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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 02, 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 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/02\hfil Existence and exponential stability]
{Existence and exponential stability of anti-periodic solutions in
cellular neural networks with time-varying delays and \\ impulsive effects}

\author[C. Xu \hfil EJDE-2016/??\hfilneg]
{Changjin Xu}

\address{Changjin  Xu \newline
Guizhou Key Laboratory of Economics System Simulation,
School of Mathematics and Statistics,
Guizhou University of Finance and Economics, Guiyang 550004, China}
\email{xcj403@126.com}

\thanks{Submitted November 4, 2014. Published January 4, 2016.}
\subjclass[2010]{34C25, 34K13, 34K25}
\keywords{Cellular neural network;    anti-periodic solution; impulse;
\hfill\break\indent exponential stability; time-varying delay}

\begin{abstract}
 In this article we study a cellular neural network with impulsive effects.
 By using differential inequality techniques, we obtain verifiable criteria
 on the existence and exponential stability  of anti-periodic solutions.
 An example is included to  illustrate the feasibility and  of our main results.
\end{abstract}

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

\section{Introduction}

 Because of the wide range of applications in neurobiology, image processing,
evolutionary theory, pattern recognition and optimization and so on,
cellular neural networks have attracted much attention in recent years \cite{9}.
 It is well known that impulsive differential equations are mathematical
apparatus for simulation of process and phenomena observed in control theory,
 physics, chemistry, population dynamics, biotechnologies, industrial robotics,
economics, etc. \cite{3,18,35}.
 Therefore many results on the existence and stability of an equilibrium point
of cellular neural networks with impulses have been reported (see
\cite{14,16,17,29,39,41,42,44,52}).
In applied sciences, the existence of
anti-periodic solutions plays a key role in characterizing the
behavior of nonlinear differential equations \cite{11,21,22,36}. For
example, high-order Hopfield neural networks can be analog voltage
transmission, and voltage transmission process can be described as
an anti-periodic process \cite{30}, anti-periodic trigonometric
polynomials play an important role in interpolation problems \cite{10},
and anti-periodic wavelets were investigated in \cite{7}, in neural
networks, the global stable anti-periodic solution can reveal the
characteristic and stability of signal \cite{37}. Recently, there are
some papers that deal with the problem of existence and stability of
anti-periodic solutions (see \cite{1,2, 8, 12,13, 15, 19, 23,24,25,26,28,
30,31,32,33,34, 38, 40,50,51}). In addition, we know that many
evolutionary processes exhibit impulsive effects which are usually
subject to short time perturbations whose durations may be neglected
in comparison with durations of the processes \cite{38}. This motivates
us to consider the existence and stability of anti-periodic
solutions for cellular neural networks with impulses. To the best of
our knowledge, very few authors have focused on the problems of
anti-periodic solutions for such impulsive cellular neural networks.
In this paper, we consider the anti-periodic solution of the
following cellular neural network with delays and impulses
\begin{gather}
\dot{x}_i(t)=-c_i(t)x_i(t)+\sum_{j=1}^na_{ij}(t)f_j(x_j(t))
+\sum_{j=1}^nb_{ij}(t)f_j(x_j(t-\tau_{ij}(t)))+u_i(t), \nonumber\\
 t\neq{t_k},   \label{e1.1}\\
{x_i}(t_k^{+})=(1+\delta_{ik})x_i(t_k),\quad k=1,2,\dots, \nonumber
\end{gather}
where $i=1,  2, \dots, n$,$x_i(t)$ represent the state vector of the
$i$th unit at time $t$,  $c_i, a_{ij}, b_{ij},f_{j} g_j, u_i,
\tau_{ij}$ are continuous functions on $R$, $c_i>0$, $a_{ij}$ are
the connection weights between $i$th unit and $j$th unit at time
$t$, $b_{ij}$ is the connection weights between $i$th unit and $j$th
unit at time $t-\tau_{ij}$, $f_j$ and $g_j$ are the activation
function, $u_i$ are external input to the $i$th unit, $\tau_{ij}$
are the time varying delay and satisfy $0\leq\tau_{ij}\leq\tau$,
$\tau$ is a positive constant, $t_k$ are the impulsive moments and
satisfy $0<t_1<t_2<\dots<t_k<\dots,
\lim_{k\to \infty}t_k=\infty$, $\delta_{ik}$ characterize the
impulsive at jumps at time $t_k$ for $i$th unit.

The main purpose of this article is to give the sufficient conditions
of existence and exponential stability of anti-periodic solution of
system \eqref{e1.1}. Some new sufficient conditions for the existence,
unique and exponential stability of anti-periodic solutions of
system \eqref{e1.1} are established. Our results not only can be applied
directly to many concrete examples of cellular neural networks, but
also extend, to a certain extent, the results in some previously
known ones. In addition, an example is presented to illustrate the
effectiveness of our main results.

For convenience, we introduce the following notation
\begin{gather*}
{a}_{ij}^{+}=\sup_{t\in{R}}|a_{ij}(t)|, \quad
{b}_{ij}^{+}=\sup_{t\in{R}}|b_{ij}(t)|, \quad
{u}_{i}^{+}=\sup_{t\in{R}}|u_{i}(t)|, \\
c_i^{-}=\min_{t\in{R}}|c_{i}(t)|, \quad
\tau=\sup_{t\in{R}}\max_{1\leq{i,j}\leq{n}}\{\tau_{ij}(t)\}.
\end{gather*}
We assume the following hypothesis:
\begin{itemize}
\item[(H1)] For $i,j=1,2,\dots,n$,
 $a_{ij}, b_{ij}, u_i, f_j, g_j: R\to {R}$,
 $c_i, \tau_{ij}: R\to [0,+\infty)$ are
continuous functions, and there exist a constant $T>0$ such that
\begin{gather*}
c_i(t+T)=c_i(t), \quad \tau_{ij}(t+T)=\tau_{ij}(t), \quad u_i(t+T)=-u_i(t),\\
a_{ij}(t+T)f_j(u)=-a_{ij}(t)f_j(-u),\quad  b_{ij}(t+T)g_j(u)=-b_{ij}(t)g_j(-u),
\end{gather*}
for all $t,u\in{R}$.

\item[(H2)] For each $j\in\{1,2,\dots,n\}$, the activation function
$f_j: R\to {R}$ is continuous and there exists an nonnegative
constant $L_j^f$ such that
$$
f_j(0)=0,\quad  |f_j(u)-f_j(v)|\leq{L_j^f}|u-v|
$$
for all $u,v\in{R}$.

\item[(H3)] $\prod_{0\leq{t_k}<t}(1+\delta_{ik})$
($i=1,2,\dots,n$, $k=1,2,\dots$) are
periodic functions of period $T$ and $\delta_{ik}>-1$.

\item[(H4)] For $i=1,2,\dots,n$,  $k=1,2,\dots$, there exist positive
constants $m$ and $M$ such that
$m\leq\prod_{0\leq{t_k}<t}(1+\delta_{ik})\leq{M}$.

