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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 50, pp. 1--11.\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-2010/50\hfil Anti-periodic solutions]
{Anti-periodic solutions for recurrent neural networks without
assuming global Lipschitz conditions}

\author[H. Zhang, Y. Wu   \hfil EJDE-2010/50\hfilneg]
{Hong Zhang, Yuanheng Wu}  

\address{Hong Zhang \newline
Department of Mathematics, Hunan University of Arts and
Science, Changde, Hunan 415000, China}
\email{hongzhang320@yahoo.com.cn}

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

\thanks{Submitted December 11, 2009. Published April 9, 2010.}
\thanks{Supported by grants 06JJ2063, 07JJ6001 from the
Scientific Research Fund of Hunan \hfill\break\indent Provincial
Natural Science Foundation,  and   08C616, 09B072 from the
Scientific Research Fund \hfill\break\indent of Hunan Provincial
Education Department of China} 
\subjclass[2000]{34C25, 34D40}
\keywords{Recurrent neural networks;
   anti-periodic; exponential stability; delay}

\begin{abstract}
 In this paper we study recurrent neural networks with time-vary\-ing
 delays and continuously distributed delays. Without assuming
 global Lipschitz conditions on the activation functions, we
 establish the 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}
\allowdisplaybreaks

\section{Introduction}

 We consider the following model for recurrent neural
networks(RNNs) with time-varying delays and continuously
distributed  delays
\begin{equation}
\begin{aligned}
x'_{i}(t) &=  -c_{i}(t)\alpha_{i}(x_{i}(t))
+\sum_{j=1}^{n}a_{ij}(t)\tilde{g}_{j}(x_{j}(t-\tau_{ij}(t)))\\
&\quad +\sum_{j=1}^{n}b_{ij}(t) \int_{0}^{\infty}K_{ij}(u)
g_{j}(x_{j}(t-u))du+I_{i}(t), \quad  i=1,  2, \dots, n,
\end{aligned}    \label{e1.1}
\end{equation}
in which $n$ corresponds to the number of units in a neural network,
$x_{i}(t)$  corresponds to the state vector of the $i$th unit at
the time $t$, $c_{i}(t)>0$  represents the rate
with which the $i$th unit will reset its potential to the resting
state in isolation when disconnected from the network and external
inputs at the time $t$.   $a_{ij}(t)$ and $ b_{ij}(t)  $ are the
connection weights at the time $t$, $\tau_{ij}(t)\geq 0$
corresponds to the transmission delay of the $i$th unit
along the axon of the $j$th unit at the time $t$, and $I_{i}(t)$
denote the external inputs at time $t$. $\tilde{g}_{j}$ and
$ g_{j}$  are activation functions of signal transmission.

As we know, RNNs is very general and includes Hopfield neural
networks, cellular neural networks and BAM neural networks. The
RNNs have been successfully applied to signal and image
processing, pattern recognition and optimization. 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 for RNNs in the literature. We refer the
reader to \cite{c1,c2,l1,h2}
 and the references cited therein.  Recently,
\cite{l2,o1,s1} obtained some sufficient conditions for the existence and
stability of  the anti-periodic solutions of RNNs. Moreover, in
the above-mentioned literature \cite{l2,o1,s1}, we observe that the
 assumption
\begin{itemize}
\item[(T0)] For each $j\in\{1,  2,  \dots,  n \}$, the
activation function $\tilde{g}_{j},g_{j}: \mathbb{R}\to \mathbb{R}$ is global
Lipschitz with  Lipschitz constants  $\tilde{L}_{j}$ and  $L_{j}$;
i.e.,
\begin{equation}
|\tilde{g}_{j}(u_{j})-\tilde{g}_{j}(v_{j})| \leq
\tilde{L}_{j}|u_{j}-v_{j}|, |g_{j}(u_{j})-g_{j}(v_{j})|\leq
L_{j}|u_{j}-v_{j}|,\label{e1.1b}
\end{equation}
for  all $u_{j},  v_{j}\in \mathbb{R}$.
\end{itemize}
has been considered as fundamental for the considered existence
and stability of anti-periodic solutions of RNNs. However,
to the best of our knowledge, few authors have considered
the problems of anti-periodic solutions of RNNs
without the  assumptions (T0).  Since the existence of
anti-periodic solutions
  play a key role in characterizing the behavior of nonlinear
differential equations (See \cite{a1,a2,c3,d1,o2,w1}). It  is worth
    while to continue to investigate the existence and
stability of anti-periodic solutions of  RNNs.

