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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 19, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/19\hfil Existence and  stability]
{Existence and  stability of almost periodic solutions for
 shunting inhibitory cellular neural networks with
 continuously  \\ distributed delays}
\author[Q. Zhou, B. Xiao, Y. Yu\hfil EJDE-2006/19\hfilneg]
{Qiyuan Zhou, Bing Xiao, Yuehua Yu}

\address{Department of Mathematics,
 Hunan University of Arts and Science,
 Changde, Hunan 415000, China}
\email[Q. Zhou]{zhouqiyuan65@yahoo.com.cn} 
\email[B. Xiao]{changde1218@yahoo.com.cn} 
\email[Y. Yu]{jinli127@yahoo.com.cn}

\date{}
\thanks{Submitted December 3, 2005. Published February 7, 2006.}
\thanks{Supported by the NNSF of China and  by grant 05JJ40009
 from Hunan Provincial \hfill\break\indent
Natural Science Foundation of China.}
\subjclass[2000]{34C25, s4K13, 34K25}
\keywords{Shunting inhibitory cellular neural networks;
   almost periodic solution; \hfill\break\indent exponential stability;
   continuously  distributed delays}

\begin{abstract}
 In this paper, we consider shunting inhibitory cellular neural
 networks (SICNNs) with continuously distributed  delays.
 Sufficient conditions for the existence and local exponential
 stability of almost periodic solutions are established
 using a fixed point theorem, Lyapunov functional method, and
 differential inequality techniques. We illustrate our results
 with an example for which our conditions are satisfied, but
 not the conditions in \cite{c1,h1,l1}.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]

\section{Introduction}

 Consider the shunting inhibitory
cellular neural networks (SICNNs) with continuously distributed
 delays
\begin{equation}
x'_{ij}(t)=-a_{ij}(t)x_{ij}(t)-\sum_{C_{kl}\in
N_{r}(i,j)}C_{ij}^{kl}f(\int_{0}^{\infty}K_{ij}(u)
x_{kl}(t-u)du)x_{ij}(t)+L_{ij}(t), \label{e1.1}
\end{equation}
where $ i=1,  2,  \dots, m$, $j=1,2,\dots,n$, $C_{ij}$ denote
the cell at the $(i,j)$ position of the lattice, the
$r$-neighborhood $N_{r}(i,j)$ of $C_{ij}$ is
$$
N_{r}(i,j)=\{C_{kl}:\max(|k-i|,|l-j|)\leq r,1\leq k\leq m,1\leq
l\leq n\}.
$$
$x_{ij}$ is the activity of the cell $C_{ij}$, $L_{ij}(t)$ is the
external input to $C_{ij}$,   $a_{ij}(t)>0$ represent the passive
decay rate of the cell activity, $C_{ij}^{kl}\geq0$ is the
connection or coupling strength of postsynaptic activity of the
cell transmitted to the cell $C_{ij}$, and the activity function
$f $ is a continuous function representing the output or firing
rate of the cell $C_{kl}$.

  Since Bouzerdout and Pinter in \cite{b1,b2,b3}   described
 SICNNs as a new cellular neural networks(CNNs), SICNNs  have been
extensively applied in psychophysics, speech, perception,
robotics, adaptive pattern recognition, vision, and image
processing. Hence, they have been the object of intensive analysis
by numerous
  authors in recent years. In particular, there have been extensive results on the
    problem of the existence and stability of periodic and almost
    periodic solutions of SICNNs  with constant time delays and
time-varying delays  in the literature. We refer the reader to
\cite{c1,h1,l1} and the references cited therein.
 Moreover,  in the above-mentioned literature, we observe that the
assumption
\begin{itemize}
\item[(T0)] there exists a nonnegative constant $M_{f}$ such
that $M_{f}=\sup_{x\in \mathbb{R}}|f(x)|$
\end{itemize}
has been considered as fundamental for the considered existence
and stability of periodic and almost periodic solutions of SICNNS.
However, to the best of our knowledge, few authors have considered
 SICNNS  without the  assumptions (T0). Thus, it is worth
while to continue to investigate the existence and stability of
almost periodic solutions of     SICNNS.

The main purpose of this paper is to obtain some sufficient
conditions for the existence  and stability and local exponential
stability of the almost periodic solutions for system
\eqref{e1.1}. By applying fixed point theorem, Lyapunov functional
method and differential inequality     techniques, we derive some
new sufficient conditions ensuring the existence and local
exponential stability  of  the almost periodic solution of system
\eqref{e1.1}, which are new and they complement previously known
    results. In particular, we do not need the assumption (T0).
Moreover, an example is also
 provided to illustrate the effectiveness of the new results.

    Throughout this paper, we set
$$\{x_{ij}(t)\}=(x_{11}(t),
\dots,x_{1n}(t),\dots,x_{i1}(t),\dots,x_{in}(t),\dots,
x_{m1}(t),\dots,x_{mn}(t)).
$$
  For all $x=\{x_{ij}(t) \} \in \mathbb{R}^{m\times n}$, we define the norm
$\|x\|=\max_{(i,j)}\{ |x_{ij}(t)|\}$. Set
\begin{align*}
B=\big\{&\varphi :\varphi=\{\varphi_{ij}(t)\}=(\varphi_{11}(t),\dots,
\varphi_{1n}(t),\dots,\\
&\varphi_{i1}(t),\dots,\varphi_{in}(t),
\dots,\varphi_{m1}(t),\dots,\varphi_{mn}(t))\big\},
\end{align*}
where $\varphi$ is an almost periodic function on $\mathbb{R}$. For
all $\varphi \in B$, we define induced module
$\|\varphi\|_{B}=\sup_{t\in \mathbb{R}}\|\varphi (t)\| $,
then $B$ is a Banach space.

