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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 99, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/99\hfil Existence and exponential stability]
{Existence and exponential stability for anti-periodic
 solutions for shunting inhibitory cellular neural networks
 with continuously distributed delays}

\author[Y. Wu,  Z. Zhou\hfil EJDE-2009/99\hfilneg]
{Yuanheng Wu, Zhan Zhou}  % in alphabetical order

\address{Yuanheng Wu \newline
College of Mathematics and Information Sciences,
Guangzhou University\\
Guangzhou 510006, China}
\email{wyhcd2006@yahoo.com.cn}

\address{Zhan Zhou \newline
College of Mathematics and Information Sciences,
Guangzhou University\\
Guangzhou 510006, China}
\email{zzhou@gzhu.edu.cn}

\thanks{Submitted July 25, 2009. Published August 19, 2009.}
\subjclass[2000]{34C25, 34K13, 34K25}
\keywords{Shunting inhibitory cellular neural networks;
 anti-periodic solution; \hfill\break\indent
exponential stability; continuously distributed delays}

\begin{abstract}
 This article concerns anti-periodic solutions for shunting
 inhibitory cellular neural networks (SICNNs), with continuously
 distributed delays, arising from the description of the neurons
 state in  delayed neural networks. Without assuming global
 Lipschitz conditions of activation functions, we obtain
 existence and local exponential stability of anti-periodic
 solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}

 \section{Introduction}

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

   It is well known that studies on SICNNs not only involve a
discussion of stability properties, but also involve many dynamic
behaviors such as periodic oscillatory behavior and almost
periodic oscillatory properties. Therefore,  considerable effort
has been devoted to study dynamic behaviors, in particular, the
existence and stability of   periodic and almost periodic solutions
of SICNNs in the literature (see, e.g.,
\cite{c1,c2,c3,h2,l1,l2}
and the references therein).
Recently, in \cite{p1,s1} the authors studied the existence and stability
of anti-periodic solutions for SICNNs with the following assumption:
\begin{itemize}
\item[(T0)] The function  $f$ is  global Lipschitz continuous;
that is, there exists a constant $\mu>0$ such that for all
$x,y \in \mathbb{R}$,
$$
|f(x)-f(y)|\leq \mu |x-y|.
$$
\end{itemize}
To the best of our knowledge, very few authors have considered
problems of anti-periodic solutions of SICNNs  \eqref{e1.1}
without Assumption (T0). Moreover, it is well known that
the existence and stability of
anti-periodic solutions play a key role in characterizing the
behavior of nonlinear differential equations; see
\cite{a1,a2,c4,w1}.
Since SICNNs can be analog voltage transmission, and voltage
transmission process often a anti-periodic process. Thus, it is
worth    while to continue to investigate the existence and
stability of anti-periodic solutions of SICNNs \eqref{e1.1} without
Assumption (T0).


The purpose of this article is to present sufficient conditions for
the existence and local exponential stability of anti-periodic solutions
of \eqref{e1.1}. Moreover, we do not assume (T0).
An example is provided to illustrate our results.

Let $u(t):\mathbb{R}\to \mathbb{R}$ be continuous in $t$.
$u(t)$ is said to be $T$-anti-periodic on $\mathbb{R}$ if,
$$
u(t+T)=-u(t) \quad \text{for  all }  t\in \mathbb{R}.
$$
In this article, for $i=1, 2, \dots, m$,
$j=1,  2,  \dots,  n$, it is assumed that
$a_{ij}, C^{kl}_{ij}:\mathbb{R}\to [0,+\infty) $,
$K_{ij} :[0, +\infty)\to \mathbb{R} $ and
$L_{ij}:\mathbb{R}\to \mathbb{R} $ are continuous  functions,
and
\begin{equation}
\begin{gathered}
a_{ij}(t+T)=a_{ij}(t), \quad
C^{kl}_{ij}(t+T)=C^{kl}_{ij}(t),  \\
f(-u)=f( u), \quad
L_{ij}(t+T) =-L_{ij}(t ) ,
\end{gathered} \label{e1.2}
\end{equation}
 for all $t,u \in \mathbb{R}$.
Let
$$
\{x_{ij}(t)\}=(x_{11}(t),
\dots,x_{1n}(t),\dots,x_{i1}(t),\dots,x_{in}(t),\dots,
x_{m1}(t),\dots,x_{mn}(t))
$$
be an element in $\mathbb{R}^{m\times n}$.
For  $x(t)=\{x_{ij}(t) \}$ in $\mathbb{R}^{m\times n}$,
define the norm
$\|x(t)\|=\max_{(i,j)}\{ |x_{ij}(t)|\}$.

