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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 121, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/121\hfil Weighted pseudo almost automorphic sequences]
{Weighted pseudo almost automorphic sequences and their applications}

\author[S. Abbas\hfil EJDE-2010/121\hfilneg]
{Syed Abbas}

\address{Syed Abbas \newline
School of Basic Sciences (Section of Mathematics), 
Indian Institute of Technology Mandi, Mandi, H.P., India}
 \email{sabbas.iitk@gmail.com}

\thanks{Submitted April 18, 2010. Published August 24, 2010.}
\subjclass[2000]{32N05, 65Q10, 92B20}
\keywords{Almost automorphic sequence; difference equations;
neural networks}

\begin{abstract}
 In this article we define the concept of weighted pseudo
 almost automorphic sequence, and establish some basic properties
 of these sequences.
 Further, as an application, we show the existence, uniqueness and
 global attractivity of weighted pseudo almost automorphic sequence
 solutions of a neural network model.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks


\section{Introduction}

Pseudo almost automorphic functions are natural generalization of
almost automorphic functions introduced by Xiao et.al.
\cite{xiao}. Recently, Blot et.al \cite{gur1} have proposed an
extension of pseudo almost automorphic functions called weighted
pseudo almost automorphic functions. The existence and uniqueness
of the pseudo almost automorphic and almost periodic solutions of
differential equations have been investigated by many authors
recently \cite{abbas2,abbas3,abbas4,abbas5,gur1,cieutat, xiao1}.


The theory of almost periodic and almost automorphic sequence
parallels those of the corresponding functions. They have been
studied by several researchers in the past \cite{araya,
cord,minh}. Unlike differential equations, solution to difference
equations result in sequences. The almost periodicity/almost
automorphy of the solution to difference equations have been
investigated in some of the recent literatures. For instance,
Cao~\cite{cao} and Huang et. al.~\cite{hwg} showed the existence
of almost periodic sequence solutions of a discrete model of
neural networks. Araya et. al.~\cite{araya} proved the existence
and uniqueness of almost automorphic sequence solutions of some
difference equations.

Difference equations play an important role in fields like
numerical methods for differential equations, finite element
methods, control theory etc. They arise as variational equations
along the orbits of discrete dynamical systems. The theory of
almost periodic sequences and their related extensions are thus
important for discrete dynamical systems in general. In this paper
we present yet another generalization of almost automorphic
sequences, the notion of weighted pseudo almost automorphy of a
sequence. Further, we explore some of the properties of these
sequences and derive the conditions under which the solution of
some difference equations would be weighted pseudo almost
automorphic. Further in the next section we consider a discrete
model of cellular neural network and show the existence of a
unique weighted pseudo almost automorphic sequence solution.


The organization of the rest of the paper is as follows: In
section $2$ the definition of weighted pseudo almost automorphic
sequence is presented. Some important results like closedness
property and composition theorem have been established. In section
$3$ we investigate the conditions for the existence and uniqueness
of weighted pseudo almost automorphic sequence solution of a
discrete model of a neural network. We further show that this
solution is globally attractive.

 \section{Preliminaries}

 Let $X$ be a real or complex Banach space endowed with the
norm $\|\cdot\|_X$.

Denote by $\mathbb{U}$ the collection of all positive functions
$\rho:\mathbb{Z} \to \mathbb{R}$. For each $\rho \in
\mathbb{U}$ define
$$
m(r,\rho)=\sum_{-r}^{r}\rho(k).
$$
Denote by $\mathbb{U}_{\infty}$ the set of all
$\rho \in \mathbb{U}$ such
that $\lim_{r \to \infty}m(r,\rho)=\infty$.
Denote by $\mathbb{U}_{b}$ the set of all bounded $\rho \in
\mathbb{U}_{\infty}$ such that $\inf_{k \in \mathbb{Z}}\rho(k)>0$.


\begin{definition} \label{aa} \rm
A function $f:\mathbb{Z}\to X$ is said to be almost automorphic
sequence if for every sequence of integer
$\{k_l\}_{l \in \mathbb{N}}$ there exists a subsequence
$\{{k_n}\}_{n \in \mathbb{N}}$ such that
$$f(k+k_n)\to g(k)\quad\text{and}\quad
g(k-k_n)\to f(k)$$
for each $k\in \mathbb{Z}$. This is also equivalent to
$$
\lim_{n\to \infty}\lim_{m\to
\infty}f(k+k_n-k_m)=f(k)
$$
for each $k\in \mathbb{Z}$.
\end{definition}

Denote by $AAS(X)$ the set of all almost automorphic sequences from
 $\mathbb{Z}$ to $X$. Then $(AAS(X),\|\cdot\|_{AAS(X)})$ is a Banach
 space with the supremum norm,
 $$
\|u\|_{AAS(X)}=\sup_{k \in \mathbb{Z}}\|u(k)\|_X.
$$

\begin{definition} \label{def2.2} \rm
A  function $f:\mathbb{Z}\times X \to X$ is said to be
almost automorphic sequence in $k$
for each $x \in X$ if for every sequence of integers
$\{k_l\}_{l\in \mathbb{N}}$ there exists a subsequence
$\{{k_n}\}_{n \in \mathbb{N}}$ such that
$$
f(k+k_n,x)\to g(k,x)\quad\text{and}\quad
g(k-k_n,x)\to f(k,x)
$$
for each $k\in \mathbb{Z}$ and $x \in X$.
The set of all such functions are denoted by
$AAS(\mathbb{Z}\times X, X)$.
\end{definition}

A bounded function $h:\mathbb{Z} \times X \to X$ is said
to be in $PAA_0S(\mathbb{Z}\times X,\rho)$ for some
$\rho \in \mathbb{U}_{\infty}$ if
$$
\lim_{r\to \infty}\frac{1}{m(r,\rho)}
\sum_{-r}^r\|h(k,x)\|_X\rho(k)=0
$$
for each $x\in X$.

\begin{definition} \label{wpaa} \rm
A function $f:\mathbb{Z}\times X\to X$ is
said to be weighted pseudo almost automorphic sequence if it can
be decomposed as a sum of two functions
$$
f=f_1+f_2,
$$
where $f_1$ is almost automorphic sequence and $f_2\in
PAA_0S(\mathbb{Z}\times X,\rho)$.
\end{definition}

The following theorem is from Blot et. al. \cite{gur1}.

\begin{theorem} \label{thm2.4}
The decomposition of a weighted pseudo almost automorphic function
is unique for any $\rho \in U_b$.
\end{theorem}

We assume that for any weighted pseudo almost automorphic
functions $f(t)$ over $\mathbb{R}$, the sequence $\{x_n\}$ define
by $x(n)=f(n)$ for $n \in \mathbb{Z}$ is weighted pseudo almost
automorphic.

