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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 59, 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 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/59\hfil Anti-periodic solutions]
{Anti-periodic solutions for high-order cellular
 neural networks with time-varying delays}

\author[Z. Huang, L. Peng, M. Xu\hfil EJDE-2010/59\hfilneg]
{Zuda Huang, Lequn Peng, Min Xu}  % in alphabetical order

\address{Department of Mathematics, Hunan University of Arts
and Science, Changde, Hunan 415000, China}
\email[Z. Huang]{yitang1972@yahoo.com.cn}
\email[L. Peng]{penglq1956@yahoo.com.cn}
\email[M. Xu]{xumincd2010@yahoo.com.cn}

\thanks{Submitted November 2, 2009. Published april 28, 2010.}
\thanks{Supported   by   the Key  Project of Chinese Ministry of 
Education (No. 2010 151),  \hfill\break\indent the grants 06JJ2063, 07JJ6001
from the Scientific Research Fund of Hunan \hfill\break\indent
Provincial Natural Science Foundation, and 08C616, 09B072 from
the Scientific Research \hfill\break\indent
Fund of Hunan Provincial Education Department of China}
\subjclass[2000]{34C25, 34K13, 34K25}
\keywords{High-order cellular neural networks;
   anti-periodic; \hfill\break\indent
  exponential stability; time-varying delays.}

\begin{abstract}
 In this article, we study  anti-periodic solutions for
 high-order cellular neural networks with time-varying delays.
 Sufficient conditions for the existence and exponential stability
 of anti-periodic solutions are presented.
\end{abstract}

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

\section{Introduction}

 In recent years, high-order cellular neural networks (HCNNs)
have attracted  attention due to their wide range of
applications in  fields such as signal and image
processing, pattern recognition, optimization, and many other
subjects. There have been many results on the problem of
global stability of equilibrium points and periodic solutions
of HCNNs in the literature (see \cite{c1,c2,l2,w1,w2,x1,x2,z1}).
However, there are
only a few references on the problem of existence and
stability of anti-periodic solutions. However,
the existence of anti-periodic solutions is important
in nonlinear differential equation (see \cite{a1,a2,l1,o1,p1}). Thus,
it is worth while to investigate the existence and stability
of anti-periodic solutions for HCNNs.

In this article, we study the anti-periodic solution of the
high-order cellular neural network medelled by
\begin{equation} \label{e1.1}
\begin{aligned}
x'_{i}(t) &=  -c_{i}(t)x_{i}(t)+
\sum_{j=1}^{n}a_{ij}(t)f_{j}(x_{j}(t-\widetilde{\tau}_{j}(t)))\\
&\quad+\sum_{j=1}^{n}\sum_{k=1}^{n}b_{ijk}(t)g_{j}(x_{j}
(t-\tau_{j}(t)))g_{k}(x_{k}(t-\tau_{k}(t)))+u_{i}(t),
\end{aligned}
\end{equation}
where $i=1,  2, \dots, n$;
$c_{i},a_{ij},b_{ijk},f_{j}, g_{j}, u_{i}$ are
continuous functions on $\mathbb{R}$;
$x=(x_{1},  x_{2},  \dots,  x_{n})^{T}\in \mathbb{R}^n$
is the state vector; $c_{i}$ is a  positive parameter;
$a_{ij}$ and $b_{ijk}$ are the first and
second order connection weights of the neural networks, respectively;
$f_{j}$ and $ g_{j}$ are the activation functions;
$u_{i}$ is  an external input to the $i$th neuron;
$\widetilde{\tau}_{j}(t)$ and $\tau_{j}(t)$ are the time-varying
delay that satisfy $0\leq\widetilde{\tau}_{j}(t)\leq\tau $ and
$0\leq\tau_{j}(t)\leq\tau $ ($\tau$ is a constant).

The initial conditions are
\begin{equation} \label{e1.2}
x_{i}(t)=\varphi_{i}(t),\quad t\in [-\tau,0],\;i=1,  2, \dots, n,
\end{equation}
where $\varphi(\cdot)=[\varphi_{1}(\cdot),\varphi_{2}(\cdot),
\dots,\varphi_{n}(\cdot)]\in
C([-\tau,0],\mathbb{R}^n)$ and $C([-\tau,0],\mathbb{R}^n)$
denotes the set of continuous functions.