The initial conditions associated with system \eqref{e1.1} are
\begin{equation}
x_{ij}(s)=\varphi_{ij}(s),\quad s\in (-\infty, 0], \; i=1,2,\dots,m,
\; j=1,2,\dots,n, \label{e1.2}
\end{equation}
where $  \varphi_{ij}(\cdot)   $ denotes real-valued bounded
continuous function defined on $(-\infty, 0]$.

    We also assume that the following conditions
\begin{itemize}
\item[(T1)]  For $i\in\{1,  2,  \dots,  m \}$, $j\in\{1,  2,
\dots,  n \}$, the delay kernels $K_{ij}:[0, \infty)\to \mathbb{R}$
are continuous, integrable and there exist nonnegative
constants $k_{ij}$ such that
$$
\int_{0}^{\infty}|K_{ij}(s)|ds\leq k_{ij}.
$$

\item[(T2)] For each $i\in\{1,  2, \dots,  m \}$,
$j\in\{1, 2, \dots,  n \}$, $L_{ij}(t)$ and $a_{ij}(t)$
    are almost periodic functions on $R$,  let
     $L^{+}_{ij}=\sup_{t\in \mathbb{R}}|L_{ij}(t)|$,   $0<a_{ij}=\inf_{t\in \mathbb{R}} a_{ij}(t) $.

\item[(T3)]  $f(0)=0$, $f:\mathbb{R}\to \mathbb{R}$
    is Lipschitz with Lipschitz constant  $\mu $, i.e.,
$$
    |f(u )-f(v )|\leq \mu |u -v |, \quad \mbox{for  all }
     u ,  v \in \mathbb{R}.
$$

\item[(T4)] there exist nonnegative constants $L, q$ and
$\delta$ such that
\begin{gather*}
L=\max_{(i,j)}\{\frac{L^{+}_{ij}}{a_{ij}}\}, \quad
 \delta=\max_{(i,j)}\Big\{
\frac{\mu k_{ij}\sum_{C_{kl}\in N_{r}(i,j)}C_{ij}^{kl}}{a_{ij}}\Big\}<1, \\
\frac{L}{(1-\delta)}\leq 1 , \quad
 q=2\delta \frac{ L}{(1-\delta)} <1.
\end{gather*}

\item[(T5)]  For $i\in\{1, 2, \dots,  m \}$, $j\in\{1, 2, \dots,  n \}$,
there exists a constant $\lambda_{0}>0$ such that
$$
\int_{0}^{\infty}|K_{ij}(s)|e^{\lambda_{0}s}ds<+\infty .
$$
\end{itemize}

\noindent {\bf Definition.} (see \cite{f1,h1})
 Let $u(t):\mathbb{R}\to    \mathbb{R}^{n}$ be continuous in $t$.
$u(t)$ is said to be almost  periodic on $\mathbb{R}$ if,
for each $\varepsilon>0$, the set $T(u,\varepsilon)=
    \{\delta:|u(t+\delta)-u(t)|<\varepsilon, \; \forall t\in \mathbb{R}\}$
is relatively dense;
  i.e., for all $\varepsilon>0$, it is possible to find a real
    number $l=l(\varepsilon)>0$, so that for any interval of length
    $l(\varepsilon)$, there exists a number
    $\delta=\delta(\varepsilon)$ in this interval such that
$|u(t+\delta)-u(t)|<\varepsilon$, for all $t\in \mathbb{R}$.


 The remaining part of this paper is organized as
follows. In Section 2, we shall derive new sufficient conditions
for checking the existence of almost periodic solutions. In
Section 3, we present some new sufficient conditions for  the
local exponential stability of the almost periodic solution of
\eqref{e1.1}. In Section 4, we shall give  an example to illustrate our
results obtained in previous sections.

\section{Existence of Almost Periodic Solutions}

\begin{theorem} \label{thm2.1}
 Under  conditions {\rm (T1)--(T4)}there exists a unique almost
periodic solution of \eqref{e1.1} in the
region $B^{*}=\{\varphi : \varphi \in B,\|\varphi-\varphi_{0}\|_{B}
\le \frac {\delta L}{1-\delta} \}$,  where
\begin{align*}
\varphi_{0}(t)&=\Big\{\int_{-\infty}^{t}e^{-\int_{s}^{t}a_{ij}(u)du})L_{ij}(s)ds
\Big\}\\
&=\Big(\int_{-\infty}^{t}e^{-\int_{s}^{t}a_{1j}(u)du}L_{11}(s)ds,\dots,
\int_{-\infty}^{t}e^{-\int_{s}^{t}a_{ij}(u)du}L_{ij}(s)ds,\\
&\quad
\dots,\int_{-\infty}^{t}e^{-\int_{s}^{t}a_{mn}(u)du}L_{mn}(s)ds
\Big).
\end{align*}
\end{theorem}