The main purpose of this paper is to give the conditions for the
existence and exponential stability of  the anti-periodic
solutions for system \eqref{e1.1}. We derive some new sufficient
conditions ensuring the existence  and local exponential stability
of the anti-periodic  solution for system \eqref{e1.1}, which are new
and complement to previously known
    results.  In particular, we do not need the assumption (T0) .
Moreover, an example is also
 provided to illustrate the effectiveness of 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,
$$
 u(t+T)=-u(t)  \quad\text{for   all  }  t\in \mathbb{R}.
$$
Throughout this article, for $i,  j=1,  2,  \dots,  n$,    it
will be assumed that $c_{i},   I_{i},   a_{ij},  b_{ij}:
\mathbb{R}\to \mathbb{R}$ and   $\tau_{ij}: \mathbb{R}\to [0, +\infty)$
are continuous $2T-$periodic functions,
and
\begin{gather}
\begin{gathered}
c_{i}(t+T)\alpha_{i}(u)=-c_{i}(t )\alpha_{i}( -u), \quad
a_{ij}(t+T)\tilde{g}_{j}(u)=-a_{ij}(t )\tilde{g}_{j}(-u), \\
\forall t,u \in \mathbb{R},
\end{gathered} \label{e1.2}
\\
\begin{gathered}
b_{ij}(t+T) = -b_{ij}(t ) ,\quad
g_{j}(u) =  g_{j}(-u)) \quad \text{or}\\
b_{ij}(t+T)  =   b_{ij}(t ),\;  g_{j}(u) = -   g_{j}(-u), \quad
\forall t,u \in \mathbb{R} ,
\end{gathered}\label{e1.3}
\\
\tau_{ij}(t+T)=\tau_{ij}(t),\quad
I_{i}(t+T)=-I_{i}(t ), \quad \forall  u \in \mathbb{R} \, .
\label{e1.4}
\end{gather}
Then, we can choose  constants $\overline{I}$  and $\tau $
    such that
\begin{equation}
\overline{I}=\max_{1\leq i \leq n}\sup_{t\in \mathbb{R}}|I_{i}(t)| ,\quad
\tau=\max_{1\leq i,j\leq n}\{\max_{t\in [0, T]}\tau_{ij}(t)\}.
 \label{e1.5}
\end{equation}
We also assume that the following conditions:
\begin{itemize}
\item[(H0)]
 For each $i\in\{1,  2,  \dots,  n \}$,
$\alpha_{i}: \mathbb{R}\to \mathbb{R}$
    are continuous function,   and there exist constants
$\underline{d}_{i}$   such that
$\alpha_{i}(0 )=0$,
$\underline{d}_{i}|u -v |\leq \mathop{\rm sign}(u-v)(\alpha_{i}(u )-\alpha_{i}(v ))$,
for all $u,  v \in \mathbb{R}$.

\item[(H1)]
 For each $j\in\{1,  2,  \dots,  n \}$, there exist
$\tilde{f}_{j}, \tilde{h}_{j}, f_{j}, h_{j}\in C(R,  R)$
and constants  $L^{\tilde{f}}_{j}, L^{\tilde{h}}_{j},
L^{f}_{j}, L^{h}_{j}  \in [0,  +\infty)$  such
that the following conditions  are  satisfied:
\begin{itemize}
 \item[(1)]  $\tilde{f}_{j}(0)=0$, $\tilde{h}_{j}(0)=0$,
 $\tilde{g}_{j}(u )=\tilde{f}_{j}(u )\tilde{h}_{j}(u )$,
 for all $u \in \mathbb{R}$;
 \item[(2)] $|\tilde{f}_{j}(u)-\tilde{f}_{j}(v)|
 \leq L^{\tilde{f}}_{j}|u-v|, |\tilde{h}_{j}(u)-\tilde{h}_{j}(v)|
 \leq L^{\tilde{h}}_{j}|u-v|$,
  for  all $u, v\in \mathbb{R}$.

\item[(3)]  $f_{j}(0)=0$, $h_{j}(0)=0$, $g_{j}(u )=f_{j}(u )h_{j}(u )$,
 for  all $u \in \mathbb{R}$;

\item[(4)]  $|f_{j}(u)-f_{j}(v)|\leq L^{f}_{j}|u-v|, |h_{j}(u)-h_{j}(v)|
 \leq L^{h}_{j}|u-v|$, for all
 $u,  v\in \mathbb{R}$.
\end{itemize}

\item[(H2)]  For $i, j\in\{1, 2,  \dots,  n \}$, the
delay kernels $K_{ij}:[0, \infty)\to \mathbb{R}$ are continuous,
integrable.

\item[(H3)] There exist constants $\eta>0,  \lambda>0$ and
$\xi_{i}>0, i=1,  2, \dots,  n$, such that for all $t>0$,
there holds
$ 0<\xi_{i}\frac{\overline{I}}{\eta}\leq 1$ and
\begin{align*}
& ( \lambda -  c _{i } (t ))\underline{d}_{i}\xi_{i} +
     \sum^n_{j=1}3|a_{ij}(t )|
    e^{\lambda  \tau  } L^{\tilde{f}}_{j} L^{\tilde{h}}_{j}\xi_{j}\\
&+ \sum^n_{j=1}3|b_{ij}(t )|\int_{0}^{\infty}|K_{ij}(u)|
     e^{\lambda u}duL^{f}_{j} L^{h}_{j}\xi_{j}
<-\eta .
\end{align*}
\end{itemize}
We will use the following notation:
$x=\{x_{j}\}=(x_{1},  x_{2}, \dots,  x_{n})^{T}\in \mathbb{R}^{n}$
to denote a column vector, in which the symbol $(^{T})$ denotes the transpose of a
vector. We let $|x|$ denote the
absolute-value vector given by
$|x|=(|x_{1}|,  |x_{2}|,  \dots,  |x_{n}|)^{T} $,
and define $\|x\|=\max_{1\leq i\leq n} |x_{i}|$.


The initial conditions associated with system \eqref{e1.1} are of the
form
\begin{equation}
x_{i}(s)=\varphi_{i}(s),\quad s\in (-\infty, 0], \; i=1,2,\dots,n,
\label{e1.6}
\end{equation}
where $\varphi_{i}(\cdot)$   denotes real-valued bounded
continuous function defined on $(-\infty, 0]$.