\item[(H5)] There exist constants $\eta>0$, $\lambda>0$, $i=1, 2, \dots, n$,
such that for all $t>0$,
$$
\lambda-c_i^{-}+\frac{M}{m}\Big[\sum_{j=1}^n(a_{ij}^{+}+b_{ij}^{+})L_j^f\Big]
e^{\lambda\tau}<-\eta<0.
$$
Let $x=(x_1,x_2,\dots,x_n)^T\in{R^n}$, in which $``T"$ denotes the
transposition. We define $|x|=(|x_1|,|x_2|,\dots,|x_n|)^T$ and
$\|x\|=\max_{1\leq{i}\leq{n}}|x_i|$. Obviously, the solution
$x(t)=(x_1(t),x_2(t),\dots,x_n(t))^T$ of \eqref{e1.1} has components
$x_i(t)$ piece-wise continuous on $(-\tau,+\infty)$, $x(t)$ is
differentiable on the open intervals $(t_{k-1},t_k)$ and
$x(t_k^{+})$ exists.
\end{itemize}

\begin{definition} \label{def1.1} \rm
 Let $u(t):R\to {R}$ be piece-wise
continuous function having countable number of discontinuous
$\{t_k\}|_{k=1}^{+\infty}$ of the first kind. It is said to be
$T$-anti-periodic on $R$ if
\begin{gather*}
u(t+T)=-u(t),\quad t\neq t_k,\\
u((t_k+T)^{+})=-u(t_k^{+}), \quad k=1,2,\dots.
\end{gather*}
\end{definition}

\begin{definition} \label{def1.2} \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
$M >1$ 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
$\varphi=(\varphi_{1}(t), \varphi_{2}(t), \dots,
\varphi_{n}(t))^{T}$,
$$
|x_{i}(t)-x^{*}_{i}(t)|\leq M \|\varphi-\varphi^{*}\|e^{-\lambda t},
\quad \text{for all }  t>0,\;  i=1, 2, \dots, n,
$$
 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}


The rest of this article is organized as follows.
In the next section, we give some preliminary results.
In Section 3, we derive the existence of $T$-anti-periodic solution,
which is globally exponential stable.
In Section 4, we present an example to
illustrate the effectiveness of our main results.

\section{Preliminaries}

In this section, we firstly establish a fundamental theorem that
enable us to reduce the existence of solution of system \eqref{e1.1} to the
corresponding problem for a delayed differential equation without
impulses. Consider the following non-impulsive delayed differential
system
\begin{equation}
\begin{aligned}
\dot{y}_i(t) & =-c_i(t)y_i(t)+\prod_{0\leq{t_k}<t}(1+\delta_{ik})^{-1}
\Big[\sum_{j=1}^na_{ij}(t)f_j\Big(\prod_{0\leq{t_k}<t}(1+\delta_{jk})y_j(t)
\Big) \\
&\quad +\sum_{j=1}^nb_{ij}(t)f_j\Big(\prod_{0\leq{t_k}<t-\tau_{ij}(t)}
 (1+\delta_{jk})y_j(t-\tau_{ij}(t))\Big)\Big] \\
&\quad +\prod_{0\leq{t_k}<t}(1+\delta_{ik})^{-1}u_i(t), \quad t>0
\end{aligned} \label{e2.1}
\end{equation}
with initial condition  $y_i(s)=\varphi_i(s)$,
$s\in[-\tau,0]$, $i=1,2,\dots,n$.

In this section, we present three important lemmas which are used to
prove our main results in Section 3.

\begin{lemma} \label{lem2.1}
 Assume that {\rm (H3)} holds.
(i) If $y=(y_1,y_2,\dots,y_n)$ is a solution of  \eqref{e2.1}, then
$$
x=\Big(\prod_{0\leq{t_k}<t}(1+\delta_{ik})y_1,
\prod_{0\leq{t_k}<t}(1+\delta_{ik})y_2, \dots,
\prod_{0\leq{t_k}<t}(1+\delta_{ik})y_n\Big)
$$
is a solution of  \eqref{e2.1}.

(ii) If $x=(x_1,x_2,\dots,x_n)$ is a solution of  \eqref{e1.1}, then
$$
y=\Big(\prod_{0\leq{t_k}<t}(1+\delta_{ik})^{-1}x_1,
\prod_{0\leq{t_k}<t}(1+\delta_{ik})^{-1}x_2, \dots,
\prod_{0\leq{t_k}<t}(1+\delta_{ik})^{-1}x_n\Big)
$$
is a solution of \eqref{e2.1}.
\end{lemma}

The proof of the above lemma is similar to that in  Li et al \cite{20}.
We omit it here.

\begin{lemma} \label{lem2.2}
Let {\rm (H1)--(H4)} hold. Suppose that
 ${y}(t)= ({y}_{1}(t), {y}_{2}(t),\dots, {y}_{n}(t))^{T} $ is a solution of
\eqref{e2.1} with initial conditions
\begin{equation} \label{e2.2}
{y}_{i}(s)={\varphi}_{i}(s), \quad |{\varphi}_{i}(s)|<\gamma, \quad
s\in [-\tau,0], \quad i=1,2,\dots,n.
\end{equation}
Then
\begin{equation} \label{e2.3}
|{y}_{i}(t)|<\gamma, \quad \forall  t\geq 0, \; i=1,2,\dots,n,
\end{equation}
where
\begin{gather}
\gamma>\frac{u_i^{+}} {mc_i^{-}-M\big[\sum_{j=1}^n(a_{ij}^{+}-b_{ij}^{+})L_j^f\big]},
\label{e2.4} \\
c_i^{-}>\frac{M}{m}\Big[\sum_{j=1}^n(a_{ij}^{+}-b_{ij}^{+})L_j^f\Big]. \nonumber
\end{gather}
\end{lemma}