   The rest of this article is organized as follows.  In Section 2,
we give some notations and preliminary knowledge. In Section
3, we present our main results. In Section 4, we present
an example to illustrate the effectiveness of
our results. Finally, we give the conclusions in Section 5.


\section{Preliminary Results}

 A continuous function $h:\mathbb{R}\to \mathbb{R}$ is said
to be $T$-anti-periodic  on $\mathbb{R}$, if
$$
h(t+T)=-h(t)\quad  \text{for  all  }  t\in \mathbb{R}.
$$
We consider \eqref{e1.1} under the following assumptions:
For $i,  j=1,  2,  \dots,  n$, it will be assumed that
\begin{equation}
\begin{gathered}
c_{i}(t+T)=c_{i}(t),\quad
\tau_{i}(t+T)=\tau_{i}(t), \quad
a_{ij}(t+T)f_{j}(v)=-a_{ij}(t)f_{j}(-v),\\
\widetilde{\tau}_{j}(t+T)=\widetilde{\tau}_{j}(t), \quad
b_{ijk}(t+T)=-b_{ijk}(t), \quad
u_{i}(t+T)=-u_{i}(t),\quad \forall t,v\in \mathbb{R}.
\end{gathered}\label{e2.1}
\end{equation}
and
\begin{equation}
\overline{u}=\max_{1\leq i\leq n}\sup_{t\in \mathbb{R}}|u_{i}(t)|.
\label{e2.2}
\end{equation}
Also we will use the assumptions.
\begin{itemize}
\item[(H1)]   For $j=1,  2,  \dots,  n $,
    there exist $\overline{g_{j}}>0$ such that
    $|g_{j}(u)|\leq\overline{g_{j}}$ for all
    $u \in \mathbb{R}$;

\item[(H2)]  for $j=1,  2,  \dots,  n $,
    there exist  $L_{j}>0$ and $M_{j}>0$ such that
\begin{gather*}
|f_{j}(u )-f_{j}(v )|  \leq L_{j}|u -v |,\quad
|g_{j}(u )-g_{j}(v )|  \leq M_{j}|u -v |,\\
f_{j}(0)=0,\quad g_{j}(0)=0,\quad  \forall
     u ,  v \in \mathbb{R}.
\end{gather*}

\item[(H3)]
There exist constants $\eta>0, \lambda>0$ and $\xi_{i}>0$,
$i=1, 2,  \dots, n$, such that for all $t>0$,
$$
[\lambda-c_{i}(t)]\xi_{i}+\Big[\sum_{j=1}^{n}|a_{ij}(t)|L_{j}\xi_{j}
+\sum_{j=1}^{n}\sum_{k=1}^{n} |b_{ijk}(t)|(\overline{g_{k}}M_{j}\xi_{j}
+\overline{g_{j}}M_{k}\xi_{k})\Big]e^{\lambda \tau}<-\eta<0.
$$
\end{itemize}

\begin{definition} \label{def2.1} \rm
Let $x^{*}(t)=\big(x^{*}_{1}(t),
x^{*}_{2}(t),\dots,x^{*}_{n}(t)\big)^{T} $ be an anti-periodic
 solution of  \eqref{e1.1} with initial value
$\varphi^{*}=(\varphi^{*}_{1}(t),  \varphi^{*}_{2}(t),  \dots,
\varphi^{*}_{n}(t))^{T} $. If there exist constants $\lambda>0$
and $M_{\varphi} >1$ such that for every solution
$x(t)=(x_{1}(t), x_{2}(t),\dots,x_{n}(t))^{T} $ of \eqref{e1.1}
with an initial value $ \varphi=(\varphi_{1}(t), \varphi_{2}(t),
\dots, \varphi_{n}(t))^{T}$, with
$$
|x_{i}(t)-x^{*}_{i}(t)|\leq M_{\varphi}
\|\varphi-\varphi^{*}\|e^{-\lambda t},\quad  \forall t>0,  \; i=1,
2,  \dots,  n,
$$
 where
$$
\|\varphi-\varphi^{*}\|=\sup_{-\tau\leq s\leq0}
\max_{1\leq i\leq n}|\varphi_{i}(s)-\varphi_{i}^{*}(s)|.
$$
Then $x^{*}(t)$  is said to be globally exponentially stable.
\end{definition}