  We shall use the following conditions
\begin{itemize}
\item[(H1)]   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 and integrable;

\item[(H2)]   there exists a function $L:\mathbb{R}^{+}\to\mathbb{R}^{+}$
 such that for each $r>0$,
\begin{equation}
|f(u )-f(v )|   \leq L(r)|u -v |,\quad |u|,|v|\leq r.
\label{e1.3}
\end{equation}

\item[(H3)]   there exists a constant $r_{0}>0$ such that
$$
D[|f(0)|r_{0}+L(r_{0})r^{2}_{0}]+L^{+}\leq r_{0} ,
$$
where
\begin{gather*}
D=\max_{(i,j)}\big\{\frac{\sum_{C_{kl}\in{N_{r}(i,j)}}
    \bar{C}_{ij}^{kl} \int_{0}^{\infty}|K_{ij}(u)|du}
{\underline{a}_{ij}}\big\}>0,\quad
 L^{+}=\max_{(i,j)}\frac{\overline{L}_{ij}}{\underline{a}_{ij}}, \\
\overline{L}_{ij}=\sup_{t\in{\mathbb{R}}}|L_{ij}(t)|, \quad
\bar{C}_{ij}^{kl}=\sup_{t\in \mathbb{R}}C_{ij}^{kl}(t), \quad
\underline{a}_{ij}=\inf_{t\in \mathbb{R}}a_{ij}(t)>0.
\end{gather*}
\end{itemize}

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{e1.4}
\end{equation}
where $ \varphi_{ij}(\cdot)$ denotes real-valued bounded
continuous function defined on $(-\infty, 0]$.

The remaining parts of this paper are organized as follows. In
Section 2, sufficient conditions are derived for the boundedness
of solution of \eqref{e1.1}. In Section 3, we present  sufficient
conditions for the existence and local exponential stability of
anti-periodic   solution of \eqref{e1.1}. In Section 4, an illustrative
example is given to show the proposed theory and method.


\section{Preliminary Results}

The following lemmas will be used to prove our main results in
    Section 3.

\begin{lemma} \label{lem2.1}
 Assume {\rm (H1)--(H3)}. Suppose that
 $\widetilde{x}(t)= \{\widetilde{x}_{ij}(t)\} $ is a
solution of \eqref{e1.1} with initial conditions
\begin{equation}
\widetilde{x}_{ij}(s)=\widetilde{\varphi}_{ij}(s), \quad
|\widetilde{\varphi}_{ij}(s)|< r_{0} , \quad  s\in (-\infty, 0],\;
ij=11,  12,  \dots, mn \label{e2.1}
\end{equation}
Then
\begin{equation}
|\widetilde{x}_{ij}(t)| <r_{0}, \quad\text{where }
 t\geq 0, ij=11,12,\dots,mn. \label{e2.2}
\end{equation}
\end{lemma}

\begin{proof}
 Assume, by way of contradiction, that \eqref{e2.2} does not hold.
Then, there exist  $ij\in \{11,  12,  \dots,    mn\}$ and $\rho>0$ such
that
\begin{equation}
\widetilde{x}_{ij}(\rho) =r_{0}, \quad
\widetilde{x}_{\bar{ij}}(t) <r_{0} \quad\text{for all }
t\in (-\infty, \rho),\; \bar{ij}=11,  12,  \dots,  mn. \label{e2.3}
\end{equation}
Calculating the the upper left
derivative of $|\widetilde{x}_{ij}(t)|$,   together with
(H1)-(H3), \eqref{e1.3} and \eqref{e2.3},  we  obtain
\begin{align*}
0 &\leq D^+(|\widetilde{x}_{ij}(\rho)|) \\
&\leq -a_{ij}(\rho)\widetilde{x}_{ij}(\rho)+|\sum_{C_{kl}\in
 N_{r}(i,j)}C_{ij}^{kl}(\rho)\int_{0}^{\infty}K_{ij}(u)
 f(\widetilde{x}_{kl}(\rho-u))du\widetilde{x}_{ij}(\rho)+L_{ij}(\rho)|\\
&\leq   -\underline{a}_{ij} r_{0}+\sum_{C_{kl}\in
 N_{r}(i,j)}\bar{C}_{ij}^{kl}\int_{0}^{\infty}|K_{ij}(u)|
 (|f(0)|+L(r_{0})|\widetilde{x}_{kl}(\rho-u) |)du r_{0}+|L_{ij}(\rho)|\\
&\leq  \underline{a}_{ij}(- r_{0}  + \frac{\sum_{C_{kl}\in
 N_{r}(i,j)}\bar{C}_{ij}^{kl}\int_{0}^{\infty}|K_{ij}(u)|}
 {\underline{a}_{ij}}(|f(0)|r_{0}+L(r_{0})r^{2} _{0} )
 +\frac{\overline{L}_{ij}}{\underline{a}_{ij}})
< 0.
\end{align*}
This is a contradiction and hence \eqref{e2.2} holds.
This completes the proof.
\end{proof}