As an example, consider the function
$$
f(k)=\operatorname{signum}(\cos2\pi k\theta)+e^{-|k|}
$$
It is well known that the function
$\operatorname{signum}(\cos(2\pi k\theta))$,
$k \in \mathbb{Z}$ is almost automorphic sequence for
$\theta$ irrational \cite{bochner3}. Now consider the weight
function $\rho_1$ defined by
\begin{equation}
\rho_1(k)=1+k^2 \quad k \in \mathbb{Z}
\end{equation}
It is easy to verify that
$$
m(r,\rho_1)=\sum_{-r}^r\rho_1(k)=\sum_{-r}^r(1+k^2)
=\frac{r(r+1)(2r+1)}{3}.
$$
Thus $\lim_{r\to \infty}m(r,\rho_1)=\infty$.
Thus $\rho_1 \in \mathbb{U}_{\infty}$.
 Further
\begin{equation}
\begin{aligned}
\lim_{r\to
\infty}\frac{1}{m(r,\rho_1)}\sum_{-r}^re^{-|k|}\rho_1(k)
&= \lim_{r\to
\infty}\frac{1}{m(r,\rho_1)}\sum_{-r}^re^{-|k|}(1+k^2)\\
&= \lim_{r\to \infty}\frac{2}{m(r,\rho_1)}
\Big(-1+\sum_{0}^re^{-k}(1+k^2)\Big)
= 0
\end{aligned}
\end{equation}
since
$$
\sum_{0}^re^{-k}k^2 \le \int_0^rk^2e^{-k}dk < \infty.
$$
 Hence $e^{-|k|} \in PAA_0S(\mathbb{Z},\rho_1)$ and so
$f(k) \in WPAAS(\mathbb{Z})$.

Further consider another weight function $\rho_2$ defined by
\[
\rho_2(k)=\begin{cases}
e^{\beta k} & k <0,\\
1 & k \geq 0
\end{cases}
\]
for some $\beta >0$. It is easy to verify that
$$
m(r,\rho_2)=\sum_{-r}^r\rho_2(k)
=\sum_{-r}^0\rho_2(k)+\sum_1^r\rho_2(k)=r+\sum_{-r}^0 e^{\beta k}.
$$
Thus $\lim_{r\to \infty}m(r,\rho_2)=\lim_{r\to
\infty}(r+\frac{1}{1-e^{-\beta}}) \to \infty$ which
implies that $\rho_2 \in \mathbb{U}_{\infty}$. Further
$$
\lim_{r\to \infty}\frac{1}{m(r,\rho_2)}\sum_{-r}^re^{-|k|}\rho_2(k)
=\lim_{r\to
\infty}\frac{1}{m(r,\rho_2)}\Big(\sum_{-r}^0e^{(\beta+1)k}
+\sum_1^re^{-k}\Big)=0
$$
for $\beta > 0$. Because $e^{-|k|} \in PAA_0S(\mathbb{Z},\rho_2)$
and so $f(k) \in WPAAS(\mathbb{Z})$.


Now we state some important properties of weighted pseudo almost
automorphic sequences.

\begin{theorem} \label{conds}
Let $u,v$ be weighted pseudo almost automorphic sequences with
same weight function $\rho$,
then the following assertions are true:
\begin{itemize}
\item[(i)] The sum $u+v$ is weighted pseudo almost automorphic
sequence;
 \item[(ii)] For every scalar $\lambda, \lambda u$ is weighted 
 pseudo almost automorphic  sequence;
\item[(iii)] The translation $u_k(l)=u(k+l)$ is weighted pseudo
almost automorphic sequence for fixed $l$ if the sequences
$\frac{\rho(k-l)}{\rho(k)}$ and $\frac{\rho(k)}{\rho(k-l)}$ are
bounded.
\end{itemize}
\end{theorem}

The proof of the above theorem is easy to verify. The proof of
assertion (iii) is similar to the proof in \cite{agarwal}.

\begin{theorem} \label{thm2.6}
Let $X$ and $Y$ be two Banach spaces. Also let $u:\mathbb{Z}
\to X$ be a weighted pseudo almost automorphic sequence.
If $T:X\to Y$ is a bounded linear continuous function,
then the composition $Tou:\mathbb{Z} \to Y$ is weighted
pseudo almost automorphic sequence.
\end{theorem}

\begin{proof}
 Let $u=u_1+u_2$, where $u_1$ is almost automorphic
and $u_2 \in PAA_0S$. Now for $\rho \in \mathbb{U}_{\infty}$,
consider
\begin{align*}
\lim_{r\to
\infty}\frac{1}{m(r,\rho)}\sum_{-r}^r\|Tou_2(k)\|\rho(k)
& \le \lim_{r\to
\infty}\frac{1}{m(r,\rho)}\sum_{-r}^r\|T\|\|u_2(k)\|\rho(k) \\
& \le \lim_{r\to
\infty}\frac{C}{m(r,\rho)}\sum_{-r}^r\|u_2(k)\|\rho(k) \to
0.
\end{align*}
To show that $Tou_1$ is almost automorphic sequence. As $u_1$ is
almost automorphic sequence, thus for any sequence $k_l$ there
exists a subsequence $k_n$ such that
$$
u_1(k+k_n) \to \bar{u}_1(k) \quad \text{and} \quad
\bar{u}_1(k-k_n) \to u_1(k)
$$
for each $k\in \mathbb{Z}$. Now for the sequence $k_n$ we
have
\begin{gather*}
\lim_{n\to \infty}T(u_1(k+k_n))=T(\lim_{n\to
\infty}u_1(k+k_n))=T(\bar{u}_1(k)), \\
\lim_{n\to \infty}T(\bar{u}_1(k-k_n))=T(\lim_{n\to
\infty}\bar{u}_1(k-k_n))=T(u_1(k))
\end{gather*}
because $T$ is continuous.
Thus $Tou_1$ is almost automorphic sequence. Hence we conclude
that $Tou$ is weighted pseudo almost automorphic sequence.
\end{proof}