Next, we present two important lemmas, to be used for proving
our main results in Section 3.

    \begin{lemma} \label{lem2.1}
Let {\rm (H1)--(H3)} hold. Suppose that
 $\widetilde{x}(t)= (\widetilde{x}_{1}(t),
\widetilde{x}_{2}(t),\dots, \widetilde{x}_{n}(t))^{T} $ is a
solution of  \eqref{e1.1} with initial conditions
\begin{equation}
\widetilde{x}_{i}(s)=\widetilde{\varphi}_{i}(s), \quad
|\widetilde{\varphi}_{i}(s)|<\xi_{i}\frac{\overline{u}+1}{\eta}, \quad
s\in [-\tau,0], \; i=1,2,\dots,n.\label{e2.3}
\end{equation}
Then
\begin{equation}
|\widetilde{\varphi}_{i}(t)|<\xi_{i}\frac{\overline{u}+1}{\eta},
 \quad \text{for  all }  t\geq 0, \; i=1,2,\dots,n.\label{e2.4}
\end{equation}
\end{lemma}

\begin{proof}
Assume, by way of contradiction, assume that \eqref{e2.4} does not hold.
 Then there must exist $i\in \{1,2,\dots,n \}$ and $\sigma>0$ such
    that
\begin{equation}
|\widetilde{x}_{i}(\sigma)| =\xi_{i}\frac{\overline{u}+1}{\eta},
\quad\text{and}\quad
|\widetilde{x}_{j}(t)| <\xi_{j}\frac{\overline{u}+1}{\eta}
 \quad   \text{for all } t\in (-\tau, \sigma),  \;
 j=1,2,\dots,n.\label{e2.5}
\end{equation}
By directly computing the  upper left derivative of
$|\widetilde{x}_{i}(t)|$. under  assumptions (H1)--(H3),
 and \eqref{e2.5}, we deduce that
\begin{equation}
\begin{aligned}
0 & \leq D^+(|\widetilde{x}_{i}(\sigma)|) \\
& \leq
-c_{i}(\sigma)|\widetilde{x}_{i}(\sigma)|+\sum_{j=1}^{n}|a_{ij}(\sigma)||f_{j}(\widetilde{x}_{j}(\sigma-\widetilde{\tau}_{j}(\sigma)))|\\
&\quad +\sum_{j=1}^{n}\sum_{k=1}^{n}|b_{ijk}(\sigma)||g_{j}(\widetilde{x}_{j}(\sigma-\tau_{j}(\sigma)))|\overline{g_{k}}+|u_{i}(\sigma)|\\
& \leq
 -c_{i}(\sigma)\xi_{i}\frac{\overline{u}+1}{\eta}+\sum_{j=1}^{n}|a_{ij}(\sigma)|L_{j}|\widetilde{x}_{j}(\sigma-\widetilde{\tau}_{j}(\sigma))|\\
&\quad +\sum_{j=1}^{n}\sum_{k=1}^{n}|b_{ijk}(\sigma)|M_{j}|\widetilde{x}_{j}(\sigma-\tau_{j}(\sigma))|\overline{g_{k}}+\overline{u}\\
& \leq
-c_{i}(\sigma)\xi_{i}\frac{\overline{u}+1}{\eta}+\sum_{j=1}^{n}|a_{ij}(\sigma)|L_{j}\xi_{j}\frac{\overline{u}+1}{\eta}
+\sum_{j=1}^{n}\sum_{k=1}^{n}|b_{ijk}(\sigma)|M_{j}\xi_{j}\frac{\overline{u}+1}{\eta}\overline{g_{k}}+\overline{u}\\
& = [-c_{i}(\sigma)\xi_{i}+\sum_{j=1}^{n}|a_{ij}(\sigma)|L_{j}\xi_{j}
+\sum_{j=1}^{n}\sum_{k=1}^{n}|b_{ijk}(\sigma)|M_{j}\xi_{j}
\overline{g_{k}}]\frac{\overline{u}+1}{\eta}+\overline{u}
< 0.
\end{aligned} \label{e2.6}
\end{equation}
which is a contradiction and implies that \eqref{e2.4} holds. This
completes the proof.
\end{proof}