\begin{remark} \label{rmk2.1} \rm
 Under Conditions (H1)--(H3),   the solution of \eqref{e1.1}
always exists \cite{h1}.
In view of the boundedness of this solution, from the theory of
functional differential equations in \cite{h1},
it follows that $\widetilde{x}(t)$ can be defined on
$(-\infty,\infty)$.
\end{remark}

\begin{lemma} \label{lem2.2}
Assume {\rm (H1)-(H3)}, and let
 $x^{*}(t)= \{x^{*}_{ij}(t)\} $ be the   solution of \eqref{e1.1}
with initial value
$\varphi^{*}=\{\varphi^{*} _{ij}(t)\}$, where
\begin{equation}
|\varphi^{*}_{ij}(s)|< r_{0} , \quad s\in (-\infty, 0],\;
 ij=11, 12,  \dots,     mn. \label{e2.4}
\end{equation}
Also assume that
\begin{itemize}
 \item[(H4)] there exists a constant  $r_{1}\geq r_{0}$ such that
 $$
|f(0)|+L(r_{0})r_{0}+L(r_{1})r_{1}<\frac{1}{D};
$$
\item[(H5)] 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}
Then there is a positive constant $\lambda$ such that for every
solution $x(t)=\{x_{ij}(t)\}$ of  \eqref{e1.1} with any initial
value $\sup_{t\in (-\infty, 0]}\|\varphi (t)\|\leq r_{1}$,
$$
\|x(t)-x^{*}(t)\|\leq M  e^{-\lambda t}, \quad \forall t>0,
$$
where $ M =\sup_{ -\infty\leq t\leq 0}\|\varphi (t)-x^{*}(t)\|$.
\end{lemma}