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

\section{Preliminary Results}

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

\begin{lemma} \label{lem2.1}
 Let {\rm (H0)--(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{I} }{\eta
 }\leq 1 , \quad s\in (-\infty, 0], \; i=1,2,\dots,n.\label{e2.1}
\end{equation}
Then
\begin{equation}
|\widetilde{x}_{i}(t)| < \xi_{i}\frac{\overline{I} }{\eta
 }, \quad \text{for  all }  t\geq 0, \; i=1,2,\dots,n. \label{e2.2}
\end{equation}
\end{lemma}

\begin{proof}
Assume, by way of contradiction, that \eqref{e2.2} does not hold.
Then, there must exist $i\in \{1,2,\dots,n \}$ and $\rho>0$ such
that
\begin{equation}
|\widetilde{x}_{i}(\rho)| =\xi_{i}\frac{\overline{I} }{\eta }, \quad
|\widetilde{x}_{j}(t)| <\xi_{j}\frac{\overline{I} }{\eta }
\quad    \text{for  all  } t\in (-\infty, \rho),  \; j=1,2,\dots,n.
\label{e2.3}
\end{equation}
 Calculating the   upper left
derivative of $|\widetilde{x}_{i}(t)|$, together with (H0),
(H1), (H2)  and (H3), \eqref{e2.3} implies
\begin{align*}
0 &\leq D^+(|\widetilde{x}_{i}(\rho)|)\\
&=-c_{i}(\rho)\alpha_{i}(\widetilde{x}_{i}(\rho))
 \mathop{\rm sgn}(\widetilde{x}_{i}(\rho))
 +\Big[\sum_{j=1}^{n}a_{ij}(\rho)\tilde{g}_{j}(\widetilde{x}_{j}
 (\rho-\tau_{ij}(\rho)))\\
&\quad +\sum_{j=1}^{n}b_{ij}(\rho) \int_{0}^{\infty}K_{ij}(u)
 g_{j}(\widetilde{x}_{j}(\rho-u))du+I_{i}(\rho)\Big]
 \mathop{\rm sgn}(\widetilde{x}_{i}(\rho))\\
&\leq -c_{i}(\rho)|\alpha_{i}(\widetilde{x}_{i}(\rho))|
 +\Big| \sum_{j=1}^{n}a_{ij}(\rho)\tilde{g}_{j}
 (\widetilde{x}_{j}(\rho-\tau_{ij}(\rho))) \\
&\quad +\sum_{j=1}^{n}b_{ij}(\rho) \int_{0}^{\infty}K_{ij}(u)
 g_{j}(\widetilde{x}_{j}(\rho-u))du+I_{i}(\rho)\Big|\\
&\leq -c_{i} (\rho)\underline{d}_{i} \xi_{i}\frac{\overline{I}}{\eta }
+\Big|\sum_{j=1}^{n}a_{ij}(\rho)\tilde{f}_{j}(\widetilde{x}_{j}
 (\rho-\tau_{ij}(\rho)))\tilde{h}_{j}(\widetilde{x}_{j}
 (\rho-\tau_{ij}(\rho)))\\
& \quad  + \sum_{j=1}^{n}b_{ij}(\rho) \int_{0}^{\infty}K_{ij}(u)f_{j}
 (\widetilde{x}_{j}(\rho-u))h_{j}(\widetilde{x}_{j}(\rho-u))du
 +I_{i}(\rho)\Big|  \\
&\leq -c_{i}(\rho)\underline{d}_{i} \xi_{i}\frac{\overline{I}}{\eta}
 + \sum_{j=1}^{n}|a_{ij}(\rho)|
 L^{\tilde{f}} _{j} \xi_{j}\frac{\overline{I} }{\eta }
 L^{\tilde{h}} _{j} \xi_{j}\frac{\overline{I} }{\eta }\\
&\quad  + \sum_{j=1}^{n}|b_{ij}(\rho)|
 \int_{0}^{\infty}|K_{ij}(u)| duL^{f}_{j} \xi_{j}
 \frac{\overline{I} }{\eta }L^{h}_{j} \xi_{j}\frac{\overline{I} }{\eta }
 +|I_{i}(\rho)|  \\
&\leq -c_{i}(\rho)\underline{d}_{i}
 \xi_{i}\frac{\overline{I} }{\eta }
 + \sum_{j=1}^{n}|a_{ij}(\rho)|
 L^{\tilde{f}} _{j} L^{\tilde{h}} _{j} \xi_{j}\frac{\overline{I} }{\eta}\\
& \quad  + \sum_{j=1}^{n}|b_{ij}(\rho)|
 \int_{0}^{\infty}|K_{ij}(u)| duL^{f}_{j} L^{h}_{j}
 \xi_{j}\frac{\overline{I} }{\eta } +|I_{i}(\rho)|\\
&\leq  [-c_{i}(\rho)\underline{d}_{i}\xi_{i}
 +\sum_{j=1}^{n}|a_{ij}(\rho)| L^{\tilde{f}} _{j} L^{\tilde{h}}_{j}\xi_{j}
 + \sum_{j=1}^{n}|b_{ij}(\rho)|
 \int_{0}^{\infty}|K_{ij}(s) |ds L^{f}_{j}
 L^{h}_{j}\xi_{j}]\frac{\overline{I} }{\eta }\\
&\quad +|I_{i}(\rho)|\\
&\leq  \Big[-c_{i}(\rho)\underline{d}_{i}\xi_{i}+
 \sum_{j=1}^{n}|a_{ij}(\rho)|e^{\lambda \tau}| L^{\tilde{f}} _{j}
 L^{\tilde{h}} _{j}\xi_{j}\\
&\quad + \sum_{j=1}^{n}|b_{ij}(\rho)|
 \int_{0}^{\infty}|K_{ij}(s)|e^{\lambda s}ds L^{f}_{j}
 L^{h}_{j}\xi_{j}\Big]\frac{\overline{I} }{\eta}
 +|I_{i}(\rho)|\\
& <  -\eta\times\frac{\overline{I} }{\eta }+|I_{i}(\rho)|\leq 0,
\end{align*}
which is a contradiction and implies that \eqref{e2.2} holds. The proof
 is complete.
\end{proof}