\begin{remark} \label{rmk2.1} \rm
It follows that the bounded
solution $\widetilde{x} (t)$ can be defined on $[0, \infty)$
according to the theory of functional differential equations in
\cite{h1}.
\end{remark}

\begin{lemma} \label{lem2.2}
 Suppose that  {\rm (H1)--(H3)} hold.
 Let $x^{*}(t)=(x^{*}_{1}(t), x^{*}_{2}(t),\dots,
x^{*}_{n}(t))^{T} $ be the solution of \eqref{e1.1} with initial
value $ \varphi^{*}=(\varphi^{*}_{1}(t),  \varphi^{*}_{2}(t),
\dots, \varphi^{*}_{n}(t))^{T} $, and
$ x(t)=(x_{1}(t), x_{2}(t),\dots,x_{n}(t))^{T} $ be the solution of
\eqref{e1.1} with initial value
$ \varphi=(\varphi _{1}(t), \varphi _{2}(t), \dots, \varphi _{n}(t))^{T}$.
Then there exist constants  $M_{\varphi}>1$ such that
$$
|x_{i}(t)-x^{*}_{i}(t)|\leq M_{\varphi}
\|\varphi-\varphi^{*}\|e^{-\lambda t}, \quad \forall t>0,  \; i=1,
2,  \dots,  n.
$$
\end{lemma}