\begin{proof}  For  $\varphi \in B$, we
consider the almost periodic solution $x_{\varphi}(t)$ of
nonlinear almost periodic differential equation
\begin{equation}
\frac{dx_{ij}}{dt}=-a_{ij}(t)x_{ij}(t)- \sum_{C_{kl}\in
N_{r}(i,j)}C_{ij}^{kl}f(\int_{0}^{\infty}K_{ij}(u)
\varphi_{kl}(t-u)du)\varphi_{ij}(t)+L_{ij}(t), \label{e2.1}
\end{equation}
Because $\varphi_{ij}(t),L_{ij}(t)$, $i=1, 2, \dots, m$,
$j=1,2,\dots,n$,
    are almost periodic functions. By \cite[P. 90-120]{h2},
\eqref{e2.1} has a
unique almost periodic solution
\begin{equation}
\begin{aligned}
x_{\varphi}(t)&=\Big\{\int_{-\infty}^{t}e^{-\int_{s}^{t}a_{ij}(u)du}
\Big[-\sum_{C_{kl}\in N_{r}(i,j)}C_{ij}^{kl}\\
&\quad\times f\big(\int_{0}^{\infty}K_{ij}(u)
\varphi_{kl}(s-u)du\big)\varphi_{ij}(s)+L_{ij}(s)\Big]ds\Big\}.
\end{aligned}\label{e2.2}
\end{equation}
 Now, we define a mapping $T:B \to B$ by setting
$$
T(\varphi)(t)=x_{\varphi}(t), \quad  \forall \varphi \in B.
$$
Since  $B^{*}=\{\varphi:\varphi \in B,\|\varphi-\varphi_{0}\|_{B}
\le \frac {\delta L}{1-\delta} \}$,
it is easy to see that   $B^{*}$ is a closed convex subset of $B$.
According to the definition of the norm of Banach space $B$, we
have
\begin{equation}
\begin{aligned}
\|\varphi_{0} \|_{B}
&=\sup_{t \in \mathbb{R}}\max_{(i,j)}
\Big\{\int_{-\infty}^{t}L_{ij}(s)e^{-\int_{s}^{t}a_{ij}(u)du}ds \Big\}\\
&\le  \sup_{t \in \mathbb{R}}\max_{(i,j)}
\big\{\frac {L^{+}_{ij}}{a_{ij}}\big\}=\max_{(i,j)}
\big\{\frac{L^{+}_{ij}}{a_{ij}}\big\}=L.
\end{aligned} \label{e2.3}
\end{equation}
Therefore, for all $\varphi \in B^{*}$, we have
\begin{equation}
\| \varphi \|_{B} \leq \| \varphi-\varphi_{0}\|_{B}
+\| \varphi_{0} \|_{B}  \leq \frac{\delta L}{1-\delta}+L
=\frac{L}{1-\delta}. \label{e2.4}
\end{equation}
 In view of (T3), we have
\begin{equation}
|f(u )|=|f(u )-f(0)|\leq \mu |u |, \quad \forall
    u  \in \mathbb{R}. \label{e2.5}
\end{equation}
 Now, we prove that the mapping $T$ is a self-mapping from $B^{*}$
to $ B^{*}$.
In fact, for all $\varphi \in B^{*}$, together with \eqref{e2.4},
\eqref{e2.5} and $\frac { L}{1-\delta}\leq  1$, we obtain
\begin{align*}
&\| T\varphi -\varphi_{0}\|_{B}\\
&=\sup_{t \in \mathbb{R}}\max_{(i,j)}\Big\{|\int_{-\infty}^{t}
e^{-\int_{s}^{t}a_{ij}(u)du}  \sum_{C_{kl}\in
N_{r}(i,j)}C_{ij}^{kl}f(\int_{0}^{\infty}K_{ij}(u)
\varphi_{kl}(s-u)du)\varphi_{ij}(s)ds| \Big\}\\
& \le  \sup_{t \in
\mathbb{R}}\max_{(i,j)}\{\int_{-\infty}^{t}e^{-\int_{s}^{t}a_{ij}(u)du}\mu
\| \varphi \|_{B}\int_{0}^{\infty}|K_{ij}(u) |du
\sum_{C_{kl} \in N_{r}(i,j)}C_{ij}^{kl}|\varphi_{ij}(s)|ds \} \\
& \leq \sup_{t \in \mathbb{R}}\max_{(i,j)}\Big\{ \mu
k_{ij}\sum_{C_{kl}\in
N_{r}(i,j)}C_{ij}^{kl}\int_{-\infty}^{t}e^{-\int_{s}^{t}a_{ij}(u)du}ds
\|
\varphi \|^{2}_{B} \Big\}\\
&\leq\max_{(i,j)}\big\{\frac {\mu k_{ij}\sum_{C_{kl}\in
N_{r}(i,j)}C_{ij}^{kl}}{a_{ij}}\big\} \| \varphi \|^{2}_{B} \\