\begin{remark} \label{rmk2.1} \rm
 In view of the boundedness of this solution, from
\cite[Theorems 2.3-2.4]{h1},
it follows that $\widetilde{x} (t)$ can be defined on $(-\infty,
\infty)$.
\end{remark}

\begin{lemma} \label{lem2.2}
  Suppose that  {\rm (H0), (H1), (H2), (H3)}  are satisfied.
 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} $,
where
\begin{equation}
 |\varphi^{*}_{i}(s)|<\xi_{i}\frac{\overline{I} }{\eta
 }\leq 1 , \quad s\in (-\infty, 0], \; i=1,2,\dots,n. \label{e2.4}
\end{equation}
Then 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 initial value $ \varphi=(\varphi_{1}(t), \varphi_{2}(t),
\dots, \varphi_{n}(t))^{T} \in  G_{1}(x^{*})$,
$$
|x_{i}(t)-x^{*}_{i}(t)|\leq M_{\varphi}
\|\varphi-\varphi^{*}\|_{1}e^{-\lambda t}, \quad \forall t>0,  \;
i=1, 2,  \dots, n,
$$
where $\|\varphi-\varphi^{*}\|_{1}=\sup_{-\infty\leq
s\leq0}\max_{1\leq i\leq
n}|\varphi_{i}(s)-\varphi_{i}^{*}(s)|$, and
\begin{align*}
G_{1}(x^{*})
=\big\{&\varphi \in C((-\infty, 0]; \ R^{n} ):
\sup_{-\infty\leq s\leq0}|\varphi_{i}(t)-\varphi^{*}_{i}(t)|
<\frac{\xi_{i}}{\max_{1\leq j\leq n }\{\xi_{j}\}},\\
&i=1, 2, \dots ,  n \big\}.
\end{align*}
\end{lemma}