 \begin{proof}
Let $y(t)=\{y_{ j}(t) \}=\{x_{ j}(t)-x^{\ast}_{ j}(t)
\}=x(t)-x^{*}(t)$. Then
\begin{align*}
y_{i}^{'}(t)
&=  -c_{i}(t)[x_{i}(t)-x^{\ast}_{i}(t)]
+\sum_{j=1}^{n}a_{ij}(t)\big[f_{j}(x_{j}(t-\widetilde{\tau}_{j}(t)))
 -f_{j}(x^{\ast}_{j}(t-\widetilde{\tau}_{j}(t)))\big]\\
&\quad +\sum_{j=1}^{n}\sum_{k=1}^{n}b_{ijk}(t)\big[g_{j}(x_{j}
  (t-\tau_{j}(t)))g_{k}(x_{k}(t-\tau_{k}(t)))\\
&\quad - g_{j}(x^{\ast}_{j}(t-\tau_{j}(t)))g_{k}(x^{\ast}_{k}
 (t-\tau_{k}(t)))\big]
\end{align*}
where $ i=1,  2, \dots,  n$.
  Next,  define a  Lyapunov functional  as
\begin{equation}
V_{i }(t) =    |y_{i }(t)|e^{\lambda t}, \quad  i=1,  2, \dots, n.
\label{e2.7}
\end{equation}
  By \eqref{e2.6} and \eqref{e2.7} it follows that
\begin{equation}
\begin{aligned}
    D^+(V_{i }(t))
& \leq  D^+(|y_{i}(t)|)e^{\lambda t}+\lambda|y_{i}(t)|e^{\lambda t}\\
& \leq (\lambda-c_{i}(t))|y_{i}(t)|e^{\lambda
  t}+\{\sum_{j=1}^{n}|a_{ij}|(t)L_{j}|y_{j}(t-\widetilde{\tau}_{j}(t))|\\
&\quad + \sum_{j=1}^{n}\sum_{k=1}^{n}|b_{ijk}(t)|[\overline{g_{k}}
  M_{j}|y_{j}(t-\tau_{j}(t))|
  +\overline{g_{j}}M_{k}|y_{k}(t-\tau_{k}(t))|]\}e^{\lambda t}
\end{aligned}  \label{e2.8}
\end{equation}
where  $i=1, 2, \dots,n$.
Let $m^{\ast}>1$ denote a real number such that
$$
m^{\ast}\xi_{i}>\|\varphi-\varphi^{*}\|
=\sup_{-\tau\leq s\leq0}\max_{1\leq j\leq n  }
 |\varphi_{ j}(s)-\varphi_{j}^{*}(s)|>0, \quad i=1, 2, \dots, n.
$$
Then by \eqref{e2.7}, we have
$$
V_{i }(t) =    |y_{i }(t)|e^{\lambda t}< m^{\ast}\xi_{i}
,\quad \text{for  all }   t\in [-\tau, 0], \; i=1,   2, \dots, n.
$$
 Thus we can claim that
\begin{equation}
 V_{i }(t) =    |y_{i }(t)|e^{\lambda t}< m^{\ast}\xi_{i}
, \quad \text{for  all }   t>0, \; i=1,   2, \dots,  n.\label{e2.9}
\end{equation}
Otherwise, there must exist $i \in \{ 1,  2,  \dots,  n \}$ and
$t_{i}>0$ such that
\begin{equation}
V_{i}(t_{i})=m^{\ast}\xi_{i},\quad V_{j}(t)<m^{\ast}\xi_{j} , \quad
\forall  t\in [-\tau,  t_{i}), \; j=1, 2, \dots,  n.\label{e2.10}
\end{equation}
Combining  \eqref{e2.8}  with  \eqref{e2.10}, we obtain
\begin{equation}
\begin{aligned}
0 &\leq   D^+(V_{i }(t_{i })-m^{\ast}\xi_{i} )\\
&=   D^+(V_{i }(t_{i })) \\
&\leq  (\lambda-c_{i}(t_{i}))|y_{i}(t_{i})|e^{\lambda
  t_{i}}+\{\sum_{j=1}^{n}|a_{ij}(t_{i})|L_{j}|y_{j}(t_{i}
 -\widetilde{\tau}_{j}(t_{i}))|\\
&\quad +
\sum_{j=1}^{n}\sum_{k=1}^{n}|b_{ijk}(t_{i})
 |[\overline{g_{k}}M_{j}|y_{j}(t_{i}-\tau_{j}(t_{i}))|
  +\overline{g_{j}}M_{k}|y_{k}(t_{i}-\tau_{k}(t_{i}))|]\}e^{\lambda
t_{i}}\\
&\leq (\lambda-c_{i}(t_{i}))m^{\ast}\xi_{i}
 +\sum_{j=1}^{n}|a_{ij}|(t_{i})L_{j}m^{\ast}\xi_{j}e^{\lambda\tau}\\
&\quad +\sum_{j=1}^{n}\sum_{k=1}^{n}|b_{ijk}(t_{i})|\overline{g_{k}}
 M_{j}m^{\ast}\xi_{j}e^{\lambda\tau}
+\sum_{j=1}^{n}\sum_{k=1}^{n}|b_{ijk}(t_{i})|\overline{g_{j}}
 M_{k}m^{\ast}\xi_{k}e^{\lambda\tau}\\
&= \{(\lambda-c_{i}(t_{i}))\xi_{i}+\big[\sum_{j=1}^{n}|a_{ij}(t_{i})
|L_{j}\xi_{j}\\
&\quad +\sum_{j=1}^{n}\sum_{k=1}^{n}|b_{ijk}(t_{i})|(\overline{g_{k}}M_{j}
\xi_{j}
+\overline{g_{j}}M_{k}\xi_{k})\big]e^{\lambda\tau}\}m^{\ast}
\end{aligned}\label{e2.11}
\end{equation}
It is clear that
$$
(\lambda-c_{i})\xi_{i}+[\sum_{j=1}^{n}|a_{ij}|(t_{i})L_{j}\xi_{j}\\
+\sum_{j=1}^{n}\sum_{k=1}^{n}|b_{ijk}(t_{i})|
(\overline{g_{k}}M_{j}\xi_{j}
+\overline{g_{j}}M_{k}\xi_{k})]e^{\lambda\tau}>0
$$
This contradicts (H3), then \eqref{e2.9} holds. Letting
  $M_{\varphi}>1$, such that
\begin{equation}
\max_{1\leq i\leq n}\{m^{\ast}\xi_i \}\leq
M_{\varphi}\|\varphi-\varphi^{*}\|, \quad  i=1,2,\dots,n.\label{e2.12}
\end{equation}
 In view of \eqref{e2.9} and \eqref{e2.12}, we obtain
$$
|x_{i}(t)-x^{*}_{i}(t)|=|y_{i}(t)|\leq \max_{1\leq i\leq
n}\{m\xi_i \}e^{-\lambda t}
 \leq M_{\varphi} \|\varphi-\varphi^{*}\|e^{-\lambda t},
$$
where $i=1,2,\dots,n$, $t>0$. This completes the proof.
\end{proof}