\begin{theorem} \label{thm2.7}
Let $u:\mathbb{Z} \to \mathbb{C}$ be weighted pseudo
almost automorphic sequence and $f:\mathbb{Z} \to X$ be
almost automorphic sequence. Then the sequence
$uf:\mathbb{Z}\to X$ defined by $(uf)(k)=u(k)f(k)$,
$k \in \mathbb{Z}$ is also weighted pseudo almost automorphic.
\end{theorem}

\begin{proof}
As $u$ is weighted pseudo almost automorphic, thus
$u$ can be written as $u=u_1+u_2$, where $u_1$ is almost
automorphic and $u_2 \in PAA_0S$. Hence we get
$u(k)f(k)=u_1(k)f(k)+u_2(k)f(k)$. Now for $\rho \in
\mathbb{U}_{\infty}$, consider
\begin{align*}
\lim_{r\to
\infty}\frac{1}{m(r,\rho)}\sum_{-r}^r\|u_2(k)f(k)\|\rho(k)
&\le \lim_{r\to \infty}\frac{1}{m(r,\rho)}
\sum_{-r}^r\|u_2(k)\|\|f(k)\|\rho(k) \\
& \le \lim_{r\to \infty}\frac{\|f\|}{m(r,\rho)}
 \sum_{-r}^r\|u_2(k)\|\rho(k) \to 0,
\end{align*}
as we know that $\|f\|=\sup_{k\in \mathbb{Z}}\|f(k)\| <\infty$
from \cite{araya}. For any sequence $k_l$ there exists a
subsequence $k_n$ such that
\begin{gather*}
u_1(k+k_n) \to \bar{u}_1(k), \quad
\bar{u}_1(k-k_n) \to u_1(k), \\
f(k+k_n) \to \bar{f}(k), \quad
\bar{f}(k-k_n) \to f(k)
\end{gather*}
for each $k\in \mathbb{Z}$. Now consider
\begin{align*}
&\|u_1(k+k)n)f(k+k_n)-\bar{u}_1(k)\bar{f}(k)\|  \\
&\le \|f(k+k_n)-\bar{f}(k)\|\|u(k+k_n)\|+\|u(k+k_n)
 -\bar{u}_1(k)\|\|\bar{f}(k)\| \to 0,
\end{align*}
as $n\to \infty$. Also we have
\begin{align*}
&\|\bar{u}_1(k-k)n)\bar{f}(k-k_n)-u_1(k)f(k)\|  \\
&\le \|\bar{f}(k-k_n)-f(k)\|\|\bar{u}(k-k_n)\|
 +\|\bar{u}(k-k_n)-u_1(k)\|\|f(k)\|  \to 0,
\end{align*}
as $n\to \infty$. Thus we conclude that $u_1(k)f(k)$ is
almost automorphic sequence and hence $u(k)f(k)$ is weighted
pseudo almost automorphic sequence.
\end{proof}

\begin{theorem} \label{thm2.8}
Let $f:\mathbb{Z} \times X \to X$ be almost automorphic
sequence which satisfies a Lipschitz condition in $x$ uniformly in
$k$, that is
$$
\|f(k,x)-f(k,y)\| \le L\|x-y\|,
$$
for each $x,y \in X$.
Assume $\phi :\mathbb{Z} \to X$ is weighted pseudo almost
automorphic sequence, then the function $f(k,\phi(k))$ is weighted
pseudo almost automorphic sequence.
\end{theorem}

\begin{proof}
 We know that $\phi$ is weighted pseudo almost
automorphic sequence, thus $\phi=\phi_1+\phi_2$ where $\phi_1$ is
almost automorphic sequence and $\phi_2 \in PAA_0S$. Writing
$$
f(k,\phi(k))=f(k,\phi_1(k)+\phi_2(k))-f(k,\phi_1(k))+f(k,\phi_1(k)).
$$
As $f$ is Lipschitz, we have
$$
\|f(k,\phi_1(k)+\phi_2(k))-f(k,\phi_1(k))\| \le L\|\phi_2(k)\|.
$$
Thus for $\rho \in \mathbb{U}_{\infty}$ we obtain
\begin{align*}
&\lim_{r\to
\infty}\frac{1}{m(r,\rho)}
\sum_{-r}^r\|f(k,\phi_1(k)+\phi_2(k))-f(k,\phi_1(k))\|\rho(k) \\
& \le \lim_{r\to \infty}\frac{L}{m(r,\rho)}
\sum_{-r}^r\|\phi_2(k)\|\rho(k)\to 0.
\end{align*}
We know that $f$ and $\phi_1$ are almost automorphic sequences. So
for each sequence $k_l$ there exist a subsequence $k_n$ such that
$$
f(k+k_n,x) \to g(k,x) \quad \text{and} \quad g(k-k_n,x) \to f(k,x)
$$
for all $k \in \mathbb{Z}, \ x \in X$.
Also we have
$$
\phi_1(k+k_n) \to \psi(k) \quad \text{and} \quad
\psi(k-k_n)\to \phi_1(k)
$$
for all $k \in \mathbb{Z}$, $x \in X$.