&=\delta\| \varphi \|^{2}_{B}\leq \delta(\frac {
L}{1-\delta})^{2}\leq \delta \frac { L}{1-\delta},
\end{align*}
which   implies $T(\varphi)(t)\in B^{*}$. So,  the mapping $T$ is
a mapping from $B^{*} $ to $B^{*}$. Next, we prove that the
mapping $T$ is a contraction mapping of the $B^{*}$. In fact,  for
all $\varphi, \psi \in B^{*}$, we have
\begin{align*}
&\|T(\varphi)-T(\psi)\|_{B}\\
&= \sup_{t \in \mathbb{R}}\|T(\varphi)(t)-T(\psi)(t)\|\\
&= \sup_{t \in \mathbb{R}}\max_{(i,j)}
\{|\int_{-\infty}^{t}e^{-\int_{s}^{t}a_{ij}(u)du}\sum_{C_{kl} \in
N_{r}(i,j)} C_{ij}^{kl} (f(\int_{0}^{\infty}K_{ij}(u)
\varphi_{kl}(s-u)du)
 \\
&\quad \times\varphi_{ij}(s)-f(\int_{0}^{\infty}K_{ij}(u)
\psi_{kl}(s-u)du) \psi_{ij}(s))ds| \}\\
&\leq  \sup_{t \in
\mathbb{R}}\max_{(i,j)}\{\int_{-\infty}^{t}e^{-\int_{s}^{t}a_{ij}(u)du}\sum_{C_{kl}
\in N_{r}(i,j)}C_{ij}^{kl} |f(\int_{0}^{\infty}K_{ij}(u)
\varphi_{kl}(s-u)du)\varphi_{ij}(s) \\
&\quad -f(\int_{0}^{\infty}K_{ij}(u) \psi_{kl}(s-u)du) \varphi_{ij}(s)
+f(\int_{0}^{\infty}K_{ij}(u) \psi_{kl}(s-u)du)
\varphi_{ij}(s)\\
&\quad -f(\int_{0}^{\infty}K_{ij}(u)\psi_{kl}(s-u)du)\psi_{ij}(s)|ds\}\,.
\end{align*}
In view of condition (T3), \eqref{e2.4}, \eqref{e2.5} and the above
inequality, we have
\begin{align*}
&\|T(\varphi)-T(\psi)\|_{B} \\
&\leq \sup_{t \in
\mathbb{R}}\max_{(i,j)}\{\int_{-\infty}^{t}e^{-\int_{s}^{t}a_{ij}(u)du}\sum_{C_{kl}\in
N_{r}(i,j)}C_{ij}^{kl}[ |f(\int_{0}^{\infty}K_{ij}(u)
\varphi_{kl}(s-u)du) \\
&\quad -f(\int_{0}^{\infty}K_{ij}(u)
\psi_{kl}(s-u)du)||\varphi_{ij}(s)|\\
&\quad + |f(\int_{0}^{\infty}K_{ij}(u)
\psi_{kl}(s-u)du)||\varphi _{ij}(s)-\psi_{ij}(s) |] ds \}
\\
&\leq \sup_{t \in
\mathbb{R}}\max_{(i,j)}\{\int_{-\infty}^{t}e^{-\int_{s}^{t}a_{ij}(u)du}\sum_{C_{kl}\in
N_{r}(i,j)}C_{ij}^{kl}[\int_{0}^{\infty}|K_{ij}(u)|\mu|
 \varphi_{kl}(s-u)  \\
&\quad - \psi_{kl}(s-u)
|du|\varphi_{ij}(s)|+\int_{0}^{\infty}|K_{ij}(u)|\mu
|\psi_{kl}(s-u)|du|\varphi _{ij}(s)-\psi_{ij}(s)|] ds \}
\\
&\leq  \sup_{t \in
\mathbb{R}}\max_{(i,j)}\{\int_{-\infty}^{t}e^{-\int_{s}^{t}a_{ij}(u)du}\sum_{C_{kl}\in
N_{r}(i,j)}C_{ij}^{kl}k_{ij}
\mu[ \|\varphi \|_{B}+\|\psi \|_{B}] \| \varphi-\psi \|_{B} ds \} \\
&\leq \max_{(i,j)} \{ \frac {\mu k_{ij}\sum_{C_{kl}\in N_{r}(i,j)}C_{ij}^{kl}}{a_{ij}}\frac {2 L}{(1-\delta)}\}  \| \varphi-\psi \|_{B}  \\
&=2\delta\frac { L}{(1-\delta)}\|\varphi-\psi\|_{B},
\end{align*}
i.e.
$$
\| T(\varphi)-T(\psi) \|_{B}\leq q \|\varphi-\psi\|_{B} .
$$
 Note that $q=2\delta\frac { L}{(1-\delta)}<1$, it is clear that
the mapping $T$ is a contraction. Therefore the mapping $T$
possesses a unique fixed point $\varphi^{*}\in B^{*}$,
 $T\varphi^{*}=\varphi^{*}$. By \eqref{e2.1}, $\varphi^{*}$ satisfies \eqref{e1.1}.
So $\varphi^{*}$ is an almost periodic solution of system \eqref{e1.1} in
$B^{*}$. The proof  is complete.
\end{proof}

