. Persidskii;
\emph{Countable systems of differential equations and stability of their
solutions III: Fundamental theorems on stability of solutions of countable
many differential equations}, Izv. Akad. Nauk. Kazach. SSR 9 (1961) 11--34.

 \bibitem{szufla} S. Szufla;
\emph{On the existence of solutions of differential equations in Banach spaces},
Bull. Acad. Polon. Sci., S\'er. Sci. Math. 30 (1982) 507--515.

 \bibitem{Voigt} A. Voigt;
 \emph{Line method approximations to the Cauchy problem for nonlinear
parabolic differential equations}, Numer. Math. 23 (1974) 23--36.

 \bibitem{Walter} W. Walter;
\emph{Differential and Integral Inequalities}, Springer, Berlin 1970.

 \bibitem{Zautykov} O. A. Zautykov;
\emph{Countable systems of differential equations and their applications},
Diff. Uravn. 1 (1965) 162--170.

 \bibitem{Zautykov2} O. A. Zautykov, K. G. Valeev;
\emph{Infinite Systems of Differential Equations},
Izdat. ``Nauka'' Kazach. SSR, Alma - Ata 1974.

\end{thebibliography}

\end{document}