\begin{remark} \label{rmk2.2} \rm
If $x^{*}(t)=(x^{*}_{1}(t), x^{*}_{2}(t),\dots,x^{*}_{n}(t))^{T}
$ is the  $T$-anti-periodic solution of  \eqref{e1.1},   it follows
from Lemma 2.2 and the Definition \ref{def2.1} that $x^{*}(t)$
is globally
exponentially stable.
\end{remark}

\section{Main Results}

In this section,we present our main result that there exists
the exponentially stable anti-periodic solution of \eqref{e1.1}.

\begin{theorem} \label{thm3.1}
   Assume that  {\rm (H1)--(H3)}  are satisfied.
Then  \eqref{e1.1} has  exactly one
$T$-anti-periodic  solution $x^{*}(t)$. Moreover, this solution is
globally exponentially stable.
\end{theorem}

\begin{proof}
Let $v(t)= (v_{1}(t), v_{2}(t),\dots, \ v_{n}(t))^{T} $ is a
solution of \eqref{e1.1} with initial conditions
\begin{equation}
v_{i}(s)=\varphi^{v}_{i}(s), \quad
|\varphi^{v}_{i}(s)|<\xi_{i}\frac{\overline{u}+1}{\eta} , \quad
s\in (-\tau, 0], \; i=1,2,\dots,n.\label{e3.1}
\end{equation}
Thus according to  Lemma 2.1, the solution $v(t)$ is bounded and
\begin{equation}
|v_{i}(t)|<\xi_{i}\frac{\overline{u}+1}{\eta} , \quad \text{for
all }  t\in \mathbb{R}, \; i=1,2,\dots,n.\label{e3.2}
\end{equation}
 From \eqref{e2.1}, we obtain
\begin{equation}
\begin{aligned}
&((-1)^{m+1}v_{i} (t + (m+1)T))'\\
&=(-1)^{m+1}\{-c_{i}(t+(m+1)T)v_{i}(t+(m+1)T)\\
&\quad +\sum_{j=1}^{n}a_{ij}(t+(m+1)T)f_{j}(v_{j}
(t+(m+1)T-\widetilde{\tau}_{j}(t+(m+1)T)))\\
&\quad +\sum_{j=1}^{n}\sum_{k=1}^{n}b_{ijk}(t+(m+1)T)
g_{j}(v_{j}(t+(m+1)T-\tau_{j}(t+(m+1)T)))\\
&\quad\times g_{k}(v_{k}(t+(m+1)T-\tau_{k}(t+(m+1)T)))
 +u_{i}(t+(m+1)T)\}\\
&= -c_{i}(t)[(-1)^{m+1}v_{i}(t+(m+1)T)]+\sum_{j=1}^{n}a_{ij}(t)
  \\
&\times f_{j}[(-1)^{m+1}v_{j}((t+(m+1)T)-\widetilde{\tau}_{j}(t+(m+1)T))]\\
&\quad +\sum_{j=1}^{n}\sum_{k=1}^{n}b_{ijk}(t)g_{j}[(-1)^{m+1}v_{j}
 ((t+(m+1)T)-\tau_{j}(t+(m+1)T))]\\
&\quad \times g_{k}[(-1)^{m+1}v_{k}((t+(m+1)T)-\tau_{k}(t+(m+1)T))]
+u_{i}(t)
\end{aligned} \label{e3.3}
\end{equation}
where $i=1, 2, \dots,n$. Thus  $(-1)^{m+1} v  (t + (m+1)T)$
are the solutions of  \eqref{e1.1} on $\mathbb{R}$ for any natural
number $m$.
Then, from Lemma 2.2, there exists a constant $M>0$ such that
\begin{equation}
\begin{aligned}
&|(-1)^{m+1}v_{i} (t + (m+1)T)-(-1)^{m} v_{i}(t + mT)|\\
&\leq    M e^{-\lambda (t + mT)}\sup_{-\tau\leq
s\leq0}\max_{1\leq i\leq n}|v_{i} (s +  T)+ v_{i} (s)|\\
&\leq   2e^{-\lambda (t + mT)} M \max_{1\leq i\leq
n}\{\xi_{i}\frac{\overline{u}+1}{\eta}\}, \quad \forall t + mT>0,
\; i=1,  2,  \dots,  n.
\end{aligned}\label{e3.4}
\end{equation}
Thus, for any natural number $m$, we have
\begin{equation}
(-1)^{m+1} v_{i} (t + (m+1)T)
  =  v_{i} (t )  +\sum_{k=0}^{m}[(-1)^{k+1}
v _{i}(t + (k+1)T)-(-1)^{k} v_{i} (t + kT)]. \label{e3.5}
\end{equation}
Hence,
\begin{equation}
     |(-1)^{m+1} v_{i} (t + (m+1)T)|
  \leq      | v_{i} (t )|  +\sum_{k=0}^{m}|
(-1)^{k+1} v _{i}(t + (k+1)T)-(-1)^{k} v_{i} (t + kT)|, \label{e3.6}
\end{equation}
where $i =1,2,\dots,n$.
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)^{m+1}
v_{i} (t + (m+1)T)-(-1)^{m } v_{i}  (t + mT)| \leq   \alpha
(e^{-\lambda T } )^{m},
 \label{e3.7}
\end{equation}
for all  $ m >N$, $i=1, 2, \dots, n$,
on any compact set of $\mathbb{R}$.
Obviously, together with \eqref{e3.5}, \eqref{e3.6} and \eqref{e3.7},
$ \{(-1)^{m} v (t + mT)\}$ uniformly converges to a continuous function
$x^{*}(t)=(x^{*}_{1}(t), x^{*}_{2}(t),\dots,x^{*}_{n}(t))^{T}$ on
any compact set of $\mathbb{R}$.