\begin{proof}
  For any given initial condition, hypotheses (H2) and (H4) guarantee the
existence and unique of $y(t)$, the solution to \eqref{e2.1} in $[-\tau, +\infty)$.
 By way of contradiction, we assume that
\eqref{e2.3} does not hold. Then there must exist $i\in \{1,2,\dots,n \}$
and $ \theta_0>0$ such that
\begin{equation} \label{e2.5}
|{y}_{i}(\theta_0)| =\gamma, \quad |{y}_{j}(\theta_0)| <\gamma
\quad \text{for all } t\in (-\tau, \theta_0),\;  j=1,2,\dots,n.
\end{equation}
By  computing the  upper left derivative of $|{y}_{i}(t)|$,
together with the  assumptions \eqref{e2.3}, \eqref{e2.4}, \eqref{e2.5},
(H2) and (H4), we have
\begin{align}
0 & \leq D^+(|{y}_{i}(\theta_0)|) \nonumber\\
& \leq -c_i(\theta_0)|y_i(\theta_0)|
+\Big|\prod_{0\leq{t_k}<\theta_0}(1+\delta_{ik})^{-1}
 \Big[\sum_{j=1}^na_{ij}(\theta_0)f_j
 \Big(\prod_{0\leq{t_k}<\theta_0}(1+\delta_{jk})y_j(\theta_0)\Big) \nonumber\\
&\quad + \sum_{j=1}^nb_{ij}(\theta_0)f_j
 \Big(\prod_{0\leq{t_k}<\theta_0-\tau_{ij}(\theta_0)}(1+\delta_{jk})
 y_j(\theta_0-\tau_{ij}(\theta_0))\Big)\Big] \nonumber\\
&\quad  +\prod_{0\leq{t_k}<\theta_0}(1+\delta_{ik})^{-1}u_i(\theta_0)\Big| \nonumber\\
& \leq -c_i^{-}|y_i(\theta_0)|+\prod_{0\leq{t_k}<\theta_0}(1+\delta_{ik})^{-1}
 \Big[\sum_{j=1}^n|a_{ij}(\theta_0)|
 \Big|f_j\Big(\prod_{0\leq{t_k}<\theta_0}
 (1+\delta_{jk})y_j(\theta_0)\Big)\Big| \nonumber\\
&\quad +\sum_{j=1}^n|b_{ij}(\theta_0)|
\Big|f_j\Big(\prod_{0\leq{t_k}<\theta_0-\tau_{ij}(\theta_0)}
 (1+\delta_{jk})y_j(\theta_0-\tau_{ij}(\theta_0))\Big)\Big|\Big] \nonumber \\
&\quad +\prod_{0\leq{t_k}<\theta_0}(1+\delta_{ik})^{-1}|u_i(\theta_0)|\nonumber\\
&\leq -c_i^{-}|y_i(\theta_0)|+\prod_{0\leq{t_k}<\theta_0}(1+\delta_{ik})^{-1}
 \Big[\sum_{j=1}^na_{ij}^{+}L_j^f\prod_{0\leq{t_k}<\theta_0}
 (1+\delta_{jk})|y_j(\theta_0)| \nonumber\\
&\quad +\sum_{j=1}^nb_{ij}^{+}L_j^f\prod_{0\leq{t_k}<\theta_0-\tau_{ij} (\theta_0)}
 (1+\delta_{jk})|y_j(\theta_0-\tau_{ij}(\theta_0))|\Big]
 +\prod_{0\leq{t_k}<\theta_0}(1+\delta_{ik})^{-1}u_i^{+}\nonumber\\
&\leq -c_i^{-}\gamma+\prod_{0\leq{t_k}<\theta_0}(1+\delta_{ik})^{-1}
 \Big[\sum_{j=1}^na_{ij}^{+}L_j^f\prod_{0\leq{t_k}<\theta_0}
 (1+\delta_{jk})\gamma \nonumber\\
&\quad +\sum_{j=1}^nb_{ij}^{+}L_j^f\prod_{0\leq{t_k}<\theta_0-\tau_{ij}
 (\theta_0)}(1+\delta_{jk})\gamma\Big]+\prod_{0\leq{t_k}<\theta_0}
 (1+\delta_{ik})^{-1}u_i^{+}\nonumber\\
&\leq -\Big[c_i^{-}-\frac{M}{m}\Big(\sum_{j=1}^n(a_{ij}^{+}-b_{ij}^{+})L_j^f\Big)
\Big]\gamma+\frac{1}{m}u_i^{+}<0, \label{e2.6}
\end{align}
which is a contradiction and implies that \eqref{e2.3} holds. This
completes the proof.
\end{proof}

\begin{lemma} \label{lem2.3}
 Suppose that  {\rm (H1)--(H5)} hold.
 Let $y^{*}(t)=(y^{*}_{1}(t), y^{*}_{2}(t),\dots,
y^{*}_{n}(t))^{T} $ be the solution of \eqref{e2.1} with initial value
$\varphi^{*}=(\varphi^{*}_{1}(t),  \varphi^{*}_{2}(t), \dots,
\varphi^{*}_{n}(t))^{T} $, and let  $ y(t)=(y_{1}(t),
y_{2}(t),\dots,y_{n}(t))^{T} $ be the solution of \eqref{e2.1} with initial
value $ \varphi=(\varphi _{1}(t), \varphi _{2}(t), \dots, \varphi
_{n}(t))^{T}$. Then there exist constants $\lambda>0$ and $M>1$ such
that
$$
|y_{i}(t)-y^{*}_{i}(t)|\leq M \|\varphi-\varphi^{*}\|e^{-\lambda t},\quad
\text{for all } t>0,\; i=1, 2, \dots,  n.
$$
\end{lemma}