\begin{proof}
In view \eqref{e2.4} and Lemma \ref{lem2.1}, we have
\begin{equation}
|x^{*}_{ij}(t)| < r_{0}, \quad\text{for  all }
 t\in \mathbb{R},\; ij=11,  12,   \dots,     mn. \label{e2.5}
\end{equation}
Set
\begin{align*}
\Gamma_{ij}(\alpha)
&=\alpha-\underline{a}_{ij}
 +\sum_{C_{kl}\in{N_{r}(i,j)}}\bar{C}_{ij}^{kl}(|f(0)|
 +L(r_{0})r_{0}\int_{0}^{\infty}|K_{ij}(s)| ds\\
&\quad +L(r_{1})r_{1}\int_{0}^{\infty}|K_{ij}(s)|e^{\alpha s}ds)
\end{align*}
where $i=1,2,\dots,m$ and $ j=1,2,\dots,n$. It is not difficult
to prove that $\Gamma_{ij}(ij=11,  12,   \dots, mn)$ are
continuous functions on $[0,\lambda_{0}]$.
Moreover, by  (H4) and (H5), we have
$$
\Gamma_{ij}(0)=-\underline{a}_{ij}
 +\sum_{C_{kl}\in{N_{r}(i,j)}}\bar{C}_{ij}^{kl}(|f(0)|
 +L(r_{0})r_{0}+L(r_{1})r_{1})\int_{0}^{\infty}|K_{ij}(s)| ds<0
$$
Thus, there exists a sufficiently small constant $\lambda>0$ such
that
\begin{equation}
\Gamma_{ij}(\lambda)<0,\quad ij=11,  12,   \dots,    mn.\label{e2.6}
\end{equation}
Take $\varepsilon>0$, and set
$$
z_{ij}(t)=|x_{ij}(t)-x^{*}_{ij}(t)|e^{\lambda t},\quad
ij=11,  12,   \dots,    mn.
$$
It follows that
$$
z_{ij}(t)\leq M<M+\varepsilon, \quad \forall t\in(-\infty,0],\;
 ij=11,  12,   \dots,     mn.
$$
Now, we claim that
\begin{equation}
z_{ij}(t)\leq M+\varepsilon,\quad \forall t>0,\; ij=11,  12,   \dots,
    mn.\label{e2.7}
\end{equation}
If this is not true, then there
exists $i_{0}\in\{1,2,\dots,m\}$ and
$j_{0}\in\{1,2,\dots,n\}$ such that
\begin{equation}
\{t>0|z_{i_{0}j_{0}}(t)> M+\varepsilon\}\neq \phi\label{e2.8}
\end{equation}
Let
 $$
t_{ij}= \begin{cases}
\inf\{t>0|z_{ij}(t)> M+\varepsilon\},
&\text{if }\{ t>0: z_{ij}(t)> M+\varepsilon\}\neq \emptyset\\
 +\infty,& \text{otherwise}.
\end{cases}
$$
Then $t_{ij}>0$ and
\begin{equation}
z_{ij}(t)\leq M+\varepsilon,\quad \forall t\in(-\infty,t_{ij}],\;
ij=11,  12,   \dots,    mn.\label{e2.9}
\end{equation}
We denote
$t_{pq}=\min_{(i,j)}t_{ij}$, where $p\in\{1,2,\dots,m\}$
and $q\in\{1,2,\dots,n\}$. In view of \eqref{e2.8}, we have
$0<t_{pq}<+\infty$. It follow from \eqref{e2.9}, we have
\begin{equation}
z_{ij}(t)\leq M+\varepsilon,\quad \forall t\in(-\infty,t_{pq}],\;
ij=11,  12,   \dots,    mn. \label{e2.10}
\end{equation}
In addition, noting that
$t_{pq}=\inf\{t>0:z_{pq}(t)> M+\varepsilon\}$, we obtain
\begin{equation}
z_{pq}(t_{pq})=M+\varepsilon,\quad
D^{+}z_{pq}(t_{pq})\geq 0.\label{e2.11}
\end{equation}
Since $x(t)$ and $x^{*}(t)$ are solutions of \eqref{e1.1}, we have
\begin{align*}
0 &\leq D^{+}z_{pq}(t_{pq})\\
&= D^{+}[|x_{pq}(t)-x^{*}_{pq}(t)|e^{\lambda t}]|_{t=t_{pq}}\\
&\leq |x_{pq}(t_{pq})-x^{*}_{pq}(t_{pq})|\lambda e^{\lambda
t_{pq}}-\underline{a}_{pq}|x_{pq}(t_{pq})-x^{*}_{pq}(t_{pq})|e^{\lambda t_{pq}}\\
&\quad +\sum_{C_{kl}\in{N_{r}(p,q)}}\bar{C}_{pq}^{kl}|\int_{0}^{\infty}K_{ij}(u)