\section{Stability of the almost periodic solution}

In this section, we establish some results for the uniqueness and
local exponential stability of the almost periodic solution of
system \eqref{e1.1} in the region $B^{*}$.

\begin{theorem} \label{thm3.1}
Let $\delta(1+2\frac{L}{1-\delta})< 1$ and suppose that
conditions {\rm (T1)--(T5)} hold.
 Then  \eqref{e1.1} has  exactly one
almost periodic solution $\varphi^{*}(t)=\{x^{*}_{ij}(t)
\}=\{\varphi^{*}_{ij}(t) \}$ in the region $B^{*}$.  Moreover,
$\varphi^{*}(t)$ is  locally exponentially stable, and the domain of
attraction of $\varphi^{*}(t)$ is the set
$$
G_{1}(\varphi^{*})=\{\varphi: \varphi\in C((-\infty, \ 0]; \
R^{mn} ), \ \|\varphi-\varphi^{*}\|=\sup_{-\infty\leq s\leq
0}\max_{(i,j)}|\varphi_{ij}(s)-\varphi_{ij}^{*}(s)|<1\};$$ namely,
there exist constants $\lambda>0$ and $M>1$ such that for every
solution $ Z(t)=\{x_{ij}(t)\} $ of system \eqref{e1.1} with any
initial value $ \varphi=\{\varphi_{ij}(t)\}\in
G_{1}(\varphi^{*})$,
$$
|x_{ij}(t)-x^{*}_{ij}(t)|\leq M \|\varphi-\varphi^{*}\|
e^{-\lambda t},
$$
 for all $t>0$,  $i= 1, 2, \dots,  m$, $j=1, 2, \dots, \ n$.
\end{theorem}