\begin{proof}
  Let $u(t)=\{u_{ i}(t) \}=\{y_{ i}(t)-y^{\ast}_{i}(t) \}=y(t)-y^{*}(t)$. Then
\begin{align}
&u_{i}'(t) \nonumber \\
&= -c_i(t)u_i(t)+\Big(\prod_{0\leq{t_k}<t}(1+\delta_{ik})^{-1}\Big)
\Big\{\sum_{j=1}^na_{ij}(t)\Big[f_j
 \Big(\prod_{0\leq{t_k}<t}(1+\delta_{jk})y_j(t)\Big) \nonumber\\
&\quad -f_j\Big(\prod_{0\leq{t_k}<t}(1+\delta_{jk})y_j^*(t)\Big)\Big]
 +\sum_{j=1}^nb_{ij}(t)\Big[f_j\Big(\prod_{0\leq{t_k}<t-\tau_{ij}(t)}
 (1+\delta_{jk})y_j(t-\tau_{ij}(t))\Big)\nonumber \\
&\quad -f_j\Big(\prod_{0\leq{t_k}<t-\tau_{ij}(t)}(1+\delta_{jk})y_j^*(t-\tau_{ij}(t))
\Big)\Big]\Big\}, \label{e2.7}
\end{align}
where $ i=1,  2, \dots,  n$.
  Next, we define a  Lyapunov functional
\begin{equation} \label{e2.8}
V_{i }(t) =|u_{i }(t)|e^{\lambda t}, \quad  i=1,  2, \dots, n.
\end{equation}
It follows from \eqref{e2.7} and \eqref{e2.8} that
\begin{align}
&D^+(V_{i }(t))\\
& \leq  D^+(|u_{i}(t)|)e^{\lambda t}+\lambda|u_{i}(t)|e^{\lambda t}\nonumber\\
& \leq (\lambda-c_i^{-})|u_{i}(t)|e^{\lambda  t}
 +\Big(\prod_{0\leq{t_k}<t}(1+\delta_{ik})^{-1}\Big)
 \Big\{\sum_{j=1}^n|a_{ij}(t)|\Big|f_j\Big(\prod_{0\leq{t_k}<t}
 (1+\delta_{jk})y_j(t)\Big) \nonumber\\
&\quad -f_j\Big(\prod_{0\leq{t_k}<t}(1+\delta_{jk})y_j^*(t)\Big)\Big| \nonumber\\
&\quad +\sum_{j=1}^n|b_{ij}(t)|\Big|f_j\Big(\prod_{0\leq{t_k}<t-\tau_{ij}(t)}
 (1+\delta_{jk})y_j(t-\tau_{ij}(t))\Big)\nonumber\\
&\quad -f_j\Big(\prod_{0\leq{t_k}<t-\tau_{ij}(t)}(1+\delta_{jk})
 y_j^*(t-\tau_{ij}(t))\Big)\Big|\Big\} e^{\lambda t} \nonumber\\
& \leq  (\lambda-c_i^{-})|u_{i}(t)|e^{\lambda   t}
 +\Big(\prod_{0\leq{t_k}<t}(1+\delta_{ik})^{-1}\Big)
 \Big[\sum_{j=1}^na_{ij}^{+}L_j^f
 \Big(\prod_{0\leq{t_k}<t}(1+\delta_{jk})\Big)|u_j(t)| \nonumber\\
&\quad +\sum_{j=1}^nb_{ij}^{+}L_j^f
 \Big(\prod_{0\leq{t_k}<t-\tau_{ij}(t)}(1+\delta_{jk})\Big)
 |u_j(t-\tau_{ij}(t))|\Big]e^{\lambda t}, \label{e2.9}
\end{align}
 where  $i=1, 2, \dots,n$.
 Let $M>1$ denote an arbitrary real number
and set
$$
\|\varphi-\varphi^{*}\|=\sup_{-\tau\leq s\leq0}\max_{1\leq j\leq n }
 |\varphi_{ j}(s)-\varphi_{j}^{*}(s)|>0, \quad j=1, 2, \dots, n.
$$
Then by \eqref{e2.9}, we have
$$
V_{i }(t) =|u_{i }(t)|e^{\lambda t}<M\|\varphi-\varphi^{*}\|\quad
\text{for all }  t\in [-\infty, 0],\; i=1,  2, \dots, n.
$$
Thus we can claim that
\begin{equation} \label{e2.10}
 V_{i }(t) =|u_{i }(t)|e^{\lambda t}< M\|\varphi-\varphi^{*}\|, \quad
\text{for  all }  t>0,  i=1,  2, \dots,  n.
\end{equation}
Otherwise, there must exist $i \in \{ 1,  2,  \dots,  n \}$ and
$t_i>0$ such that
\begin{equation} \label{e2.11}
V_{i}(t_i)=M\|\varphi-\varphi^{*}\|, \quad
V_{j}(t)<M\|\varphi-\varphi^{*}\|\quad  \text{for all }  t\in [-\tau,
t_i),\; j=1, 2, \dots,  n.
\end{equation}
Combining  \eqref{e2.9} with  \eqref{e2.11}, we obtain
\begin{align*}
0 &\leq    D^+(V_{i }(t_i)-M\|\varphi-\varphi^{*}\|)
=   D^+(V_{i }(t_i)) \nonumber\\
&\leq  (\lambda-c_i^{-})|u_{i}(t_i)|e^{\lambda   t_i} \nonumber \\
&\quad +\Big(\prod_{0\leq{t_k}<t_i}(1+\delta_{ik})^{-1}\Big)
 \Big[\sum_{j=1}^na_{ij}^{+}L_j^f\Big(\prod_{0\leq{t_k}<t_i}
 (1+\delta_{jk})\Big)|u_j(t_i)|e^{\lambda \tau_0} \nonumber\\
&\quad +\sum_{j=1}^nb_{ij}^{+}L_j^f
 \Big(\prod_{0\leq{t_k}<t_i-\tau_{ij}(t_i)}(1+\delta_{jk})\Big)
 |u_j(t_i-\tau_{ij}(t_i))|e^{\lambda t_i}\Big]\nonumber\\
&= (\lambda-c_i^{-})|u_{i}(t_i)|e^{\lambda   t_i} \nonumber \\
&\quad +\Big(\prod_{0\leq{t_k}<t_i}(1+\delta_{ik})^{-1}\Big)
 \Big[\sum_{j=1}^na_{ij}^{+}L_j^f\Big(\prod_{0\leq{t_k}<t_i}(1+\delta_{jk})\Big)
 |u_j(t_i)|e^{\lambda t_i} \nonumber\\
&\quad +\sum_{j=1}^nb_{ij}^{+}L_j^f\Big(\prod_{0\leq{t_k}<\tau_0-\tau_{ij}(t_i)}
 (1+\delta_{jk})\Big)|u_j(t_i-\tau_{ij}(t_i))|e^{\lambda
(t_i-\tau_{ij}(t_i))}e^{\lambda\tau_{ij}(t_i)}\Big] \nonumber\\
&\leq (\lambda-c_i^{-})M\|\varphi-\varphi^{*}\| \nonumber\\
&\quad +\Big(\prod_{0\leq{t_k}<t_i}(1+\delta_{ik})^{-1}\Big)
 \Big[\sum_{j=1}^na_{ij}^{+}L_j^f\Big(\prod_{0\leq{t_k}<t_i}(1+\delta_{jk})\Big)
 M\|\varphi-\varphi^{*}\| \nonumber\\
&\quad +\sum_{j=1}^nb_{ij}^{+}L_j^f
 \Big(\prod_{0\leq{t_k}<t_i-\tau_{ij}(t_i)}(1+\delta_{jk})\Big)
 e^{\lambda\tau}M\|\varphi-\varphi^{*}\|\Big]\nonumber\\
&= \Big\{(\lambda-c_i^{-})+\Big(\prod_{0\leq{t_k}<t_i}(1+\delta_{ik})^{-1}\Big)
 \Big[\sum_{j=1}^na_{ij}^{+}L_j^f\Big(\prod_{0\leq{t_k}<t_i}(1+\delta_{jk})\Big)
 \nonumber\\
&\quad + \sum_{j=1}^nb_{ij}^{+}L_j^f
 \Big(\prod_{0\leq{t_k}<t_i-\tau_{ij}(t_i)}(1+\delta_{jk})\Big)
 e^{\lambda\tau}\Big]\Big\}M\|\varphi-\varphi^{*}\|\nonumber\\
&\leq \Big\{(\lambda-c_i^{-})+\frac{M}{m}
 \Big[\sum_{j=1}^n(a_{ij}^{+}+b_{ij}^{+})L_j^f\Big]e^{\lambda\tau}\Big\}
 M\|\varphi-\varphi^{*}\|.
\end{align*} % \label{e2.12}
Then
$$
\lambda-c_i^{-}+\frac{M}{m}\Big[\sum_{j=1}^n(a_{ij}^{+}+b_{ij}^{+})L_j^f\Big]
 e^{\lambda\tau}>0,
$$
which contradicts (H5), then \eqref{e2.11} holds. In view of \eqref{e2.10}, we
know that
$$
V_i(t)=|u_i(t)|e^{\lambda t}<M\|\varphi-\varphi^{*}\|,i=1,2,\dots,n.
$$
Namely,
$$
|y_i(t)-y_i^*(t)|=|u_{i}(t)|<M\|\varphi-\varphi^{*}\|\quad \text{for all } t>0,\;
i=1,2,\dots,n.
$$
 This completes the proof.
\end{proof}