\begin{proof}
 In view \eqref{e2.4} and Lemma \ref{lem2.1},
\begin{equation}
|x^{*}_{i}(t)| < \xi_{i}\frac{\overline{I} }{\eta}\leq 1, \quad
 \text{for  all }  t\in \mathbb{R}, \; i=1,2,\dots,n. \label{e2.5}
\end{equation}
Let $y(t)=\{y_{ j}(t) \}=\{x_{ j}(t)-x^{*}_{ j}(t)\}=x(t)-x^{*}(t)$.
Then
\begin{equation}
\begin{aligned}
y_{i}'(t)&=-c_{i}(t)[\alpha_{i}(x_{i}(t))-\alpha_{i}(x^{*}_{i}(t))]\\
&\quad +\sum^n_{j=1}a_{ij}(t)
    (\tilde{g}_{j}(y_{j}(t-\tau_{ij}(t))+x^{*}_{j}(t-\tau_{ij}(t)))
   -\tilde{g}_{j}(x^{*}_{j}(t-\tau_{ij}(t))))\\
&\quad +\sum^n_{j=1}b_{ij}(t)\int_{0}^{\infty}K_{ij}(u)
    (g_{j}(y_{j}(t-u)+x^{*}_{j}(t-u))-g_{j}(x^{*}_{j}(t-u)))du,
\end{aligned}\label{e2.6}
\end{equation}
where $i=1, 2, \dots, n$.
  We consider the Lyapunov functional
 \begin{equation}
V_{i }(t) =    |y_{i }(t)|e^{\lambda t}, \quad i=1,   2, \dots, n.
 \label{e2.7}
\end{equation}
Calculating the left right
derivative of $V_{i }(t)$ along the solution $y(t)=\{y_{j }(t)\}$
of system \eqref{e2.6} with the initial value
$\bar{\varphi}=\varphi-\varphi^{*}$, from   \eqref{e2.6},  we have
\begin{align}
&D^+(V_{i }(t)) \notag \\
&\leq  - c_{i }(t) \underline{d}_{i}|y_{i }(t)|e^{\lambda t}+
     \sum^n_{j=1}|a_{ij}(t)
    (\tilde{g}_{j}(y_{j}(t-\tau_{ij}(t))+x^{*}_{j}(t-\tau_{ij}(t))) \notag \\
&\quad -\tilde{g}_{j}(x^{*}_{j}(t-\tau_{ij}(t))))|e^{\lambda t}+
     \sum^n_{j=1}|b_{ij}(t)\int_{0}^{\infty}K_{ij}(u)
    (g_{j}(y_{j}(t-u)+x^{*}_{j}(t-u)) \notag \\
&\quad -g_{j}(x^{*}_{j}(t-u)))du  |e^{\lambda t}
     +\lambda |y_{i }(t)|e^{\lambda t} \notag \\
&\leq ( \lambda - c_{i }(t) \underline{d}_{i})|y_{i }(t)|e^{\lambda t}+
     \sum^n_{j=1}|a_{ij}(t)|(|\tilde{f}_{j}(y_{j}(t-\tau_{ij}(t)) \notag \\
&\quad +x^{*}_{j}(t-\tau_{ij}(t)))
  \tilde{h}_{j}(y_{j}(t-\tau_{ij}(t)) +x^{*}_{j}(t-\tau_{ij}(t)))\notag\\
&\quad -\tilde{f}_{j}( x^{*}_{j}(t-\tau_{ij}(t)))
      \tilde{h}_{j}(y_{j}(t-\tau_{ij}(t))
 +x^{*}_{j}(t-\tau_{ij}(t)))| \notag\\
&\quad +|\tilde{f}_{j}( x^{*}_{j}(t-\tau_{ij}(t)))
   \tilde{h}_{j}(y_{j}(t-\tau_{ij}(t))+x^{*}_{j}(t-\tau_{ij}(t)))\notag\\
&\quad -\tilde{f}_{j}(x^{*}_{j}(t-\tau_{ij}(t))))
   \tilde{h}_{j}(x^{*}_{j}(t-\tau_{ij}(t)))|)e^{\lambda t} \notag\\
&\quad + \sum^n_{j=1}|b_{ij}(t)|\int_{0}^{\infty}|K_{ij}(u)|
    (|f_{j}(y_{j}(t-u)+x^{*}_{j}(t-u))h_{j}(y_{j}(t-u)+x^{*}_{j}(t-u)) \notag\\
&\quad  -f_{j}(y_{j}(t-u)+x^{*}_{j}(t-u))h_{j}( x^{*}_{j}(t-u))|
 +|f_{j}(y_{j}(t-u) \notag \\
&\quad +x^{*}_{j}(t-u))  h_{j}( x^{*}_{j}(t-u))
 -f_{j}(x^{*}_{j}(t-u))h_{j}(x^{*}_{j}(t-u))|)du
 e^{\lambda t} \notag \\
&\leq  ( \lambda - c_{i }(t)\underline{d}_{i} )|y_{i }(t)|e^{\lambda t}+
     \sum^n_{j=1}|a_{ij}(t)|L^{\tilde{f}}_{j} L^{\tilde{h}}_{j}|
     y_{j}(t-\tau_{ij}(t))|(
     |y_{j}(t-\tau_{ij}(t))  \notag \\
&\quad +x^{*}_{j}(t-\tau_{ij}(t))|
  +| x^{*}_{j}(t-\tau_{ij}(t))|)  e^{\lambda t} \notag\\
&\quad + \sum^n_{j=1}|b_{ij}(t)|\int_{0}^{\infty}K_{ij}(u) L^{f}_{j} L^{h}_{j}
    |y_{j}(t-u)|(|y_{j}(t-u)+x^{*}_{j}(t-u)|\notag\\
&\quad  +|x^{*}_{j}(t-u)|) du  e^{\lambda t}, \label{e2.8}
\end{align}
where  $i=1,  2, \dots,n$.
In view  of the definition of $\varphi \in G_{1}(Z^{*})$,
$$
 V_{i }(t)  =    |y_{i }(t)|e^{\lambda t}
< \frac{\xi_{i}}{\max_{1\leq j\leq n }\{\xi_{j}\}}, \quad
\text{for  all }     t\in (-\infty,  0], \; j=1, 2,\dots, n.
$$
We claim that
\begin{equation}
V_{i }(t) =    |y_{i }(t)|e^{\lambda t}
< \frac{\xi_{i}}{\max_{1\leq j\leq n }\{\xi_{j}\}}, \quad
\text{for  all }  t>0, \quad i=1,   2, \dots,  n.
\label{e2.9}
\end{equation}
Contrarily, there must exist $i \in \{ 1,  2, \dots, n \}$
and $t_{i}>0$ such that
$$