f( x_{kl}(t_{pq}-u))dux_{pq}(t_{pq})\\
&\quad -\int_{0}^{\infty}K_{ij}(u) f(x^{*}_{kl}(t_{pq}-u))du
x^{*}_{pq}(t_{pq})|e^{\lambda t_{pq}}
\\
&= |x_{pq}(t_{pq})-x^{*}_{pq}(t_{pq})|\lambda e^{\lambda
t_{pq}}-\underline{a}_{pq}|x_{pq}(t_{pq})-x^{*}_{pq}(t_{pq})|e^{\lambda t_{pq}}\\
&\quad +\sum_{C_{kl}\in{N_{r}(p,q)}}\bar{C}_{pq}^{kl}|\int_{0}^{\infty}K_{ij}(u)
f(x^{*}_{kl}(t_{pq}-u) du (x_{pq}(t_{pq})-x^{*}_{pq}(t_{pq})) \\
&\quad +\int_{0}^{\infty}K_{ij}(u)(f(x _{kl}(t_{pq}-u))-
f(x^{*}_{kl}(t_{pq}-u)))dux _{pq}(t_{pq}) |e^{\lambda
t_{pq}}\\
&\leq (\lambda-\underline{a}_{pq})z_{pq}(t_{pq})
 +\sum_{C_{kl}\in{N_{r}(p,q)}}\bar{C}_{pq}^{kl}
\int_{0}^{\infty}|K_{ij}(u)|| f(x^{*}_{kl}(t_{pq}-u)|du\\
&\quad\times |x_{pq}(t_{pq})-x^{*}_{pq}(t_{pq})|e^{\lambda t_{pq}}
\\
&\quad +\sum_{C_{kl}\in{N_{r}(p,q)}}\bar{C}_{pq}^{kl}
\int_{0}^{\infty}|K_{ij}(u)||f(x _{kl}(t_{pq}-u))-
f(x^{*}_{kl}(t_{pq}-u))|du\\
&\quad \times |x_{pq}(t_{pq})|e^{\lambda t_{pq}}. %  \eqref{e2.12}
\end{align*}
Now, combining the above inequality, \eqref{e2.10}, \eqref{e2.11}, (H2)
 and (H3), we deduce
\begin{align*}
0 &\leq   D^+z_{pq }(t_{pq })\\
&\leq (\lambda-\underline{a}_{pq})(M+\varepsilon)
 +\sum_{C_{kl}\in{N_{r}(p,q)}}\bar{C}_{pq}^{kl}
(|f(0)|+L(r_{0})r_{0})\int_{0}^{\infty}|K_{ij}(u)|du\cdot z_{pq}(t_{pq}) \\
& \quad +  \sum_{C_{kl}\in{N_{r}(p,q)}}\bar{C}_{pq}^{kl}
L(r_{1})\int_{0}^{\infty}|K_{ij}(u)|| x _{kl}(t_{pq}-u) -
 x^{*}_{kl}(t_{pq}-u)|\\
&\quad\times e^{\lambda (t_{pq}-u)}e^{\lambda u}du\cdot r_{1}\\
&\leq (\lambda-\underline{a}_{pq})(M+\varepsilon)
 +\sum_{C_{kl}\in{N_{r}(p,q)}}\bar{C}_{pq}^{kl}
(|f(0)|+L(r_{0})r_{0})\cdot (M+\varepsilon) \\
& \quad +  \sum_{C_{kl}\in{N_{r}(p,q)}}\bar{C}_{pq}^{kl}
L(r_{1})r_{1}\int_{0}^{\infty}|K_{ij}(u)| e^{\lambda u}du (M+\varepsilon)\\
&\leq
(\lambda-\underline{a}_{pq})(M+\varepsilon)+\sum_{C_{kl}\in{N_{r}(p,q)}}\bar{C}_{pq}^{kl}
(|f(0)|\\
&\quad +L(r_{0})r_{0})\int_{0}^{\infty}|K_{ij}(u)|  du\cdot (M+\varepsilon) \\
& \quad +  \sum_{C_{kl}\in{N_{r}(p,q)}}\bar{C}_{pq}^{kl}
L(r_{1})r_{1}\int_{0}^{\infty}|K_{ij}(u)| e^{\lambda u}du\cdot
(M+\varepsilon)\,.
\end{align*}
It follow that
\begin{align*}
&\lambda-\underline{a}_{pq}+\sum_{C_{kl}\in{N_{r}(p,q)}}
 \bar{C}_{pq}^{kl}\cdot (|f(0)|\\
&+L(r_{0})r_{0}\int_{0}^{\infty}|K_{ij}(u)| du
+L(r_{1})r_{1}\int_{0}^{\infty}|K_{ij}(u)| e^{\lambda u}du)
\geq 0
\end{align*}
which  contradicts with \eqref{e2.6}. Hence, \eqref{e2.7} holds; i.e.
$$
|x_{ij}(t)-x^{*}_{ij}(t)|e^{\lambda t}=z_{ij}(t)\leq M+\varepsilon,
\quad \forall t>0,\; i=1,2,\dots,m, \; j=1,2,\dots,n.
$$
Therefore,
$$
\|x(t)-x^{*}(t)\|=\max_{(i,j)}|x_{ij}(t)-x^{*}_{ij}(t)|\leq
(M+\varepsilon)e^{-\lambda t},\quad \forall t>0.
$$
Letting $\varepsilon\to0^+$, we obtain
$$
\|x(t)-x^{*}(t)\|\leq M e^{-\lambda t}, \quad \forall t>0.
$$
The proof is complete.
\end{proof}