\begin{remark} \label{rmk2.1} \rm
If $y^{*}(t)=(y^{*}_{1}(t), y^{*}_{2}(t),\dots,y^{*}_{n}(t))^{T} $ is a
$T$-anti-periodic solution of \eqref{e2.1},  it follows from Lemma \ref{lem2.2} and
Definition \ref{def1.2} that $y^{*}(t)$ 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)--(H5)}  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 $v(t)= (v_{1}(t), v_{2}(t),\dots, \ v_{n}(t))^{T}$
be a solution of \eqref{e2.1} with initial conditions
\begin{equation}
v_{i}(s)=\varphi^{v}_{i}(s), |\varphi^{v}_{i}(s)|<\gamma, \quad
s\in(-\tau, 0], \; i=1,2,\dots,n.
\end{equation}%%%%\eqref{e3.1)
Thus according to  Lemma \ref{lem2.2}, the solution $v(t)$ is bounded and
\begin{equation}
|v_{i}(t)|<\gamma\quad  \text{for all } t\in{R},\; i=1,2,\dots,n.
\end{equation}%%%%%%\eqref{e3.2)
 From \eqref{e2.1}, we obtain
\begin{align}
&\big((-1)^{p+1}v_{i} (t + (p+1)T)\big)'\nonumber\\
&=(-1)^{p+1}\bigg\{-c_i(t+(p+1)T)v_i(t + (p+1)T)\nonumber\\
&\quad+\prod_{0\leq{t_k}<t+(p+1)T}(1+\delta_{ik})^{-1}
 \Big[\sum_{j=1}^na_{ij}(t+(p+1)T) \nonumber\\
&\quad\times f_j
  \Big(\prod_{0\leq{t_k}<t+(p+1)T}(1+\delta_{jk})v_j(t+(p+1)T)\Big) \nonumber\\
&\quad+\sum_{j=1}^nb_{ij}(t+(p+1)T)f_j
 \Big(\prod_{0\leq{t_k}<t+(p+1)T-\tau_{ij}(t+(p+1)T)}
 (1+\delta_{jk})v_j(t+(p+1)T\nonumber \\
&\quad -\tau_{ij}(t+(p+1)T))\Big)\bigg] \nonumber\\
&\quad +\prod_{0\leq{t_k}<t+(p+1)T}(1+\delta_{ik})^{-1}u_i(t+(p+1)T)\Big\}\nonumber\\
&=-c_i(t)(-1)^{p+1}v_i(t + (p+1)T) \nonumber\\
&\quad +\prod_{0\leq{t_k}<t}(1+\delta_{ik})^{-1}
 \Big[\sum_{j=1}^na_{ij}(t)f_j\Big(\prod_{0\leq{t_k}<t}(1+\delta_{jk})
 (-1)^{p+1}v_j(t+(p+1)T)\Big) \nonumber\\
&\quad + \sum_{j=1}^nb_{ij}(t)f_j
\Big(\prod_{0\leq{t_k}<t-\tau_{ij}(t)}(1+\delta_{jk})(-1)^{p+1}v_j
 (t+(p+1)T-\tau_{ij}(t))\Big)\Big] \nonumber\\
&\quad +\prod_{0\leq{t_k}<t}(1+\delta_{ik})^{-1}u_i(t), \label{e3.3}
\end{align}
 where $i=1, 2, \dots,n$.  Thus  $(-1)^{p+1}
v(t +(p+1)T)$ are the solutions of  \eqref{e2.1} on $R$ for any natural
number $p$. Then, from Lemma \ref{lem2.3}, there exists a constant $M>1$ such that
\begin{equation}
\begin{aligned}
&|(-1)^{p+1}v_{i} (t + (p+1)T)-(-1)^{k} v_{i}(t + pT)| \\
&\leq   M e^{-\lambda (t + pT)}\sup_{-\tau\leq
s\leq0}\max_{1\leq i\leq n}|v_{i} (s +  T)+ v_{i} (s)| \\
&\leq    2e^{-\lambda (t + pT)} M\gamma,
\end{aligned} \label{e3.4}
\end{equation}
where $i=1,2,\dots,n$. Thus, for any natural number $q$, we have
\begin{equation} \label{e3.5}
(-1)^{q+1} v_{i} (t + (q+1)T)
  =  v_{i} (t )  +\sum_{k=0}^{q}[(-1)^{k+1}
v _{i}(t + (k+1)T)-(-1)^{k} v_{i} (t + kT)].
\end{equation}
Hence
\begin{equation} \label{e3.6}
\begin{aligned}
&|(-1)^{q+1} v_{i} (t + (q+1)T)| \\
&\leq      | v_{i} (t )|  +\sum_{k=0}^{q}|
(-1)^{k+1} v _{i}(t + (k+1)T)-(-1)^{k} v_{i} (t + kT)|,
\end{aligned}
\end{equation}
where $i =1,2,\dots,n$. From \eqref{e3.4}, \eqref{e3.6} it follows
that $(-1)^{q+1}v_i(t+(q+1)T)$ is a fundamental sequence on any compact
set of $R$. Obviously, $\{(-1)^{q} v (t + qT)\}$ converges uniformly
 to a piece-wise continuous function
$y^{*}(t)=(y^{*}_{1}(t), y^{*}_{2}(t),\dots, y^{*}_{n}(t))^{T}$ on
any compact set of ${R}$.

Now we  show that $y^{*}(t)$ is $T$-anti-periodic solution of \eqref{e2.1}.
Firstly, $y^{*}(t)$ is $T$-anti-periodic, since
\begin{equation}
\begin{aligned}
y^{*}(t+T)
&= \lim_{q\to \infty}(-1)^{q } v (t +T+ qT) \\
&= -\lim_{(q+1)\to \infty}(-1)^{q+1 } v (t +(q +1)T)=-y^{*}(t ).
\end{aligned} \label{e3.7}
\end{equation}
In the sequel, we prove that $y^{*}(t)$ is a solution of \eqref{e2.1}.
Noting that the right-hand side of \eqref{e2.1} is piece-wise continuous,
\eqref{e3.3} implies that $ \{((-1)^{q+1} v (t +(q+1)T))'\}$ uniformly
converges to a piece-wise continuous function on any compact subset
of ${R}$. Thus, letting $q \to\infty$, we can easily obtain
\begin{equation}
\begin{aligned}
\dot{y}_i^*(t)
&=-c_i(t)y_i^*(t)+\prod_{0\leq{t_k}<t}(1+\delta_{ik})^{-1}
\Big[\sum_{j=1}^na_{ij}(t)f_j\Big(\prod_{0\leq{t_k}<t}(1+\delta_{jk})y_j^*(t)
\Big) \\
&\quad + \sum_{j=1}^nb_{ij}(t)f_j
 \Big(\prod_{0\leq{t_k}<t-\tau_{ij}(t)}(1+\delta_{jk})y_j^*(t-\tau_{ij}(t))\Big)
 \Big] \\
&\quad +\prod_{0\leq{t_k}<t}(1+\delta_{ik})^{-1}u_i(t), \quad t>0,
\end{aligned} \label{e3.8}
\end{equation}
where $i=1,2,\dots,n$. Therefore, $y^{*}(t)$ is a solution of \eqref{e2.1}.
Applying Lemma \ref{lem2.1}, Definition \ref{def1.2} and  Lemma \ref{lem2.3}, we can easily
check that $x^{*}(t)$ is globally exponentially stable.
The proof  is complete.
\end{proof}

 Shi and Dong \cite{38} investigated the following
Hopfield neural networks with impulses:
\begin{equation}
\begin{gathered}
\dot{x}_i(t)=-c_i(t)x_i(t)+\sum_{j=1}^nb_{ij}(t)f_j(x_j(t))+I_i(t),\quad t\neq t_k, \\
 x_i(t_k^{+})=(1+d_{ik})x_i(t_k), \quad k=1,2,\dots,
\end{gathered} \label{e3.9}
\end{equation}
where $i=1,2,\dots,n$. About the manning of the parameters, one can
see \cite{38}. By some analytical technique and by upper left derivative
of the Lyapunov functional with $t\neq t_k$ and $t=t_k,$ Shi and
Dong \cite{38} obtained some sufficient conditions which ensure the
existence and the global exponential stability of anti-periodic
solution of system \eqref{e3.9}. In this paper, we consider a more general
neural networks with delays and impulses. Moreover, the research
technique is different from that of \cite{38}. By transforming the neural
networks with impulses into an equivalent form without impulses and
constructing the Lyapunov functional, we obtain the sufficient
conditions which ensure the existence and global exponential
stability of anti-periodic solution of the model. From this
viewpoint, we say that the results obtained in this paper complement
the previous results in \cite{38}.