Now we  show that $x^{*}(t)$ is $T$-anti-periodic solution of
\eqref{e1.1}. Firstly, $x^{*}(t)$ is $T$-anti-periodic, since
\begin{align*}
x^{*}(t+T)
&=\lim_{m\to \infty}(-1)^{m } v (t +T+  m  T)\\
&=-\lim_{(m+1)\to \infty}(-1)^{m+1 } v (t  + (m +1)T)=-x^{*}(t ).
\end{align*}
secondly, we prove that $x^{*}(t)$ is
a solution of \eqref{e1.1}. Because of
the continuity of the right-hand side of \eqref{e1.1},  \eqref{e3.3}
implies that $ \{((-1)^{m+1} v  (t +(m+1)T))'\}$   uniformly
converges to a continuous function on any
compact subset of $\mathbb{R}$.  Thus, letting $m \to\infty$, we
can easily obtain
\begin{equation}
\begin{aligned}
\frac{d}{dt}\{x^{*}_{i}(t)\}
&=  -c_{i}(t)x^{*}_{i}(t)+
\sum_{j=1}^{n}a_{ij}(t)f_{j}(x^{*}_{j}(t-\widetilde{\tau}_{j}(t)))\\
&\quad +\sum_{j=1}^{n}\sum_{k=1}^{n}b_{ijk}(t)g_{j}(x^{*}_{j}
(t-\tau_{j}(t)))
g_{k}(x^{*}_{k}(t-\tau_{k}(t)))+u_{i}(t),
\end{aligned} \label{e3.8}
\end{equation}
where $i=1,  2, \dots, n$.
Therefore, $x^{*}(t)$ is a  solution of \eqref{e1.1}.