V_{i}(t_{i})=\frac{\xi_{i}}{\max_{1\leq j\leq n}\{\xi_{j}\}}, \quad
V_{j}(t)<\frac{\xi_{j}}{\max_{1\leq j\leq n }\{\xi_{j}\}} ,
$$
for all  $t\in (-\infty,  t_{i})$, $j=1, 2, \dots,  n$,
which, together with $\varphi \in G_{1}(Z^{*})$,  implies
\begin{equation}
V_{i}(t_{i})-\frac{\xi_{i}}{\max_{1\leq j\leq n
}\{\xi_{j}\}}=0,  \quad
V_{j}(t)-\frac{\xi_{j}}{\max_{1\leq j\leq n }\{\xi_{j}\}}<0,
\label{e2.10}
\end{equation}
for all  $t\in (-\infty,  t_{i})$, $j=1,   2, \dots,  n$,
and
\begin{equation}
|y_{j}(t)| \leq 1 , \quad \forall  t\in (-\infty, \ t_{i}), \; j=1,   2,
\dots,  n. \label{e2.11}
\end{equation}
Together with  \eqref{e2.5},  \eqref{e2.8}, \eqref{e2.10} and
\eqref{e2.11}, we obtain
\begin{align}
0 &\leq   D^+(V_{i }(t_{i })-m\xi_{i} ) \notag\\
&=  D^+(V_{i }(t_{i })) \notag\\
&\leq   ( \lambda -  c _{i } (t_{i
 })\underline{d}_{i})|y_{i }(t_{i })|e^{\lambda t_{i }}+
     \sum^n_{j=1}|a_{ij}(t_{i })|L^{\tilde{f}}_{j} L^{\tilde{h}}_{j}|
     y_{j}(t_{i }-\tau_{ij}(t_{i }))|(
     |y_{j}(t_{i }-\tau_{ij}(t_{i })) \notag \\
&\quad  +x^{*}_{j}(t_{i }-\tau_{ij}(t_{i }))|+| x^{*}_{j}(t_{i }
   -\tau_{ij}(t_{i }))|)  e^{\lambda t_{i }} \notag\\
&\quad +  \sum^n_{j=1}|b_{ij}(t_{i })|\int_{0}^{\infty}|K_{ij}(u)| L^{f}_{j} L^{h}_{j}
    |y_{j}(t_{i }-u)| \notag\\
&\quad  \times \Big(|y_{j}(t_{i }-u)+x^{*}_{j}(t_{i }-u)|+|x^{*}_{j}(t_{i }-u)|\Big) du
 e^{\lambda t_{i }} \notag\\
&\leq    ( \lambda -  c _{i } (t_{i
  })\underline{d}_{i})|y_{i }(t_{i })|e^{\lambda t_{i }}+
     \sum^n_{j=1}3|a_{ij}(t_{i })|L^{\tilde{f}}_{j} L^{\tilde{h}}_{j}|
     y_{j}(t_{i }-\tau_{ij}(t_{i }))|  e^{\lambda t_{i }} \notag\\
&\quad +  \sum^n_{j=1}3|b_{ij}(t_{i })|\int_{0}^{\infty}|K_{ij}(u)| L^{f}_{j} L^{h}_{j}
    |y_{j}(t_{i }-u)|  du
  e^{\lambda t_{i }}\notag \\
& =     ( \lambda -  c _{i } (t_{i })\underline{d}_{i})|y_{i
  }(t_{i })|e^{\lambda t_{i }} \notag\\
&\quad + \sum^n_{j=1}3|a_{ij}(t_{i })|L^{\tilde{f}}_{j} L^{\tilde{h}}_{j}||y_{j}(t_{i }-\tau_{ij}(t_{i }))|e^{\lambda (t_{i }-\tau_{ij}(t_{i }))}
    e^{\lambda  \tau_{ij}(t_{i }) }\notag \\
&\quad + \sum^n_{j=1}3|b_{ij}(t_{i })|\int_{0}^{\infty}|K_{ij}(u)|
    L^{f}_{j} L^{h}_{j}|y_{j}(t_{i }-u)|
  e^{\lambda (t_{i }-u)}e^{\lambda u}du \notag\\
&\leq  \Big[( \lambda -  c _{i } (t_{i})\underline{d}_{i})\xi_{i} +
     \sum^n_{j=1}3|a_{ij}(t_{i })|
    e^{\lambda  \tau  } L^{\tilde{f}}_{j} L^{\tilde{h}}_{j}\xi_{j} \notag\\
&\quad +     \sum^n_{j=1}3|b_{ij}(t_{i })|\int_{0}^{\infty}|K_{ij}(u)|
     e^{\lambda u}duL^{f}_{j} L^{h}_{j}\xi_{j}\Big]
  \frac{1}{\max_{1\leq j\leq n}\{\xi_{j}\}} . \label{e2.12}
\end{align}
Thus,
\begin{equation} \label{e2.13}
\begin{aligned}
0 &\leq ( \lambda -  c _{i } (t_{i})\underline{d}_{i})\xi_{i} +
     \sum^n_{j=1}3|a_{ij}(t_{i })|
    e^{\lambda  \tau  } L^{\tilde{f}}_{j} L^{\tilde{h}}_{j}\xi_{j}\\
&\quad +     \sum^n_{j=1}3|b_{ij}(t_{i })|\int_{0}^{\infty}|K_{ij}(u)|
     e^{\lambda u}duL^{f}_{j} L^{h}_{j}\xi_{j},
\end{aligned}
\end{equation}
which contradicts (H3). Hence, \eqref{e2.9} holds.  Letting
\[
 \|\varphi-\varphi^{*}\|_{1} =\sup_{-\infty\leq
s\leq0}\max_{1\leq j\leq n  }|\varphi_{ j}(s)-\varphi_{
j}^{*}(s)|>0, \quad i=1, 2, \dots, n,
\]
 and
$ M_{\varphi}>1$ such that
\begin{equation}
\frac{\xi_{i}}{\max_{1\leq j\leq n }\{\xi_{j}\}}\leq
M_{\varphi}\|\varphi-\varphi^{*}\|_{1}, \quad  i=1,2,\dots,n.
\label{e2.14}
\end{equation}
 In view of \eqref{e2.9} and \eqref{e2.13},
$$
|x_{i}(t)-x^{*}_{i}(t)|=|y_{i}(t)|\leq \max_{1\leq i\leq
n}\{m\xi_i \}e^{-\lambda t}
 \leq M_{\varphi} \|\varphi-\varphi^{*}\|_{1}e^{-\lambda t},
$$
where $i=1,2,\dots,n$, $t>0$. This completes the proof.
\end{proof}