In \cite{13,22,23,30,33,34,36,47,48,51}, authors
considered the anti-periodic solution of neural networks without
impulses. In \cite{31,37,45,46,49}, authors investigated the global
exponential stability of anti-periodic solution of neural networks
with impulses by upper left derivative of the Lyapunov functional
with $t\neq t_k$ and $t=t_k$. In \cite{21}, author studied the existence
and global exponential stability anti-periodic solution of neural
networks with impulses by the method of coincidence degree theory
and Lyapunov functions. In this paper, we firstly transform the
neural networks with impulses into an equivalent neural networks
without impulses, then consider the existence and global exponential
stability of anti-periodic solution of the equivalent model by
constructing a suitable Lyapunov functional. To the best of our
knowledge, there are very few papers that deal with this aspect.
Moreover, all the results in \cite{13,22,23,30,31,33,34,36,37,45,46,47,48,49,51}
and the references therein cannot applicable to system \eqref{e1.1} to
obtain the existence and global exponential stability of
anti-periodic solutions. Therefore the results obtained in this
paper are essentially new and complement the previous publications.


\section{An example}

In this section, we  illustrate the results
obtained in previous sections. Let $n=2$, consider the cellular
neural networks with time-varying delays and impulsive effects
\begin{equation}
\begin{gathered}
\begin{aligned}
\dot{x}_1(t)&=-c_1(t)x_1(t)+\sum_{j=1}^2a_{1j}(t)f_j(x_j(t))\\
&\quad +\sum_{j=1}^2b_{1j}(t)f_j(x_j(t-\tau_{1j}(t)))+u_1(t), \quad
 t\neq{t_k}, 
\end{aligned}\\
\begin{aligned}
\dot{x}_2(t)&=-c_2(t)x_2(t)+\sum_{j=1}^2a_{2j}(t)f_j(x_j(t))\\
&\quad +\sum_{j=1}^2b_{2j}(t)f_j(x_j(t-\tau_{2j}(t)))+u_2(t), \quad
 t\neq{t_k},
\end{aligned}\\
{x_1}(t_k^{+})=(1+\delta_{1k})x_i(t_k),\quad k=1,2,\dots,\\
{x_2}(t_k^{+})=(1+\delta_{2k})x_i(t_k),\quad k=1,2,\dots,
\end{gathered} \label{e4.1}
\end{equation}
which is  equivalent to
\begin{gather}
\begin{aligned}
\dot{y}_1(t)&=-c_1(t)y_1(t)+\prod_{0\leq{t_k}<t}(1+\delta_{1k})^{-1}
\Big[\sum_{j=1}^2a_{ij}(t)f_j\Big(\prod_{0\leq{t_k}<t}(1+\delta_{jk})y_j(t)\Big)
\\
&\quad +\sum_{j=1}^2b_{1j}(t)f_j
\Big(\prod_{0\leq{t_k}<t-\tau_{1j}(t)}(1+\delta_{jk})y_j(t-\tau_{1j}(t))\Big)\Big]\\
&\quad +\prod_{0\leq{t_k}<t}(1+\delta_{1k})^{-1}u_1(t),\quad t>0
\end{aligned} \nonumber\\
\begin{aligned}
\dot{y}_2(t)&=-c_2(t)y_2(t)+\prod_{0\leq{t_k}<t}(1+\delta_{2k})^{-1}
\Big[\sum_{j=1}^2a_{2j}(t)f_j\Big(\prod_{0\leq{t_k}<t}(1+\delta_{jk})y_j(t)\Big)
\\
&\quad +\sum_{j=1}^2b_{2j}(t)f_j
\Big(\prod_{0\leq{t_k}<t-\tau_{2j}(t)}(1+\delta_{jk})y_j(t-\tau_{2j}(t))\Big)\Big]\\
&\quad +\prod_{0\leq{t_k}<t}(1+\delta_{2k})^{-1}u_2(t),\quad t>0, 
\end{aligned} \label{e4.2}
\end{gather} 
where $f_j(u)=\frac{1}{2}(|u+1|-|u-1|)$ ($j=1,2$), $u_1(t)=0.1\sin t$,
$u_2(t)=0.2\cos t$ and
\begin{gather*}
    \begin{bmatrix}
      c_1(t) & c_2(t) \\
      u_1(t) & u_2(t)
    \end{bmatrix}
=\begin{bmatrix}
              3+|\cos t| & 3+|\sin t| \\
              2\sin t &  3\sin t
            \end{bmatrix}
\\
\begin{bmatrix}
      a_{11}(t) & a_{12}(t) \\
      a_{21}(t) & a_{22}(t)
    \end{bmatrix}
=\begin{bmatrix}
              \frac{1}{5}|\sin t| &\frac{1}{5}|\cos t| \\
              \frac{1}{4}|\cos t| & \frac{1}{4}|\sin t|
            \end{bmatrix},
\\
\begin{bmatrix}
      b_{11}(t) & b_{12}(t) \\
      b_{21}(t) & b_{22}(t)
    \end{bmatrix}
=\begin{bmatrix}
              \frac{1}{4}|\sin t| &  \frac{1}{4}|\cos t| \\
              \frac{1}{5}|\cos t| & \frac{1}{5}|\sin t|
            \end{bmatrix},
\\
    \begin{bmatrix}
      \tau_{11}(t) & \tau_{12}(t) \\
      \tau_{21}(t) & \tau_{22}(t)
    \end{bmatrix}
= \begin{bmatrix}
              0.05|\sin t| &  0.05|\sin t| \\
               0.04|\cos t| &  0.04|\cos t| \\
            \end{bmatrix}
\end{gather*}
Then $L_j^f=1, c_1^{-}=c_2^{-}=2$, $a_{1j}^{+}=0.2$,
$a_{2j}^{+}=0.25$, $b_{1j}^{+}=0.25$, $b_{2j}^{+}=0.2$,
$\tau=0.05$.
 Let  $\eta=0.6$,
$\lambda=0.5$, $m=1$ and $M=2$. Then
$$
\lambda-c_i^{-}+\frac{M}{m}\Big[\sum_{j=1}^2(a_{ij}^{+}+b_{ij}^{+})L_j^f\Big]
e^{\lambda\tau}<0.5-3+0.9\times2\times{e^{0.05\times0.5}}
=-0.6544<-0.6<0,
$$
which implies that system \eqref{e4.2} satisfies all the conditions in
Theorem \ref{thm3.1}. Thus we can conclude that  \eqref{e4.1} has exactly one
$\pi$-anti-periodic solution. Moreover, this solution is globally
exponentially stable. The results are verified by the numerical
simulations in Figure \ref{fig1}.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1}
\end{center}
\caption{Time response of state variables $x_1(t)$ (red) and
$x_2(t)$ (blue).}
\label{fig1}
\end{figure}