\begin{proof}
 From Theorem \ref{thm2.1}, system \eqref{e1.1} has exactly one almost periodic
solution $\varphi^{*}(t)=\{x^{*}_{ij}(t) \}=\{\varphi^{*}_{ij}(t)
\}$ in the region $B^{*}$. Let $ Z(t)=\{x_{ij}(t) \} $ be an
arbitrary solution of system \eqref{e1.1} with initial value $
\varphi=\{\varphi_{ij}(t) \} \in G_{1}(\varphi^{*})$.  Set
$y(t)=\{y_{ij}(t) \}=\{x_{ij}(t)-x^{*}_{ij}(t)
\}=Z(t)-\varphi^{*}(t)$. Then
\begin{equation}
\begin{aligned}
y'_{ij}(t)&=-a_{ij}(t)y_{ij}(t)-\sum_{C_{kl}\in
N_{r}(i,j)}C_{ij}^{kl}[f(\int_{0}^{\infty}K_{ij}(u)
x_{kl}(t-u)du)x_{ij}(t) \\
&\quad -f(\int_{0}^{\infty}K_{ij}(u) x^{*}_{kl}(t-u)du)x^{*}_{ij}(t)],
\end{aligned}\label{e3.1}
\end{equation}
for $i=1, 2, \dots, m$, $j=1,2,\dots,n$.
Since $\delta<1, \ \delta(1+2\frac{L}{1-\delta})< 1$, we can
easily obtain
\begin{align*}
a_{ij}&>\mu k_{ij}\sum_{C_{kl}\in N_{r}(i,j)}C_{ij}^{kl}+\mu
k_{ij}\sum_{C_{kl}\in N_{r}(i,j)}C_{ij}^{kl}\frac{
L}{1-\delta}\\
&\quad +\mu k_{ij}\sum_{C_{kl}\in
N_{r}(i,j)}C_{ij}^{kl}\frac{ L}{1-\delta},
\end{align*} %\label{e3.2}
where $i=1, 2, \dots, m$, $j=1,2,\dots,n$.
 Set
\begin{align*}
\Gamma_{ij}(\omega)
& =\omega- a_{ij}  +     \sum_{C_{kl}\in
N_{r}(i,j)}C_{ij}^{kl} [\mu\int_{0}^{\infty}|K_{ij}(u)|e^{\omega
u}du+\mu k_{ij} \frac{ L}{1-\delta}\\
&\quad +\mu\int_{0}^{\infty}|K_{ij}(u)|e^{\omega u}du   \frac {
L}{1-\delta}],
\end{align*}  % \label{e3.3}
  where $i=1,   2, \dots,m$, $j=1,2,\dots,n$.
 Clearly, $\Gamma_{ij}(\omega)$, $i=1,   2, \dots, m$, $j=1,2,\dots,n$,
are continuous functions on $[0, \lambda_{0}]$.
 Since
\begin{align*}
 \Gamma_{ij}(0) &=  -a_{ij}  +     \sum_{C_{kl}\in
N_{r}(i,j)}C_{ij}^{kl} [\mu\int_{0}^{\infty}|K_{ij}(u)|du+\mu
k_{ij} \frac{ L}{1-\delta} \\
&\quad +\mu\int_{0}^{\infty}|K_{ij}(u)|du \frac { L}{1-\delta}] \\
&\leq -a_{ij}+\mu k_{ij}\sum_{C_{kl}\in N_{r}(i,j)}C_{ij}^{kl}
 +\mu k_{ij}\sum_{C_{kl}\in N_{r}(i,j)}C_{ij}^{kl}\frac{L}{1-\delta} \\
&\quad +\mu k_{ij}\sum_{C_{kl}\in N_{r}(i,j)}C_{ij}^{kl}\frac{ L}{1-\delta}
 <0,
\end{align*} %{ e3.4}
where $i=1, 2, \dots,
m, j=1,2,\dots,n$. It follows that we can choose a positive
constant $\lambda\in [0, \lambda_{0}]$ such that
\begin{equation}
\begin{aligned}
 \Gamma_{ij}(\lambda)& =(\lambda- a_{ij} ) +  \sum_{C_{kl}\in
N_{r}(i,j)}C_{ij}^{kl} [\mu\int_{0}^{\infty}|K_{ij}(u)|e^{\lambda
u}du+\mu k_{ij} \frac{ L}{1-\delta} \\
&\quad +\mu\int_{0}^{\infty}|K_{ij}(u)|e^{\lambda u}du   \frac {
L}{1-\delta}]<0,
\end{aligned} \label{e3.5}
\end{equation}
 where $i=1,  2, \dots,m$, $j=1,2,\dots,n$.
 We consider the Lyapunov functional
\begin{equation}
 V_{ij}(t) =   |y_{ij}(t)|e^{\lambda t}, \quad i=1,   2, \dots, m,\;
 j=1,2,\dots,n.
 \label{e3.6}
\end{equation}
Calculating the upper right
derivative of $V_{ij}(t)$ along the solution $y(t)=\{y_{ij}(t)\}$
of system \eqref{e3.1} with the initial value
$\bar{\varphi}=\varphi-\varphi^{*}$, we have
\begin{equation}
\begin{aligned}
& D^+(V_{ij}(t)) \\
& \leq   - a_{ij} |y_{ij}(t)|e^{\lambda t}+  \sum_{C_{kl}\in
N_{r}(i,j)}C_{ij}^{kl}|f(\int_{0}^{\infty}K_{ij}(u)
x_{kl}(t-u)du)x_{ij}(t)     \\
& \quad -f(\int_{0}^{\infty}K_{ij}(u)
x^{*}_{kl}(t-u)du)x^{*}_{ij}(t)|e^{\lambda t}
     +\lambda |y_{ij}(t)|e^{\lambda t}\\
& = (\lambda- a_{ij} )|y_{ij}(t)|e^{\lambda t}+  \sum_{C_{kl}\in
N_{r}(i,j)}C_{ij}^{kl}|f(\int_{0}^{\infty}K_{ij}(u)
x_{kl}(t-u)du)y_{ij}(t)
      \\
  & \quad + [f(\int_{0}^{\infty}K_{ij}(u)x_{kl}(t-u)du)
-f(\int_{0}^{\infty}K_{ij}(u)x^{*}_{kl}(t-u)du)]
x^{*}_{ij}(t)|e^{\lambda t} ,
 \end{aligned}\label{e3.7}
\end{equation}
where  $i=1, 2, \dots, m$, $j=1,2,\dots,n$.
Let
$$
\|\varphi-\varphi^{*}\| =\sup_{-\infty\leq
s\leq0}\max_{(i,j)}|\varphi_{ij}(s)-\varphi_{ij}^{*}(s)|>0.
$$
Since $\|\varphi-\varphi^{*}\|<1$, we can choose a positive constant
$M>1$ such that
\begin{equation}
M\|\varphi-\varphi^{*}\| <1, \quad
(M\|\varphi-\varphi^{*}\|)^{2} <M\|\varphi-\varphi^{*}\|. \label{e3.8}
\end{equation}
It follows from \eqref{e3.6} that