\begin{remark} \label{rmk2.2} \rm
If $x^{*}(t)=(x^{*}_{1}(t), x^{*}_{2}(t),\dots,x^{*}_{n}(t))^{T}$
be the  $T$-anti-periodic solution of system \eqref{e1.1},   it follows
from Lemma \ref{lem2.2}   that $x^{*}(t)$ is  locally exponentially stable.
\end{remark}


\section{Main Results}

The following is our main result.

\begin{theorem} \label{thm3.1}
 Suppose that {\rm (H0)-(H3)}  are satisfied.
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_{1}(t),v_{2}(t),\dots,  v_{n}(t))^{T} $ be 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{I} }{\eta} , \quad
s\in (-\infty, 0], \; i=1,2,\dots,n. \label{e3.1}
\end{equation}
By Lemma \ref{lem2.1}, the solution $v(t) $ is bounded and
\begin{equation}
|v_{i}(t)|<\xi_{i}\frac{\overline{I} }{\eta} , \quad
\text{for all }  t\in \mathbb{R}, \; i=1,2,\dots,n.
\label{e3.2}
\end{equation}
  From \eqref{e1.1}--\eqref{e1.4},
\begin{equation}
\begin{aligned}
&((-1)^{k+1} v_{i} (t + (k+1)T))'\\
&= (-1)^{k+1} v'_{i} (t + (k+1)T)\\
&= (-1)^{k+1}\{-c_{i}(t+ (k+1)T)\alpha_{i}(v_{i}(t+ (k+1)T))\\
&\quad +\sum_{j=1}^{n}a_{ij}(t+ (k+1)T) \tilde{g}_{j}(v_{j}(t+
 (k+1)T-\tau_{ij}(t+ (k+1)T)))\\
&\quad + \sum_{j=1}^{n}b_{ij}(t+ (k+1)T)\int_{0}^{\infty}K_{ij}(u)
  g_{j}(v_{j}(t+ (k+1)T-u))du\\
&\quad +I_{i}(t+ (k+1)T)\} \\
&= (-1)^{k+1}\{-c_{i}(t+ (k+1)T)\alpha_{i}(v_{i}(t+ (k+1)T))\\
&\quad +\sum_{j=1}^{n}a_{ij}(t+ (k+1)T) \tilde{g}_{j}(v_{j}(t+
 (k+1)T-\tau_{ij}(t )))\\
&\quad + \sum_{j=1}^{n}b_{ij}(t+ (k+1)T)
 \int_{0}^{\infty}K_{ij}(u) g_{j}(v_{j}(t+ (k+1)T-u))du\\
&\quad +I_{i}(t+ (k+1)T)\}\\
 &=  -c_{i}(t )\alpha_{i}((-1)^{k+1}v_{i}(t+ (n+1)T))\\
&\quad +\sum_{j=1}^{n}a_{ij}(t )
  \tilde{g}_{j}((-1)^{k+1} v_{j}(t+(k+1)T-\tau_{ij}(t )))\\
&\quad + \sum_{j=1}^{n}b_{ij}(t ) \int_{0}^{\infty}K_{ij}(u)
  g_{j}((-1)^{k+1} v_{j}(t+ (k+1)T-u))du+I_{i}(t ),
\end{aligned}\label{e3.3}
\end{equation}
$i=1, 2, \dots,n$.
Thus, for any natural number $k$, $(-1)^{k+1} v  (t + (k+1)T)$
are the solutions of \eqref{e1.1} on $\mathbb{R}$.
 Then, by Lemma \ref{lem2.2}, there exists a constant $M>0$ such that
\begin{equation}
\begin{aligned}
&|(-1)^{k+1} v_{i} (t + (k+1)T)-(-1)^{k } v_{i}  (t + kT)|\\
&\leq    M e^{-\lambda (t + kT)}\sup_{-\infty\leq
s\leq0}\max_{1\leq i\leq n}|v_{i} (s +  T)+ v_{i} (s)|\\
&\leq   e^{-\lambda (t + kT)} M 2\max_{1\leq i\leq n}
 \{\xi_{i}\frac{\overline{I} }{\eta}\}, \forall t + kT>0,  \quad
i=1,  2,  \dots,  n.
\end{aligned}  \label{e3.4}
\end{equation}
Thus, for any natural number $m $, we obtain
\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}
Then,
\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)^{k+1} v_{i} (t + (k+1)T)-(-1)^{k } v_{i}  (t + kT)|
\leq   \alpha (e^{-\lambda T } )^{k}  ,\quad
 \forall   k >N, \; i=1,  2,  \dots, n,
 \label{e3.7}
\end{equation}
on any compact subset of $\mathbb{R}$. It
follows from \eqref{e3.5}--\eqref{e3.7} that
$ \{(-1)^{m} v (t +mT)\}$  converges uniformly to a continuous function
$x^{*}(t)=(x^{*}_{1}(t), x^{*}_{2}(t),\dots,x^{*}_{n}(t))^{T}$ on
any compact subset of $\mathbb{R}$.