Finally, by  applying Lemma 2.2, it is easy to check that $x^{*}(t)$
is globally exponentially stable.  This completes the proof.
\end{proof}

\section{An Example}

In this section, a simple  example is provided to illustrate
our results.
 Consider the high-order cellular neural network with delays
 \begin{equation} \label{e4.1}
\begin{aligned}
 x_{1}'(t)
&= -x_{1}(t)+\frac{1}{4}|\sin t|f_{1}(x_{1}(t-1))
 +\frac{1}{36}|\cos t|f_{2}(x_{2}(t-2))\\
&\quad + \frac{1}{72}\sin t g^{2}_{1}(x_{1}(t-1))
 +\frac{1}{36}\cos t g_{1}(x_{1}(t-1))g_{2}(x_{2}(t-2))\\
 &\quad+\frac{1}{72}\cos t g^{2}_{2}(x_{2}(t-2))+\frac{1}{9}\sin t ,\\
 x_{2}'(t)
&= -x_{2}(t)+\frac{1}{36}|\cos t|f_{1}(x_{1}(t-1))
 +\frac{1}{4}|\sin t|f_{2}(x_{2}(t-2))\\
&\quad + \frac{1}{72}\cos t g^{2}_{1}(x_{1}(t-1))
 +\frac{1}{36}\cos t g_{1}(x_{1}(t-1))g_{2}(x_{2}(t-2))\\
&\quad +\frac{1}{72}\sin t g^{2}_{2}(x_{2}(t-2))+\frac{1}{9}\sin t\,,
\end{aligned}
\end{equation}
where $f_{1}(x)=f_{2}(x)= x$,
$g_{1}(x)=g_{2}(x)= \arctan x$,
$c_{1}(t)=c_{2}(t)=1, u_{1}(t)=\frac{1}{9}\sin t$,
$u_{2}(t)=\frac{1}{9}\sin t$,
$a_{11}(t)=a_{22}(t)=\frac{1}{4}|\sin t|$,
$a_{12}(t)=a_{21}(t)=\frac{1}{36}|\cos t|$,
$b_{111}(t)=b_{222}(t)=\frac{1}{72}\sin t$,
$b_{112}(t)=b_{121}(t)=b_{122}(t)=b_{211}(t)=b_{212}(t)=b_{221}(t)=
\frac{1}{72}\cos t$.
Noting that
$$
L_{1}=L_{2}=M_{1}=M_{2}=1,\quad
\overline{g_{1}}=\overline{g_{2}}=\frac{\pi}{2}.
$$
Therefore,   there  exist constants
 $\eta=\frac{1}{2}$, $
\lambda=\frac{1}{1800}$ and $\xi_{1}=\xi_{2}=1$,
such that for all $t>0$, $i=1,2$, there  holds
$$
 [\lambda-c_{i}(t)]\xi_{i}+[\sum_{j=1}^{2}a_{ij}(t)L_{j}\xi_{j}
+\sum_{j=1}^{2}\sum_{k=1}^{2} b_{ijk}(t)(\overline{g_{k}}M_{j}\xi_{j}
+\overline{g_{j}}M_{k}\xi_{k})]e^{\lambda
\tau}<-\eta,
$$
which implies that system \eqref{e4.1} satisfy all the
conditions in Theorem 3.1. Hence,  \eqref{e4.1} has exactly one
$\pi$-anti-periodic solution. Moreover, this solution
is globally exponentially stable.

This fact is verified in the numerical simulation in Figure 1.

\begin{figure}[ht] \label{fig1}
\begin{center}
\includegraphics[width=0.48\textwidth]{fig1}
\includegraphics[width=0.48\textwidth]{fig2}
\end{center}
\caption{Numerical solution $(x_{1 }(t),x_{2}(t))$ of system
 \eqref{e4.1} for $(\varphi_{1}(s),\varphi_{2}(s) 
)=(0.5,0.8)$.}
\end{figure}

We remark that \eqref{e4.1} is a very simple form of high-order cellular
neural networks with  delays.  However, the results in
the references  can  not be  applicable for obtaining existence
and exponential stability of  the anti-periodic solutions. This makes
our results new.

\subsection*{Acknowledgements}
The authors  would like to express the  sincere appreciation to
the anonymous referee  for his/her helpful comments in improving the
presentation and quality of this paper.

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\end{document}