$$
V_{ij}(t) =  |y_{ij}(t)|e^{\lambda t}< M \|\varphi-\varphi^{*}\|,
$$
 for  all $t\in (-\infty, 0]$, $i=1,   2, \dots,m$, $j=1,2,\dots,n$.
We claim that
\begin{equation}
V_{ij}(t)  =  |y_{ij}(t)|e^{\lambda t}< M \|\varphi-\varphi^{*}\|,
\label{e3.9}
\end{equation}
for all $t>0$, $i=1,   2, \dots, m$, $j=1,2,\dots,n$.
Contrarily, there must exist $ij\in \{ 11,  12,  \dots,
    1n, \dots, m1,  m2,  \dots,     mn \}$ and $t_{ij}>0$ such that
\begin{equation}
V_{ij}(t_{ij})=M \|\varphi-\varphi^{*}\|,\quad
V_{\overline{ij }}(t)<M \|\varphi-\varphi^{*}\| , \forall  t\in
(-\infty, \ t_{ij}), \label{e3.10}
\end{equation}
where $\overline{ij } \in \{ 11,  12,  \dots,
    1n, \dots, m1,  m2,  \dots,     mn \}$.
It follows from  \eqref{e3.10} that
\begin{equation}
V_{ij}(t_{ij})-M \|\varphi-\varphi^{*}\| =0 ,\quad
V_{\overline{ij } }(t)-M \|\varphi-\varphi^{*}\| <0, \label{e3.11}
\end{equation}
for all $t\in (-\infty, t_{ij})$, where
$\overline{ij } \in \{ 11,  12,  \dots,  1n, \dots, m1,  m2,  \dots, mn \}$.
 From  \eqref{e2.4},  \eqref{e2.5}, \eqref{e3.7} ,  \eqref{e3.8}
and \eqref{e3.11}, we obtain
\begin{align*}
&0\leq   D^+(V_{ij}(t_{ij})-M \|\varphi-\varphi^{*}\| ) \\
&=  D^+(V_{ij}(t_{ij})) \\
& \leq   (\lambda- a_{ij} )|y_{ij}(t_{ij})|e^{\lambda t_{ij}}+
     \sum_{C_{kl}\in
N_{r}(i,j)}C_{ij}^{kl}|
f(\int_{0}^{\infty}K_{ij}(u)x_{kl}(t_{ij}-u)du)y_{ij}(t_{ij})
      \\
& \quad +
[f(\int_{0}^{\infty}K_{ij}(u)x_{kl}(t_{ij}-u)du)-f(\int_{0}^{\infty}K_{ij}(u)x^{*}_{kl}
(t_{ij}-u)du)]x^{*}_{ij}(t_{ij})|e^{\lambda t_{ij}}
\\
& \leq   (\lambda- a_{ij} )M \|\varphi-\varphi^{*}\| + \!\sum_{C_{kl}\in
 N_{r}(i,j)}C_{ij}^{kl}[\mu\int_{0}^{\infty}|K_{ij}(u)||x_{kl}(t_{ij}-u)|
 du|y_{ij}(t_{ij})|e^{\lambda t_{ij}}      \\
& \quad +\int_{0}^{\infty}|K_{ij}(u)|e^{\lambda u}
  \mu|y_{kl}(t_{ij}-u)|e^{\lambda
 (t_{ij}-u)}du|x^{*}_{ij}(t_{ij})|]\\
& \leq   (\lambda- a_{ij} )M \|\varphi-\varphi^{*}\| +  \sum_{C_{kl}\in
 N_{r}(i,j)}C_{ij}^{kl}[\mu\int_{0}^{\infty}|K_{ij}(u)|(|x_{kl}(t_{ij}-u)-x^{*}_{ij}(t_{ij}-u)|\\
& \quad +|x^{*}_{ij}(t_{ij}-u)|) du|y_{ij}(t_{ij})|e^{\lambda
t_{ij}} \\
&\quad +\int_{0}^{\infty}|K_{ij}(u)|e^{\lambda u}
\mu|y_{kl}(t_{ij}-u)|e^{\lambda
(t_{ij}-u)}du|x^{*}_{ij}(t_{ij})|]\\
& \leq  (\lambda- a_{ij} )M
\|\varphi-\varphi^{*}\| +
     \sum_{C_{kl}\in
N_{r}(i,j)}C_{ij}^{kl}[\mu\int_{0}^{\infty}|K_{ij}(u)|e^{\lambda
u}|y_{kl}(t_{ij}-u)|e^{\lambda
(t_{ij}-u)}du\\
     & \quad  \times |y_{ij}(t_{ij})|e^{\lambda t_{ij}}e^{-\lambda
t_{ij}}+ \mu\int_{0}^{\infty}|K_{ij}(u)||x^{*}_{ij}(t_{ij}-u)|
du|y_{ij}(t_{ij})|e^{\lambda
t_{ij}}\\
      & \quad +\int_{0}^{\infty}|K_{ij}(u)|e^{\lambda u}
\mu|y_{kl}(t_{ij}-u)|e^{\lambda (t_{ij}-u)}du|x^{*}_{ij}(t_{ij})|]\\
& \quad   (\lambda- a_{ij} )M
\|\varphi-\varphi^{*}\| +      \sum_{C_{kl}\in
N_{r}(i,j)}C_{ij}^{kl}[\mu\int_{0}^{\infty}|K_{ij}(u)|e^{\lambda
u}du(M \|\varphi-\varphi^{*}\| )^{2}e^{-\lambda
t_{ij}}\\
      & \quad   + \mu\int_{0}^{\infty}|K_{ij}(u)|
du \frac { L}{1-\delta}  M
\|\varphi-\varphi^{*}\|+\mu\int_{0}^{\infty}|K_{ij}(u)|e^{\lambda u}
 du \frac { L}{1-\delta} M
\|\varphi-\varphi^{*}\|]\\
& \quad  \{(\lambda- a_{ij} ) +   \sum_{C_{kl}\in
N_{r}(i,j)}C_{ij}^{kl} [\mu\int_{0}^{\infty}|K_{ij}(u)|e^{\lambda
u}du+\mu k_{ij} \frac{ L}{1-\delta} \\
   & \quad  +\mu\int_{0}^{\infty}|K_{ij}(u)|e^{\lambda
u}du   \frac { L}{1-\delta}] \}M \|\varphi-\varphi^{*}\| .
\end{align*} %\eqref{e3.12}
Therefore,
\begin{align*}
0 &\leq(\lambda- a_{ij} ) +  \sum_{C_{kl}\in
N_{r}(i,j)}C_{ij}^{kl} [\mu\int_{0}^{\infty}|K_{ij}(u)|e^{\lambda
u}du+\mu k_{ij} \frac{ L}{1-\delta}\\
&\quad +\mu\int_{0}^{\infty}|K_{ij}(u)|e^{\lambda u}du   \frac {
L}{1-\delta}],
\end{align*}
which contradicts \eqref{e3.5}. Hence, \eqref{e3.9} holds.
It follows that
$$
|x_{ij}(t)-x^{*}_{ij}(t)|=|y_{ij}(t)|<M
\|\varphi-\varphi^{*}\|e^{-\lambda t}, %\eqno \eqref{e3.13}
$$
for $ t>0$,  $i=1, 2,\dots, m$, $j=1,2,\dots,n$.
This completes the proof.
\end{proof}