Now consider
\begin{align*}
&f(k+k_n,\phi_1(k+k_n))-g(k,\psi(k))\\
&=f(k+k_n,\phi_1(k+k_n))-f(k+k_n,\psi(k))+f(k+k_n,\psi(k))
-g(k,\psi(k)),
\end{align*}
we have
\begin{align*}
& \|f(k+k_n,\phi_1(k+k_n))-g(k,\psi(k))\|  \\
& \le \|f(k+k_n,\phi_1(k+k_n))-f(k+k_n,\psi(k))\|
+\|f(k+k_n,\psi(k))-g(k,\psi(k))\| \\
& \le L\|\phi(k+k_n)-\psi(k)\|+\|f(k+k_n,\psi(k))-g(k,\psi(k))\|
\to 0,
\end{align*}
as $n\to \infty$. Thus we conclude that
$$
f(k+k_n,\phi_1(k+k_n))\to g(k,\psi(k)).
$$
Next assume
\begin{align*}
&g(k-k_n,\psi(k-k_n))-f(k,\phi_1(k))\\
&=g(k-k_n,\psi(k-k_n))-g(k-k_n,\phi_1(k))+g(k-k_n,\phi_1(k))
-f(k,\phi_1(k)),
\end{align*}
we obtain
\begin{align*}
& \|g(k-k_n,\psi(k-k_n))-f(k,\phi_1(k))\|  \\
& \le \|g(k-k_n,\psi(k-k_n))-g(k-k_n,\phi_1(k))\|
+\|g(k-k_n,\phi_1(k))-f(k,\phi_1(k))\| \\
&\leq L\|\psi(k-k_n)-\phi_1(k)\|+\|g(k-k_n,\phi_1(k))
-f(k,\phi_1(k))\| \to 0,
\end{align*}
as $n \to \infty$. Hence
$$
g(k-k_n,\psi(k-k_n))\to f(k,\phi_1(k)).
$$
It is easy to verify that $g$ satisfies
the same Lipschitz condition as $f$. By the above analysis one can
conclude that $f(k,\phi_1(k))$ is almost automorphic sequence.
Hence $f(k,\phi)$ is weighted pseudo almost automorphic sequence.

Define the operator $\Delta u(k)=u(k+1)-u(k)$. Now it is easy to
verify that for any weighted pseudo almost automorphic sequence
$\{u(k)\}_{k\in \mathbb{Z}}$, $\Delta u(k)$ is weighted pseudo
almost automorphic sequence.
\end{proof}

The next result is important to study weighted pseudo almost
automorphic sequence solutions of difference equations. For the
theorems listed below we assume that the function $\rho$ satisfy
condition $(iii)$ of theorem \ref{conds}.

\begin{theorem} \label{thm2.9}
Assume $v: \mathbb{Z} \to \mathbb{C}$ is a summable
function, that is, $\sum_{k \in \mathbb{Z}}|v(k)|<\infty$. Then
for any weighted pseudo almost automorphic sequence $u:\mathbb{Z}
\to X$ the following function
$$
w(k)=\sum_{l \in \mathbb{Z}}v(l)u(k-l), \quad k\in \mathbb{Z}
$$
is weighted pseudo almost automorphic sequence.
\end{theorem}

\begin{proof}
Consider $w_2(k)=\sum_{l \in
\mathbb{Z}}v(l)u_2(k-l)$, where $u_2 \in PAA_0S$ is the second
component of $u$. We have $$\|w_2(k)\| \le \sum_{l \in
\mathbb{Z}}|v(l)|\|u_2(k-l)\| \le \sum_{l \in
\mathbb{Z}}|v(l)|\|u_2\| \le M\|u_2\|,$$ for some positive
constant $M$. Thus one gets
$$
\lim_{r\to \infty}\frac{1}{m(r,\rho)}
\sum_{-r}^r\|w_2(k)\|\rho(k) \le
\lim_{r\to \infty}\frac{M}{m(r,\rho)}\sum_{-r}^r\|u_2(k)\|\rho(k)
 \to 0.
$$
Let $w_1(k)=\sum_{l \in \mathbb{Z}}v(l)u_1(k-l)$,
where $u_1$ is the almost automorphic component of $u$.
The almost automorphy
of $w_1$ follows from \cite[theorem 2.13]{araya}.
\end{proof}

The theorems stated below are easy generalization of
\cite[theorems 3.1 and 4.1]{araya}.
Consider the difference equation
\begin{equation}
u(n+1)=\lambda u(n)+f(n), \label{difference1}
\end{equation}
where $\lambda$ is any real or complex number. Now denote the
region $\mathbb{D}=\{z\in \mathbb{C}: |z|=1\}$.

\begin{theorem} \label{thm2.10}
Let $X$ be a Banach space. If $\lambda \in
\mathbb{C}\backslash\mathbb{D}$ and $f:\mathbb{Z} \to X$
is weighted pseudo almost automorphic sequence, then the weighted
pseudo almost automorphic sequence solution of equation
\eqref{difference1} is given by
\begin{itemize}
\item[(i)] $u(n)=\sum_{k=-\infty}^n\lambda^{n-k}f(k-1)$ if
$|\lambda|<1$;

\item[(ii)] $u(n)=-\sum_{k=n}^{\infty}\lambda^{n-k-1}f(k)$ if $|\lambda|>1$.
\end{itemize}
\end{theorem}

We have the following theorem for the difference equation
\begin{equation}
u(n+1)=\lambda u(n)+f(n,u(n)). \label{difference2}
\end{equation}

\begin{theorem} \label{thm2.11}
Let $X$ be a Banach space. If $\lambda \in
\mathbb{C}\backslash\mathbb{D}$ and $f:\mathbb{Z}\times X
\to X$ be a weighted pseudo almost automorphic sequence
that satisfies the following Lipschitz condition
$$
\|f(k,x)-f(k,y)\| \le L\|x-y\|
$$
for each $x,y \in X$ and $k\in \mathbb{Z}$.
Then the weighted pseudo almost automorphic sequence solution of
difference equation \eqref{difference2} is given by
\begin{itemize}
\item[(i)] $u(n)=\sum_{k=-\infty}^n\lambda^{n-k}f(k-1,u(k-1))$
if $|\lambda|<1-L$;
\item[(ii)] $u(n)=-\sum_{k=n}^{\infty}\lambda^{n-k-1}f(k,u(k))$
if $|\lambda|>1+L$.
\end{itemize}
\end{theorem}