\begin{remark} \label{rmk2.2} \rm
Let $x^{*}(t)=(x^{*}_{11}(t), x^{*}_{12}(t),\dots,x^{*}_{mn}(t))^{T}
$ is the  $T$-anti-periodic solution of \eqref{e1.1}. It follows
from Lemma \ref{lem2.2}  that $x^{*}(t)$ is globally exponentially stable.
\end{remark}

\section{ Main Results}

\begin{theorem} \label{thm3.1}
Assume {\rm (H1)-(H5)}.  Then  \eqref{e1.1} has  at least one
$T$-anti-periodic  solution $x^{*}(t)$. Moreover, $x^{*}(t)$ is
 locally exponentially stable.
\end{theorem}

\begin{proof}.
 Let $v(t)= (v_{11}(t), v_{12}(t),\dots, v_{mn}(t))^{T} $ be
a solution of \eqref{e1.1} with initial conditions
\begin{equation}
v_{ij}(s)=\varphi^{v}_{i}(s), \quad
|\varphi^{v}_{ij}(s)|<r_{0},\quad
 s\in (-\infty, 0],\; ij=11,12,\dots,mn. \label{e3.1}
\end{equation}
By Lemma \ref{lem2.1}, the solution $v(t) $ is bounded and
\begin{equation}
|v_{ij}(t)|<r_{0} , \quad\text{ for  all  } \ t\in \mathbb{R},\;
ij=11,12,\dots,mn. \label{e3.2}
\end{equation}
 From \eqref{e1.1}  and \eqref{e1.2}, we have
\begin{equation}
\begin{aligned}
&((-1)^{k+1} v_{ij} (t + (k+1)T))'\\
&=(-1)^{k+1} v'_{ij} (t + (k+1)T)\\
&=(-1)^{k+1}\{-a_{ij}(t + (k+1)T))v_{ij}(t+ (k+1)T)-\sum_{C_{kl}\in
N_{r}(i,j)}C_{ij}^{kl}(t + (k+1)T))\\
& \quad \times \int_{0}^{\infty}K_{ij}(u)
f(v_{kl}(t+ (k+1)T-u))du v_{ij}(t + (k+1)T)+L_{ij}(t+ (k+1)T)\} \\
&= -a_{ij}(t  ))(-1)^{k+1}v_{ij}(t+ (k+1)T)-\sum_{C_{kl}\in
N_{r}(i,j)}C_{ij}^{kl}(t  ))\int_{0}^{\infty}K_{ij}(u) \\
& \quad\times f((-1)^{k+1}v_{kl}(t+ (k+1)T-u))du
(-1)^{k+1}v_{ij}(t + (k+1)T)+L_{ij}(t ) ,
\end{aligned}\label{e3.3}
\end{equation}
where $ij=11, 12, \dots, mn$. Thus, for any natural number $k$,
$(-1)^{k+1} v (t+ (k+1)T)$  are the solutions of \eqref{e1.1}.
Then, by Lemma \ref{lem2.2}, we get
\begin{equation}
\begin{aligned}
&|(-1)^{k+1} v_{ij} (t + (k+1)T)-(-1)^{k } v_{ij}  (t + kT)|\\
&\leq  e^{-\lambda (t + kT)}\sup_{-\infty\leq
s\leq0}\max_{(i,j)}|v_{ij} (s +  T)+ v_{ij} (s)| \\
&\leq   e^{-\lambda (t + kT)}   2r_{0}, \quad \forall t + kT>0, \;
ij=11, 12, \dots, mn.
\end{aligned} \label{e3.4}
\end{equation}
Thus, for any natural number $p $, we obtain
\begin{equation}
\begin{aligned}
&(-1)^{p+1} v_{ij} (t + (p+1)T)\\
&=       v_{ij} (t )  +\sum_{k=0}^{p}[(-1)^{k+1}
v _{ij}(t + (k+1)T)-(-1)^{k} v_{ij} (t + kT)].
\end{aligned} \label{e3.5}
\end{equation}
Then
\begin{equation}
\begin{aligned}
& |(-1)^{p+1} v_{ij} (t + (p+1)T)|\\
& \leq | v_{ij} (t )|  +\sum_{k=0}^{p}|
(-1)^{k+1} v _{ij}(t + (k+1)T)-(-1)^{k} v_{ij} (t + kT)|,
\end{aligned}\label{e3.6}
\end{equation}
where $ij =11,12,\dots,mn$.