\section{Example }

To illustrate the results obtained in previous sections we present the
following example.
Consider the  shunting inhibitory cellular neural
 network with  delays
$$
\frac{dx_{ij}}{dt}=-a_{ij}(t)x_{ij}- \sum_{C_{kl}\in
N_{r}(i,j)}C_{ij}^{kl}f(\int_{0}^{\infty}K_{ij}(u)
x_{kl}(t-u)du)x_{ij}+L_{ij}(t), \label{e4.1}
$$
where $i=1,2,3$, $j=1,2,3$,
\begin{gather*}
\begin{bmatrix}
a_{11}(t)&a_{12}(t)&a_{13}(t)\\
a_{21}(t)&a_{22}(t)&a_{23}(t)\\
a_{31}(t)&a_{32}(t)&a_{33}(t)
\end{bmatrix}
=
\begin{bmatrix}
 1+|\sin t|&1+|\sin t|&3+|\sin t|\\
 3+|\sin t|&1+|\sin t|&3+|\sin t|\\
 3+|\sin t|&1+|\sin t|&3+|\sin t|
\end{bmatrix}, \\ %\eqno (4.2)
\begin{bmatrix}
C_{11}&C_{12}&C_{13}\\
C_{21}&C_{22}&C_{23}\\
C_{31}&C_{32}&C_{33}
\end{bmatrix}
=
\begin{bmatrix}
0.1&0.2&0.1\\
0.2&0&0.2\\
0.1&0.2&0.1
\end{bmatrix} ,\\ % \eqno (4.3)
\begin{bmatrix}
L_{11}&L_{12}&L_{13}\\
L_{21}&L_{22}&L_{23}\\
L_{31}&L_{32}&L_{33}
\end{bmatrix}
=
\begin{bmatrix}
0.5\sin t&0.5\cos t&0.2\sin t\\
0.4\cos t&0.2\sin t&0.3\sin t\\
0.4\cos t&0.6\sin t&0.2\cos t
\end{bmatrix}.
\end{gather*}\\ % \eqno (4.4)
Set $r=1$,  $K_{ij}(u)=(\sin u)e^{-u}$, $i=1,2,3$, $j=1,2,3$, and
$f(x)=\frac{1}{10}x$. Clearly
$\mu=0.1$, $\sum_{C_{kl}\in N_{1}(1,1)}C_{11}^{kl}=0.5$,
\begin{gather*}
\sum_{C_{kl}\in N_{1}(1,2)}C_{12}^{kl}=0.8\,,\quad
\sum_{C_{kl}\in N_{1}(1,3)}C_{13}^{kl}=0.5\,,\\
\sum_{C_{kl}\in N_{1}(2,1)}C_{21}^{kl}=0.8\,,\quad
\sum_{C_{kl}\in N_{1}(2,2)}C_{22}^{kl}=1.2\,,\\
\sum_{C_{kl}\in N_{1}(2,3)}C_{23}^{kl}=0.8\,,\quad
\sum_{C_{kl}\in N_{1}(3,1)}C_{31}^{kl}=0.5\,,\\
\sum_{C_{kl}\in N_{1}(3,2)}C_{32}^{kl}=0.8\,,\quad
\sum_{C_{kl}\in N_{1}(3,3)}C_{33}^{kl}=0.5\,,\\
\sum_{(i,j)}\sum_{C_{kl}\in N_{1}(i,j)}C_{ij}^{kl}=6.4\,,
\end{gather*}
 $k_{ij}=1$, $i=1,2,3$, $j=1,2,3$,
\begin{gather*}
\delta  =  \max_{(i,j)}
\big\{\frac{\mu\sum_{C_{kl}\in N_{1}(i,j)}C_{ij}^{kl}}{a_{ij}}\big\}=0.12<1 ,\\
L =  \max_{(i,j)}\big\{ \frac{L^{+}_{ij}}{a_{ij}}\big\}=0.6, \quad
 \frac { L}{(1-\delta)}=\frac{0.6}{1-0.12}<1, \\
q= 2\delta\frac { L}{(1-\delta)}= 2\times 0.12
\times\frac{0.6}{1-0.12}<1, \\
\delta(1+2\frac{L}{1-\delta})=0.12(1+2\times 0.12
\times\frac{0.6}{1-0.12})< 1\,.
\end{gather*}
By theorem \ref{thm3.1},   the system \eqref{e4.1} has a unique
almost periodic solution $\varphi^{*}(t)$ in the region
$\| \varphi-\varphi_{0}\|_{B} \leq 0.08128$.
Moreover, $\varphi^{*}(t)$ is  locally exponentially stable, the
domain of attraction of $\varphi^{*}(t)$ is the set
 $G_{1}(\varphi^{*})$.

We remark that   System \eqref{e4.1} is a very simple form of SICNNs,
and that it does not not satisfy the condition (T0).
Therefore, the results in \cite{c1,h1,l1} can not be applied to this system.
This implies that the results of this paper are essentially new.

\subsection*{Conclusion}
The shunting inhibitory cellular neural networks
with continuously distributed  delays have been studied.
Some sufficient conditions  for the existence and local exponential
stability of  almost periodic solutions have been
established. The obtained results  are new and  complement  previously
known results. Moreover, an example  is given to illustrate
our results.

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\end{document}