\section{Weighted pseudo almost automorphic sequence in
neural networks}

In this section we consider a discrete model of cellular neural
network and show the existence of a unique weighted pseudo almost
automorphic sequence solution. A cellular neural network is a
nonlinear dynamic circuit consisting of many processing units
called cells arranged in two or three dimensional array. This is
very useful in the areas of signal processing, image processing,
pattern classification and associative memories. Hence, the
application of cellular networks is of great interest to many
researchers. In \cite{cao,cz,ch,mg}, the authors have dealt with
the global exponential stability and the existence of a periodic
solution of a cellular neural network with delays using the
general method of Lyapunov functional. The discrete analogue of
continuous time cellular network models are very important for
theoretical analysis as well as for implementation. Thus, it is
essential to formulate a discrete time analogue of continuous time
network. A most acceptable method is to discretize the continuous
time network. For detailed analysis on the discretization method
the reader may consult Mohamad and Gopalsamy \cite{mg}, Stewart
\cite{st}.

Consider the following model of cellular neural network consisting
of $m$ interconnected cells
$$
\frac{dx_i(t)}{dt}=-a_i([t])x_i(t)
+\sum_{j=1}^{m}b_{ij}([t])f_j(x_j([t]))+I_i([t]),
$$
where $i=1,2,\dots ,m$ and $[\cdot]$ denote the greatest integer
function. The differential equations of above kind are called
equations with piecewise constant argument. The functions $x_i(t)$
denotes the potential of the cell $i$ at time $t$. The terms
$a_i(t)$ denotes the rate which the cell $i$ resets its potential
to the resting state when isolated from other cells and inputs.
$b_{ij}(t)$ denotes the strengths of connectivity between the
$j$th cell and the $i$th cell. The functions $f_i$ and $I_i$
denote the nonlinear output function and external input source
introduced from outside the network to $i$th component,
respectively.

The discrete analogue of the above model is given by
\begin{equation}
x_i(n+1)=x_i(n)e^{-a_i(n)}+\frac{1-e^{-a_i(n)}}{a_i(n)}
\Big\{\sum_{j=1}^m
b_{ij}(n)f_j(x_j(n))+I_i(n)\Big\}, \label{meq}
\end{equation}
where $i=1,2,\dots ,m$, $n \in \mathbb{Z}$. The existence of
weighted pseudo almost automorphic sequence solutions of equation
\eqref{meq} has been established in this section. For more details
on the model we refer to Huang et. al. \cite{hwg} in which the
authors have proved the existence of an almost periodic sequence
solution of equations \eqref{meq}. The present author has shown
the existence of $k$
pseudo almost periodic sequence solutions of a neural network
model \cite{abbas}.

The assumptions described below are necessary to show the
existence of weighted almost automorphic solutions of equation
\eqref{meq}.

 \subsection*{Assumptions}
\begin{itemize}
\item[(A1)] $a_i(n)>0$ is almost automorphic sequence and
$b_{ij}(n), \ I_i(n)$ are weighted pseudo almost automorphic
sequence for $i,j=1,2,\dots,m$.

\item[(A2)] There exist positive constants $M_j$ and $L_i$ such
that $|f_j(x)| \le M_j$ and $|f_i(x)-f_i(y)|\le L_i|x-y|$ for each
$x,y \in \mathbb{R}$ and $j=1,2,\dots,m$; $i=1,2,\dots,m$.
\end{itemize}
For the discrete equation \eqref{meq}, define
$$
C_i(n)=e^{-a_i(n)}, \quad
D_{ij}(n)= b_{ij}(n)\frac{1-e^{-a_i(n)}}{a_i(n)},\quad
F_i(n)=I_i(n)\frac{1-e^{-a_i(n)}}{a_i(n)}.
$$
Using the above notation, \eqref{meq} can be written as,
\begin{equation}
x_i(n+1)=C_i(n)x_i(n)+\sum_{j=1}^{m}D_{ij}(n)f_j(x_j(n))+F_i(n),
\label{de1}
\end{equation}
for $i=1,2,\dots,m$.
Denote:
\begin{gather*}
C_i^{\ast}=\sup_{n\in \mathbb{Z}}|C_i(n)|, \quad
I_i^{\ast}=\sup_{n\in \mathbb{Z}}|I_i(n)|,\\
D_{ij}^{\ast}=\sup_{n\in \mathbb{Z}}|D_{ij}(n)|, \quad
F_i^{\ast}=\sup_{n\in \mathbb{Z}}|F_i(n)|,\\
b_{ij}^{\ast}=\sup_{n\in \mathbb{Z}}|b_{ij}(n)|, \quad
a_i^{\ast}=\inf_{n\in \mathbb{Z}}a_i(n),\quad
P_i=\sum_{j=1}^{m}D_{ij}^{\ast}M_j+F_i^{\ast}.
\end{gather*}

\begin{definition} \label{def2.12} \rm
A solution $x(\nu)=(x_1(\nu),\dots, x_m(\nu))^T$ of \eqref{de1}
is said to be globally attractive if for any other solution
$y(\nu)=(y_1(\nu),\dots, y_m(\nu))^T$ of \eqref{de1}, we have
$$
\lim_{\nu \to \infty}|x_i(\nu)-y_i(\nu)|=0.
$$
\end{definition}