In view of \eqref{e3.4}, we can choose a sufficiently large constant
$N>0$ and a positive constant $\alpha $ such that
\begin{equation}
|(-1)^{k+1} v_{ij} (t + (k+1)T)-(-1)^{k } v_{ij}  (t + kT)|
\leq   \alpha (e^{-\lambda T } )^{k}  , \label{e3.7}
\end{equation}
for all  $k >N$, $i=1, 2, \dots, n$,
on any compact set of $\mathbb{R}$. It
follows from \eqref{e3.6} and \eqref{e3.7} that
$ \{(-1)^{p} v (t + pT)\}$
uniformly converges to a continuous function
$x^{*}(t)=(x^{*}_{11}(t), x^{*}_{12}(t),\dots,x^{*}_{mn}(t))^{T}$
on any compact set of $\mathbb{R}$.

Now we show that $x^{*}(t)$ is $T$-anti-periodic solution of \eqref{e1.1}.
First, $x^{*}(t)$ is $T$-anti-periodic, since
$$
x^{*}(t+T)=\lim_{p\to \infty}(-1)^{p } v
(t +T+  p  T)=-\lim_{(p+1)\to \infty}(-1)^{p+1 } v
(t  + (p +1)T)=-x^{*}(t ).
$$
Next, we prove that $x^{*}(t)$ is a
solution of \eqref{e1.1}. In fact,  together with
the continuity of the right side of \eqref{e1.1},  \eqref{e3.3}
implies that $ \{((-1)^{p+1} v  (t + (p+1)T))'\}$  uniformly converges
to a continuous function on any
compact set of $\mathbb{R}$.  Thus,  letting $p \to\infty$,
we obtain
\begin{equation}
\begin{aligned}
\frac{d}{dt}\{x^{*}_{ij}(t)\}
&= -a_{ij}(t)x^{*}_{ij}(t )\\
&\quad -\sum_{C_{kl}\in N_{r}(i,j)}C_{ij}^{kl}(t)
\int_{0}^{\infty}K_{ij}(u) f(x^{*}_{kl}(t-u))du  x^{*}_{ij}(t ) +
L_{ij }(t ).
\end{aligned}    \label{e3.8}
\end{equation}
Therefore, $x^{*}(t)$ is a   solution of \eqref{e1.1}.
Finally, by Lemma \ref{lem2.2}, we can prove that $x^{*}(t)$ is  locally
exponentially stable.  This completes the proof.
\end{proof}