\subsection*{Conclusion}
In this paper, we investigated the asymptotical behavior of a
cellular neural networks with time-varying delays and impulsive
effects. Applying the fundamental theorem, we reduce the existence
of solution of system \eqref{e1.1} to the corresponding problem for a
delayed differential equation without impulses and derive a series
of new sufficient conditions to guarantee the existence and global
exponential stability of an anti-periodic solution for the cellular
neural networks with time-varying delays and impulsive effects.  The
obtained conditions are easily to check in practice. Finally, an
example is given to illustrative the feasibility and effectiveness.
To the best of our knowledge, there are only few papers that focus
on the anti-periodic solution problem of cellular neural networks
with impulsive effects by reducing the impulsive cellular neural
networks to the cellular neural networks without impulse. Thus our
work is new and an excellent complement of previously known results.

\subsection*{Acknowledgments}
This work was supported by the National Natural Science Foundation of China
 (Nos. 11261010,11201138, 11101126), by the Natural Science and
Technology Foundation of Guizhou Province (J[2015]2025), by the
125 Special Major Science and Technology of Department of Education of Guizhou
Province ([2012]011), and by the Natural Science Innovation Team Project of
Guizhou Province ([2013]14).

\begin{thebibliography}{00}

\bibitem{1} A. R. Aftabizadeh, S. Aizicovici, N. H. Pavel;
\emph{On a class of second-order anti-periodic boundary value problems},
Journal of Mathematical Analysis and Applications  171 (2) (1992), 301-320.

\bibitem{2} S. Aizicovici, M. McKibben, S. Reich;
\emph{Anti-periodic solutions to nonmonotone evolution equations with
discontinuous nonlinearities}, Nonlinear Analysis: Theory, Methods \& Applications
 43 (2) (2001), 233-251.

\bibitem{3} M. U. Akhmet;
\emph{On the general problem of stability for
impulsive differential equations}, Journal of Mathematical Analysis
and Applications 288 (1) (2003), 182-196.

\bibitem{4} J. D. Cao;
\emph{New results concerning exponential stability and periodic solutions
of delayed cellular neural networks}, Physics Letters A  307 (2-3) (2003),  136-147.

\bibitem{5} J. D. Cao, D. W. C. Ho, X. Huang;
\emph{LMI-based criteria for global robust stability of bidirectional
associative memory networks with time delay}, Nonlinear Analysis  66
(7) (2007), 1558-1572.

\bibitem{6} J. D. Cao, J. Wang;
\emph{Global exponential stability and periodicity of recurrent neural networks
with time delays}, IEEE Transactions on Circuits and Systems. Part I:
Regular Papers  52 (5) (2005), 920-931.

\bibitem{7} H. L. Chen;
\emph{antiperioidc wavelets}, Journal of Computational Mathematics 14 (1) (1996),
 32-39.

\bibitem{8} Y. Q. Chen, J. J. Nieto, D. O'Regan;
\emph{Anti-periodic solutions for fully nonlinear first-order differential
equations}, Mathematical and Computer Modelling 46 (9-10) (2007), 1183-1190.

\bibitem{9} M. A. Cohen, S. Grossberg;
\emph{Absolute stability of global
pattern formation and parallel memory by competetive neural
networks}, IEEE Transactions on Systems, Man, and Cybernetics
13 (5) (1983), 815-826.

\bibitem{10} J. Y. Du, H. L. Han, G. X. Jin;
\emph{On trigonometric and paratrigonometric Hermite interpolation},
Journal of Approximation Theory 131 (1) (2004), 74-99.

\bibitem{11} Q. Y. Fan, W. T. Wang, X. J. Yi;
\emph{Anti-periodic solutions for a class of nonlinear $n$th-order differential
equations with delays}, Journal of Computational and Applied
Mathematics 230 (2) (2009), 762-769.

\bibitem{12} Q. Y. Fan, W. T. Wang, X. J. Yi, L. H. Huang;
\emph{Anti-periodic solutions for a class of third-order nonlinear
differential equations with deviating argument}, Electric Journal of
Qualitative Theory of Differential Equations 8 (12) (2011), 1-12.

\bibitem{13} S. H. Gong;
\emph{Anti-periodic solutions for a class of Cohen-Grossberg neural
networks}, Computers \& Mathematics with Applications 58 (2)
(2009), 341-347.

\bibitem{14} Z. J. Gui, W. G. Ge;
\emph{Periodic solutions of nonautonomous cellular neural networks with impulses,
 Chaos}, Solitons and Fractals  32 (5) (2007), 1760-1771.

\bibitem{15} Z. D. Huang, L. Q. Peng, M. Xu;
\emph{Anti-periodic solutions for high-order cellular neural netowrks with
time-varying delays},
Electric Journal of Differential Equations 2010 (5) (2010), 1-9.

\bibitem{16} Z. T. Huang, Q. G. Yang;
\emph{Exponential stability of impulsive high-order cellular neural networks
with time-varying delays}, Nonlinear Analysis: Real World Applications 11 (1)
(2010), 592-600.

\bibitem{17} H. Y. Kang, H. Zhou, B. Li;
\emph{Exponential Stability of Solution for Delayed Cellular Neural
Networks with Impulses}, Procedia Engineering 15 (2011), 1626-1631.

\bibitem{18} V. Lakshmikantham, D. D. Bainov, P. S. Simeonov;
\emph{Theory of Impulsive Differential Equations}, World Scientific, Singapore,
1989.

\bibitem{19} Y. K. Li;
\emph{Anti-periodic solutions to impulsive
shunting inhibitory cellular neural networks with distributed delays
on time scales}, Communications in Nonlinear Science and Numerical
Simulation 16 (8) (2011), 3326-3336.

\bibitem{20} Y. K. Li, W. Y. Xing, L. H. Lu;
\emph{Existence and global exponential stability of periodic
solution of a class of neural networks with impulses}, Chaos,
Solitons and Fractals 27 (2) (2006), 437-445.

\bibitem{21} Y. K. Li, E. L. Xu, T. W. Zhang;
\emph{Existence and stability of anti-periodic solution for a class of generalized
neural networks with impulsives and arbitrary delays on time scales},
Journal of Inequalities and Applications,  Volume 2010, Article ID
132790, 19 pages.

\bibitem{22} Y. K. Li, L. Yang;
\emph{Anti-periodic solutions for Cohen-Grossberg neural netowrks with
bounded and unbounded dealys},
Communications in Nonlinear Science and Numerical Simulation 14 (7) (2009),
3134-3140.