\begin{lemma} \label{lemma1}
Suppose assumption {\rm (A1)} holds, then $C_i \in AAS$
and $D_{ij}, F_i \in WPAAS$ for $i,j=1,2,\dots,m$.
\end{lemma}
\begin{proof} From the assumption (A1) we know that
$a_i(n)$ is almost automorphic sequence. Thus for any sequence
$k_l$ there exists a subsequence $k_m$ such that
$$
a_i(n+k_m) \to a_{i1}(n) \ \text{and} \ a_{i1}(n-k_m) \to a_i(n).
$$
Denoting $C_{i1}(n)=e^{-a_{i1}(n)}$, we have
\[
|C_i(n+k_m)-C_{i1}(n)|
=|e^{-a_{i}(n+k_m)}-e^{-a_{i1}(n)}|
\le |a_{i}(n+k_m)-a_{i1}(n)|
\to 0,
\]
 as $m \to \infty$. Also
\[
|C_{i1}(n-k_m)-C_{i}(n)|
=|e^{-a_{i1}(n-k_m)}-e^{-a_{i}(n)}| \le |a_{i1}(n-k_m)-a_{i}(n)|
\to 0,
\]
as $m \to \infty$. Thus one can conclude
that $C_i(n)$ are almost automorphic. Now since $b_{ij}$ and $I_i$
are weighted pseudo almost automorphic, we can decompose them into
two parts
$$
b_{ij}=b_{ij1}+b_{ij2} \quad \text{and} \quad I_i=I_{i1}+I_{i2}
$$
such that $b_{ij1}, I_{i1} \in AAS$ and
$b_{ij2}, I_{i2} \in PAA_0S$.
We have
\begin{gather*}
b_{ij1}(n+k_m) \to \bar{b}_{ij1}(n),\quad
\bar{b}_{ij1}(n-k_m) \to b_{ij1}(n),\\
I_{i1}(n+k_m) \to \bar{I}_{i1}(n), \quad
\bar{I}_{i1}(n-k_m) \to I_{i1}(n).
\end{gather*}
Also for $\rho \in \mathbb{U}_{\infty}$ we obtain
\begin{gather*}
\lim_{r \to \infty}\frac{1}{m(\rho,r)}
 \sum_{n=-r}^r|b_{ij2}(n)|\rho(n)=0,\\
\lim_{r \to \infty}\frac{1}{m(\rho,r)}
 \sum_{n=-r}^r|I_{i2}(n)|\rho(n)=0.
\end{gather*}
Denote $D_{ij1}(n)=b_{ij1}(n)\frac{1-e^{-a_{i}(n)}}{a_{i}(n)}$ and
$\bar{D}_{ij1}(n)=\bar{b}_{ij1}(n)\frac{1-e^{-a_{i1}(n)}}{a_{i1}(n)}$
we have
\begin{align*}
&|D_{ij1}(n+k_m)-\bar{D}_{ij1}(n)|\\
&= \Big|b_{ij1}(n+k_m)\frac{1-e^{-a_{i}(n+k_m)}}{a_{i}(n+k_m)}
 -\bar{b}_{ij1}(n)\frac{1-e^{-a_{i1}(n)}}{a_{i1}(n)}\Big|\\
&\leq |b_{ij1}(n+k_m)-\bar{b}_{ij1}(n)|\times\Big|
 \frac{1-e^{-a_{i}(n+k_m)}}{a_{i}(n+k_m)}\Big| \\
&\quad +|\bar{b}_{ij1}(n)|\times\Big|
 \frac{1-e^{-a_{i}(n+k_m)}}{a_{i}(n+k_m)}
 -\frac{1-e^{-a_{i1}(n)}}{a_{i1}(n)}\Big|\\
&\to \infty, \quad \text{as }  m \to \infty.
\end{align*}
Also
\begin{align*}
&|\bar{D}_{ij1}(n-k_m)-D_{ij1}(n)|\\
&= \Big|\bar{b}_{ij1}(n-k_m)
 \frac{1-e^{-a_{i1}(n-k_m)}}{a_{i1}(n-k_m)}-b_{ij1}(n)
 \frac{1-e^{-a_{i}(n)}}{a_{i}(n)}\Big| \\
&\leq |\bar{b}_{ij1}(n-k_m)-b_{ij1}(n)|\times\Big|\frac{1-e^{-a_{i1}(n-k_m)}}{a_{i1}(n-k_m)}\Big|
\\
&\quad +|b_{ij1}(n)|\times\Big|\frac{1-e^{-a_{i1}(n-k_m)}}{a_{i1}(n-k_m)}-
\frac{1-e^{-a_{i}(n)}}{a_{i}(n)}\Big| \\
& \to \infty, \quad  \text{as }  m \to \infty.
\end{align*}
Considering
$D_{ij2}(n)=b_{ij2}(n)\frac{1-e^{-a_{i}(n)}}{a_{i}(n)}$ one obtains
\begin{align*}
& \lim_{r \to
\infty}\frac{1}{m(r,\rho)}\sum_{n=-r}^r|D_{ij2}(n)|\rho(n)
 \\ &\leq  \lim_{r \to
\infty}\frac{1}{m(r,\rho)}\sum_{n=-r}^r|b_{ij2}(n)|\times
\Big|\frac{1-e^{-a_{i}(n)}}{a_{i}(n)}\Big|\rho(n)  \\
&\leq  \lim_{r \to
\infty}\frac{1}{a_i^*m(r,\rho)}\sum_{n=-r}^r|b_{ij2}(n)|\rho(n).
\end{align*}
Since $b_{ij2}$ are weighted pseudo almost automorphic sequences,
we have
$$
\lim_{r \to \infty}\frac{1}{m(r,\rho)}\sum_{n=-r}^r|D_{ij2}(n)|\rho(n)
 = 0.$$
Thus $D_{ij}$ are weighted pseudo almost automorphic sequences. By
the similar analysis one can easily show that $F_i$ are also
weighted pseudo almost automorphic sequences.
\end{proof}

\begin{lemma} \label{lemma2}
Under assumptions {\rm (A1), (A2)}, every solution of
\eqref{de1} is bounded.
\end{lemma}