\section{An Example}
In this section, we give an example to illustrate the results
obtained in previous sections.
Consider the following 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}(t)
 \int_{0}^{\infty}K_{ij}(u)
f(x_{kl}(t-u))dux_{ij}(t)+L_{ij}(t), \label{e4.1}
\end{equation}
 where    $ i=1, 2, 3$, $j=1, 2, 3$,  $K_{ij}(u)=(\sin u)e^{-u}$,
$f(x)=\frac{x^{4}+1}{6}$,
\begin{gather*}
\begin{pmatrix}
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{pmatrix}
=\begin{pmatrix}
5+|\sin t| & 5 +|\sin  t|& 9+|\sin t|\\
6 +|\sin t|& 6 +|\sin t|& 7+|\sin t|\\
8 +|\sin t|& 8+|\sin t| & 5+|\sin  t|\end{pmatrix},
\\
\begin{pmatrix}
C_{11}(t)&C_{12}(t)&C_{13}(t)\\
C_{21}(t)&C_{22}(t)&C_{23}(t)\\
C_{31}(t)&C_{32}(t)&C_{33}(t)\end{pmatrix}
=\begin{pmatrix}
0.1 |\sin  t|& 0.3|\sin \sqrt{3}t| & 0.5|\sin  t|\\
0.2 |\sin  t|& 0.1 |\sin  t|& 0.2|\sin  t|\\
0.1|\sin  t| & 0.2 |\sin  t|& 0.1|\sin
\sqrt{3}t|\end{pmatrix},\\
\begin{pmatrix}
L_{11}(t)&L_{12}(t)&L_{13}(t)\\
L_{21}(t)&L_{22}(t)&L_{23}(t)\\
L_{31}(t)&L_{32}(t)&L_{33}(t)\end{pmatrix}
=\begin{pmatrix}
\sin t & \sin t & \cos t\\
\sin  t & \cos t & \cos t\\
\cos t & \sin  t & \sin t\end{pmatrix}\,.
\end{gather*}
 Obviously, (H1) holds. Set $L(r)=\frac{2}{3}r^{3}$. Since
$f'(x)=\frac{2}{3}x^{3}$, (H2) holds. Next, let us check (H3).
Clearly, we have $f(0)=\frac{1}{6}$,
\begin{gather*}
\sum_{C_{kl}\in{N_{1}(1,1)}}\bar{C}_{11}^{kl}=0.7,\quad
\sum_{C_{kl}\in{N_{1}(1,2)}}\bar{C}_{12}^{kl}=1.4, \quad
\sum_{C_{kl}\in{N_{1}(1,3)}}\bar{C}_{13}^{kl}=1.1,\\
\sum_{C_{kl}\in{N_{1}(2,1)}}\bar{C}_{21}^{kl}=1,\quad
\sum_{C_{kl}\in{N_{1}(2,2)}}\bar{C}_{22}^{kl}=1.8, \quad
\sum_{C_{kl}\in{N_{1}(2,3)}}\bar{C}_{23}^{kl}=1.4, \\
\sum_{C_{kl}\in{N_{1}(3,1)}}\bar{C}_{31}^{kl}=0.6,\quad
\sum_{C_{kl}\in{N_{1}(3,2)}}\bar{C}_{32}^{kl}=0.9,\quad
\sum_{C_{kl}\in{N_{1}(3,3)}}\bar{C}_{33}^{kl}=0.6,\\
D=\max_{(i,j)}\big\{\frac{\sum_{C_{kl}\in{N_{r}(i,j)}}
\bar{C}_{ij}^{kl}\int_{0}^{\infty}|K_{ij}(u)|du}{\underline{a}_{ij}}\big\}
\leq \max_{(i,j)}\big\{\frac{\sum_{C_{kl}\in{N_{r}(i,j)}}
\bar{C}_{ij}^{kl}}{\underline{a}_{ij}}\big\}\leq 0.3,\\
L^{+}=\max_{(i,j)}\frac{\overline{L_{ij}}}{\underline{a}_{ij}}=0.2
\end{gather*}
Take $r_{0}=1$. Then
$$
D[|f(0)|r_{0}+L(r_{0})r^{2}_{0}]+L=0.3\cdot
(\frac{1}{6}+\frac{2}{3})+0.2=0.45<1=r_{0}
$$
and
$$
D|f(0)|+2DL(r_{0})r_{0}=0.005+0.4=0.45<1.
$$
Thus, (H3) holds for $r_{0}=1$.

Take $r_{1}=(7/2)^{1/4}$. Then
$$
D|f(0)|+DL(r_{0})r_{0}+
\max_{(i,j)}\big\{\frac{\sum_{C_{kl}\in{N_{r}(i,j)}}
\bar{C}_{ij}^{kl}}{\underline{a}_{ij}}\big\}L(r_{1})r_{1}
=0.25+0.7=0.95<1
$$
So (H4) holds. By Theorem \ref{thm3.1},  system \eqref{e4.1} has a
$\pi$-anti-periodic solution $x^{*}(t)$ with initial value
$\sup_{t\in [-1, 0]}\|\varphi^{*}\|< 1$. Moreover, all
solutions  of \eqref{e4.1} with initial value
$\sup_{t\in (-\infty, 0]}\|\varphi (t)\|\leq
 (7/2)^{1/4}$ converge exponentially to $x^{*}(t)$
as $t\to+\infty$.

\begin{remark} \label{rmk4.1} \rm
SICNNs \eqref{e4.1} is a very simple form of shunting inhibitory
cellular neural networks with  continuously distributed delays.
Since $f(x)=\frac{x^{4}+1}{6} $. One can observe that the
condition (T0) is  not satisfied. Therefore,  the results in
\cite{p1,s1} and the     references therein can not be applicable to
 \eqref{e4.1}. This  implies that our results  are essentially new.
\end{remark}

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\end{document}