\bibitem{23} Y. K. Li, L. Yang, W. Q. Wu;
\emph{Anti-periodic solutions for a class of Cohen-Grossberg neural
networks with time-varying on time scales}, International Journal
of System Science 42 (7) (2011), 1127-1132.

\bibitem{24} Y. Q. Li, L. H. Huang;
\emph{Anti-periodic solutions for a class of Lienard-type systems with
continuously distributed delays}, Nonlinear Analysis: Real World Applications
10 (4) (2009), 2127-2132.

\bibitem{25} B. W. Liu;
\emph{An anti-periodic LaSalle oscillation theorem for a class of
functional differential equations}, Journal of Computational and
Applied Mathematics 223 (2) (2009), 1081-1086.

\bibitem{26} B. W. Liu;
\emph{Anti-periodic solutions for forced Rayleigh-type
equations}, Nonlinear Analysis: Real World Applications 10 (5) (2009),
2850-2856.

\bibitem{27} X. Liu, J. D. Cao;
\emph{Exponential stability of anti-periodic solutions for
neural networks with multiple discrete and distributed delays,
Proceedings of the Institution of Mechanical Engineers, Part I},
Journal of Systems and Control Engineering  223 (3) (2009), 299-308.

\bibitem{28} X. Lv, P. Yan, D.J. Liu;
\emph{Anti-periodic solutions for a class of nonlinear second-order Rayleigh
equations with delays}, Communications in Nonlinear Science and Numerical
Simulation 15 (11) (2010), 3593-3598.

\bibitem{29} S. Mohamad;
\emph{Exponential stability in Hopfield-type neural networks with impulses},
Chaos, Solitons and Fractals 32 (2) (2007), 456-467.

\bibitem{30} C. X. Ou;
\emph{Anti-periodic solutions for high-order Hopfield neural networks},
Computers $\&$ Mathematics with Applications  56 (7) (2008),
1838-1844.

\bibitem{31} L. J. Pan, J. D. Cao;
\emph{Anti-periodic solution for delayed
cellular neural networks with impulsive effects}, Nonlinear Analysis:
Real World Applications 12 (6) (2011), 3014-3027.

\bibitem{32} J. Y. Park, T. G. Ha;
\emph{Existence of anti-periodic solutions for quasilinear parabolic
hemivariational inequalities}, Nonlinear Analysis: Theory,
Methods \& Applications 71 (7-8) (2009), 3203-3217.

\bibitem{33} G. Q. Peng, L. H. Huang;
\emph{Anti-periodic solutions for shunting inhibitory cellular neural
networks with continuously distributed delays}, Nonlinear Analysis:
Real World Applications 10 (40) (2009) 2434-2440.

\bibitem{34} L. Peng, W. T. Wang;
\emph{Anti-periodic solutions for shunting inhibitory cellular neural networks
with time-varying delays in leakage terms}, Neurocomputing 111 (2013), 27-33.

\bibitem{35} A. M. Samoilenko, N. A. Perestyuk;
\emph{Impulsive Differential Equations}, World Scientific, Singapore, 1995.

\bibitem{36} J. Y. Shao;
\emph{Anti-periodic solutions for shunting
inhibitory cellular neural networks with time-varying delays},
Physics Letters A 372 (30) (2008), 5011-5016.

\bibitem{37} P. L. Shi, L. Z. Dong;
\emph{Anti-periodic solutions for neural
networks with delays and impulses}, Mathematical and Computational
Applications 18 (1) (2013), 50-61.

\bibitem{38} P.L. Shi, L. Z. Dong;
\emph{Existence and exponential stability of anti-periodic solutions
of Hopfield neural networks with impulses}, Applied Mathematics and Computation
216 (2) (2010), 623-630.

\bibitem{39} J. T. Sun, Q. G. Wang, H. Q. Gao;
\emph{Periodic solution for nonautonomous cellular neural networks with impulses},
Chaos, Solitons and Fractals 40 (3) (2009), 1423-1427.

\bibitem{40} W. Wang, J. Shen;
\emph{Existence of solutions for anti-periodic boundary value problems},
Nonliear Analysis 70 (2) (2009), 598-605.

\bibitem{41} Y. X. Wang, W. M. Xiong, Q. Y. Zhou, B. Xiao, Y. H. Yu;
\emph{Global exponential stability of cellular neural networks with
continuously distributed delays and impulses}, Physics Letters A 350
(1-2) (2006), 89-95.

\bibitem{42} Y. H. Xia, J. D. Cao, Z. K. Huang;
\emph{Existence and exponential stability of almost periodic solution for
shunting inhibitory cellular neural networks with impulses},
Chaos, Solitons and Fractals 34 (5) (2007), 1599-1607.

\bibitem{43} H. J. Xiang, J. D. Cao;
\emph{Almost periodic solution of recurrent neural networks with continuously
distributed delays}, Nonlinear Analysis 71 (12) (2009), 6097-6108.

\bibitem{44} W. M. Xiong, Q. Y. Zhou, B. Xiao, Y. H. Yu;
\emph{Global exponential stability of cellular neural networks with mixed
delays and impulses}, Chaos, Solitons and Fractals 34 (3) (2007)
896-902.

\bibitem{45} C.J. Xu, Y. S. Wu;
\emph{Anti-periodic solutions for high-order cellular neural networks
 with mixed delays and impulses}, Advances in Difference Equations 161 (2015) doi:
 10.1186/s13662-015-0497-4.

\bibitem{46} C. J. Xu, Q. M. Zhang;
\emph{Existence and exponential stability of
anti-periodic solutions for a high-order delayed Cohen-Grossberg
neural networks with impulsive effects},  Neural Processing Letters 40
(3) (2014), 227-243.

\bibitem{47} C. J. Xu, Q. M. Zhang;
\emph{Existence and global exponential stability of anti-periodic
 solutions of high-order bidirectional  associative memory (BAM)
networks with time-varying delays on time scales},
  Journal of Computational Science  8 (2015) 48-61.

\bibitem{48} C. J. Xu, Q. M. Zhang;
\emph{Existence and stability of pseudo almost periodic solutions for
shunting inhibitory cellular neural networks with neutral type delays
and time-varying leakage delays}, Network: Computation in Neural Systems
25 (4) (2014), 168-192.

\bibitem{49} C.J. Xu, Q. M. Zhang;
\emph{On anti-periodic solutions for Cohen-Grossberg shunting
 inhibitory neural networks with time-varying delays and impulses},
Neural Computation  26 (10) (2014), 2328-2349


\bibitem{50} Y. H. Yu, J. Y. Shao, G. X. Yue;
\emph{Existence and uniqueness of anti-periodic solutions for a kind
of Rayleigh equation with two deviating arguments},
Nonlinear Analysis: Theory, Methods \& Applications
71 (10) (2009), 4689-4695.

\bibitem{51} A. P. Zhang;
\emph{Existence and exponential stability of
anti-periodic solutions for HCNNs with time-varying leakage delays},
Advances in Difference Equations  162 (2013) doi:10.1186/1687-1847-2013-162.

\bibitem{52} Y. P. Zhang;
\emph{Stationary oscillation for cellular neural networks with time delays
and impulses}, Mathematics and Computers in Simulation 79 (10) (2009), 3174-3178.

\end{thebibliography}

\end{document}