\begin{proof}
One can easily observe that the relation
$$
C_i(n)x_i(n)-R_i \le x_i(n+1)\le C_i(n)x_i(n)+R_i,
$$
where $R_i=\sum_{j=1}^{m}D_{ij}^{\ast}+F_i^{\ast}$ holds. Consider
the  difference equations
$$
\bar{x}_i(n+1)=C_i(n)\bar{x}_i(n)+R_i,
$$
where $\bar{x}_i(0)=x_i(0)$. Using induction we have
\begin{align*}
\bar{x}_i(n)
&= \prod_{k=1}^nC_i(k)\bar{x}_i(0)+R_i
 \Big(\sum_{l=1}^{n-1}\prod_{k=1}^lC_i(k)+1\Big) \\
&\leq e^{-na_i^{\ast}}\bar{x}_i(0)+R_i(\sum_{l=1}^{n-1}
 e^{-la_i^{\ast}}+1) \\
&\leq |\bar{x}_i(0)|+\frac{R_i}{1-e^{-a_{i}^{\ast}}}.
\end{align*}
One can easily observe that $x_i(n) \le \bar{x}_i(n)$. Now using
the difference equation
$$
\tilde{x}_i(n+1)=C_i(n)\tilde{x}_i(n)-R_i
$$
 and doing  similar calculation we obtain
$$
\tilde{x}_i(n) \ge -|\bar{x}_i(0)|-\frac{R_i}{1-e^{-a_{i}^{\ast}}}.
$$
Combining the two inequalities above, we have the estimate
$$
-|x_i(0)|-\frac{R_i}{1-e^{-a_{i}^{\ast}}} \le x_i(n)\le
|x_i(0)|+\frac{R_i}{1-e^{-a_{i}^{\ast}}}.
$$
Thus $x_i$ are bounded.
\end{proof}

Now consider the  difference equations
\begin{equation}
x_i(n+1)=C_i(n)x_i(n)+F_i(n). \label{de2}
\end{equation}

\begin{lemma} \label{lemma3}
Under assumption {\rm (A1)}, there exists a weighted pseudo
almost automorphic sequence solution of \eqref{de2}.
\end{lemma}

\begin{proof}
Using an induction argument, one obtain
\begin{align*}
x_i(n+1)&=  \prod_{k=0}^n
C_i(k)x_i(0)+\sum_{l=0}^{n}\prod_{k=n-l+1}^n C_i(k)F_i(n-l) \\
& =e^{-\sum_{k=0}^n
a_i(k)}x_i(0)+\sum_{l=0}^{n}I_i(n-l)\frac{1-e^{-a_i(n-l)}}{a_i(n-l)}e^{-\sum_{k=n-l+1}^n
a_i(k)}.
\end{align*}
Consider the sequence
$$
\hat{x}_i(n)=\sum_{l=0}^{\infty}I_i(n-l)
\frac{1-e^{-a_i(n-l)}}{a_i(n-l)}e^{-\sum_{k=n-l+1}^n
a_i(k)}.
$$
Since
$$
|\hat{x}_i(n)|\le \sum_{l=0}^{\infty}I_i^*
\frac{1-e^{-a_i^*}}{a_i^*}e^{-(l-1)a_i^*}=
\sum_{l=0}^{\infty}e^{a_i^*}\frac{1-e^{-a_i^*}}{a_i^*}e^{-la_i^*}\le
\frac{I_i^*e^{a_i^*}}{a_i^*},
$$
the sequence $\hat{x}_i(n)$ is well defined.
It is easy to verify that
$$
\hat{x}_i(n+1)=C_i(n)\hat{x}_i(n)+F_i(n).
$$
Hence the sequence $\hat{x}_i=\{\hat{x}_i(n)\}$ is bounded. Now
define
$$
\hat{x}_{i1}(n)=\sum_{l=0}^{\infty}I_{i1}(n-l)
\frac{1-e^{-a_i(n-l)}}{a_i(n-l)}e^{-\sum_{k=n-l+1}^n a_i(k)}
$$
and
$$
\hat{y}_{i1}(n)=\sum_{l=0}^{\infty}\bar{I}_{i1}(n-l)
\frac{1-e^{-a_{i1}(n-l)}}{a_{i1}(n-l)}e^{-\sum_{k=n-l+1}^n
a_{i1}(k)}.
$$
Also let
$$
\hat{x}_{i2}(n)=\sum_{l=0}^{\infty}I_{i2}(n-l)
\frac{1-e^{-a_i(n-l)}}{a_i(n-l)}e^{-\sum_{k=n-l+1}^n a_i(k)},
$$
where $I_{i1}$ and $I_{i2}$ are two component of $I_i$.
For any sequence $k_l$ there exists a sequence $k_m$ such that
\begin{align*}
&|\hat{x}_{i1}(n+k_m)-\hat{y}_{i1}(n)|  \\&= \Big|
\sum_{l=0}^{\infty}I_{i1}(n+k_m-l)\frac{1-e^{-a_i(n+k_m-l)}}{a_i(n+k_m-l)}
 e^{-\sum_{k=n-l+1}^n
a_i(k+k_m)}\\
&\quad -\sum_{l=0}^{\infty}\bar{I}_{i1}(n-l)\frac{1-e^{-a_{i1}(n-l)}}{a_{i1}(n-l)}
e^{-\sum_{k=n-l+1}^n a_{i1}(k)}\Big|  \\
&\le \sum_{l=0}^{\infty}
\Big(|I_{i1}(n+k_m-l)-\bar{I}_{i1}(n-l)|
\frac{1-e^{-a_i(n+k_m-l)}}{|a_i(n+k_m-l)|}e^{-\sum_{k=n-l+1}^n
|a_i(k+k_m)|} \\
&\quad +\Big|\bar{I}_{i1}(n-l)\Big| \times
\Big|\frac{1-e^{-a_i(n+k_m-l)}}{a_i(n+k_m-l)}e^{-\sum_{k=n-l+1}^n
a_i(k+k_m)}\\
&\quad -\frac{1-e^{-a_{i1}(n-l)}}{a_{i1}(n-l)}e^{-\sum_{k=n-l+1}^n
a_{i1}(k)}\Big|\Big)\\
&\le \epsilon
\frac{e^{a_i^*}}{a_i^*} +\epsilon \frac{I_i^*e^{a_i^*}}{{a_i^*}^2}
 \\
&\leq   \frac{e^{a_i^*}}{a_i^*}\epsilon+
\frac{I_i^*e^{a_i^*}}{{a_i^*}^2}\epsilon.
\end{align*}
The above calculations imply that
$$
\hat{x}_{i1}(n+k_m)\to \hat{y}_{i1}(n) \quad m \to \infty.
$$
Similarly one can show that
$$
\hat{y}_{i1}(n-k_m)\to \hat{x}_{i1}(n) \quad m \to \infty.
$$
Hence $x_{i1}$ is almost automorphic sequence.

For $\rho \in \mathbb{U}_{\infty}$, we have
\begin{align*}
\sum_{n=-r}^{r}|\hat{x}_{i2}(n)|\rho(n)
&\leq
\sum_{n=-r}^{r}\sum_{l=0}^{\infty}|I_{i2}(n-l)|
\frac{1-e^{-a_i(n-l)}}{|a_i(n-l)|}e^{-\sum_{k=n-l+1}^n
|a_i(k)|} \rho(n) \\
&\leq
\sum_{n=-r}^{r}\sum_{l=0}^{\infty}|I_{i2}(n-l)|
\frac{e^{-(l-1)a_i^*}}{a_i^*}\rho(n) \\
&\leq \frac{e^{a_i^*}}{a_i^*}\sum_{n=-r}^{r}|I_{i2}(n)|\rho(n).
\end{align*}
Now we can easily observe that
\[
\lim_{r \to \infty} \frac{1}{m(r,\rho)}
\sum_{n=-r}^{r}|\hat{x}_{i2}(n)|\rho(n) \le
\frac{e^{a_i^*}}{a_i^*}\lim_{r \to \infty}
\frac{1}{m(r,\rho)}|I_{i2}(n)|\rho(n)=0,
\]
because $I_{i2} \in PAA_0S$. We can easily see that
$\hat{x}_i=\hat{x}_{i1}+\hat{x}_{i2}$. Hence $\hat{x}_i$ are
weighted pseudo almost automorphic sequences \eqref{de2}.
\end{proof}

\begin{theorem} \label{thm1}
Assume {\rm (A1), (A2)} hold.
There exists a unique  weighted pseudo almost automorphic
sequence solution of \eqref{de1} which is globally attractive, if
$$
\max_{1\le i \le m}\{C_i^{\ast}+\sum_{j=1}^{m}D_{ij}^{\ast}L_j\}<1.
$$
\end{theorem}

\begin{proof}
Denote a metric $d:WPAAS \times WPAAS \to
\mathbb{R}^+$, by
$$
d(x,y)=\sup_{n\in \mathbb{Z}}\max_{1\le i \le m}|x_i(n)-y_i(n)|.
$$
Now define a mapping $F:WPAAS \to WPAAS$ by $Fx=y$, where
$$
Fx=(F_1x,F_2x,\dots, F_mx)^T
$$
such that $F_ix=y_i$ and $y_i=\{y_i(n)\}$. Define
$$
y_i(n+1)=C_i(n)x_i(n)+\sum_{j=1}^{m}D_{ij}(n)f_j(x_j(n))
+F_i(n),
$$
where $\hat{x}_i$ is weighted pseudo almost automorphic
sequence solution of \eqref{de2}. Using lemma \ref{lemma1} and
assumption (A2), one can observe that $F$ maps weighted pseudo
almost automorphic sequences into weighted pseudo almost
automorphic sequences. Now denote
$$
\max_{1\le i \le m}\{C_i^{\ast}+\sum_{j=1}^{m}D_{ij}^{\ast}L_j\}=r<1.
$$
For $x,y \in WPAAS$, we have
\begin{align*}
\|Fx-Fy\|
&= \sup_{n\in \mathbb{Z}}\max_{1\le i \le m}
\sum_{j=1}^{m}\Big|[(D_{ij}(n)(f_j(x_j(n))-f_j(y_j(n)))]\Big|  \\
&\le  \sup_{n\in \mathbb{Z}}\max_{1\le i \le m}
\sum_{j=1}^{m}D_{ij}^*L_j|x_j(n)-y_j(n)|\\
& \le  r\|x-y\|.
\end{align*}
Hence $F$ is a contraction. It follows that equation \eqref{de1}
has a unique weighted pseudo almost automorphic sequence $x$.
Let $y$ be any sequence satisfying equation \eqref{de1}. Consider
$Q(n)=x(n)-y(n)$, then we obtain
$$
Q_i(n+1)=C_i(n)Q_i(n)+\sum_{j=1}^{m}D_{ij}(n)(f_j(x_j(n))-f_j(y_j(n))).
$$
Taking modulus of both sides one has
$$
|Q_i(n+1)| \le C_i^*|Q_i(n)|+\sum_{j=1}^{m}D_{ij}^*L_j|Q_j(n)|.
$$
Define $Q(n)=\max_{1\le i \le m}|Q_i(n)|$, we have
\[
|Q(n+1)| \le C_i^*|Q(n)|+\sum_{j=1}^{m}D_{ij}^*L_jQ(n)\le r Q(n).
\]
By induction we have
$Q(n) \le r^nQ(0)$.
 Hence
$$
|x_i(n)-y_i(n)|\to 0 \quad \text{as }n \to \infty.
$$
Thus $x$ is a unique weighted pseudo almost automorphic sequence
solution of \eqref{de1} which is globally attractive.
\end{proof}

\noindent\textbf{Remark:}
For $\rho=1$ the solution is pseudo almost
automorphic and is denoted by $PAAS$. The results presented in
this paper can also be applied to more general models like delayed
model of neural networks.


\subsection*{Acknowledgments} The author would like to thanks the
anonymous referee for his/her suggestions for the better exposition of
the paper.

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\end{document}