Now we  show that $x^{*}(t)$ is $T$-anti-periodic solution of
system \eqref{e1.1}. First, $x^{*}(t)$ 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*}
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)^{m+1} v  (t +(m+1)T))'\}$   uniformly
converges to a continuous function on any
compact set of $\mathbb{R}$.  Thus,  letting $m \to\infty$,
we obtain
\begin{equation}
 \begin{aligned}
\frac{d}{dt}\{x^{*}_{i}(t)\}
&=  -c_{i}(t)\alpha_{i}(x^{*}_{i}(t))+
\sum_{j=1}^{n}a_{ij}(t)\tilde{g}_{j}(x^{*}_{j}(t-\tau_{ij}(t)))\\
&\quad + \sum_{j=1}^{n}b_{ij}(t) \int_{0}^{\infty}K_{ij}(u)
g_{j}(x^{*}_{j}(t-u))du+I_{i}(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 globally
exponentially stable.  This completes the proof.
\end{proof}

\section{An Example}

In this section, we give an example to demonstrate the results
obtained in previous sections.
Consider the recurrent neutral network
\begin{equation}
\begin{gathered}
\begin{aligned}
 x_{1}'(t)
&=-( x_{1}(t)+x_{1}^{3}(t) )+\frac{1}{24}(\cos t)
 \tilde{g}_{1}(x_{1}(t-22))+\frac{1}{108} (\cos t)
 \tilde{g}_{2}(x_{2}(t-11))\\
&\quad +\frac{1}{24} (\sin t ) g_{1}(\int_{0}^{\infty}
|\sin u |e^{-u}x_{1}(t-u)du)\\
&\quad +\frac{1}{108} (\sin t)
 g_{2}(\int_{0}^{\infty}|\cos u |e^{-u} x_{2}(t-u)du)+I_{1}(t),
\end{aligned}\\
\begin{aligned}
 x_{2}'(t)&=-( x_{2}(t)+x_{2}^{3}(t) )+\frac{1}{12} ( \cos t )
\tilde{g}_{1}(x_{1}(t-6))+\frac{1}{24} (\cos t) \tilde{g}(x_{2}(t-8))\\
&\quad +  \frac{1}{12}(\sin t) g_{1}(\int_{0}^{\infty}|\cos u |
e^{-u}x_{1}(t-u)du)\\
&\quad +\frac{1}{24} (\sin t)
g_{2}(\int_{0}^{\infty}|\sin u |e^{-u} x_{2}(t-u)du)+I_{2}(t),
\end{aligned}
\end{gathered} \label{e4.1}
\end{equation}
where $\tilde{g}_{1}(x)=\tilde{g}_{2}(x) =|\sin x|x,
g_{1}(x)=g_{2}(x)=x^{2}=x\times x$,
  $I_{1}(t)=\frac{1}{24}  \cos t $,
$I_{2}(t)= \frac{1}{24}  \sin t $.

Noting that $c_{1}=c_{2}= L^{\tilde{f}}_{j}=
L^{\tilde{h}}_{j}=L^{f}_{j} =L^{h}_{j} =1$,
$a^{+}_{11} = b^{+}_{11} =\frac{1}{24}$,
$a^{+}_{12} = b^{+}_{12} = \frac{1}{108}$,
$a^{+}_{21} = b^{+}_{21} =\frac{1}{12}$,
$a^{+}_{22} = b^{+}_{22} =\frac{1}{24}$,
$ \underline{d}_{i}=1$,
$\int_{0}^{\infty}K_{ij}(s)ds \leq 1  $, where
$i,j=1,2, \beta^{+}=\sup_{t\in \mathbb{R}} \beta(t)$.
Then
$$
d_{ij}=c^{-1}_{i}\underline{d}_{i}( 3a^{+}_{ij}L^{\tilde{f}}_{j}
L^{\tilde{h}}_{j} + 3b^{+}_{ij}
L^{f}_{j} L^{h}_{j}) \quad    i,j=1,2,\quad
D =(d_{ij})_{2\times 2}= \begin{pmatrix}
 1/4& 1/18 \\
 1/2& 1/4 \end{pmatrix}.
$$
 From the theory of  $M$-matrices in \cite{b1},  we
can choose  constants $\eta=\frac{1}{4} $ and $\xi_{i}=1$ such that
for all $t>0$,
\begin{align*}
&- c_{i} \underline{d}_{i} \xi_{i}+
\sum_{j=1}^{2}3a^{+}_{ij}  L^{\tilde{f}}_{j} L^{\tilde{h}}_{j}
\xi_{j}+ \sum_{j=1}^{2}3b^{+}_{ij} \int_{0}^{\infty}K_{ij}(s) ds
 L^{f}_{j}  L^{h}_{j}\xi_{j}\\
&<- c_{i} \underline{d}_{i} \xi_{i}+ \sum_{j=1}^{2}a^{+}_{ij}
L^{\tilde{f}}_{j} L^{\tilde{h}}_{j}\xi_{j}+
\sum_{j=1}^{2}b^{+}_{ij}
  L^{f}_{j}  L^{h}_{j}\xi_{j}<-\eta,
\end{align*}
where $i=1, 2$, which implies that  \eqref{e4.1} satisfy all the
conditions in Theorem \ref{thm3.1}. Hence,  \eqref{e4.1} has at least
 one    $\pi$-anti-periodic solution $x^{*}(t)$. Moreover, $x^{*}(t)$
is  locally exponentially stable, the domain of attraction of
$Z^{*}(t)$ is the set $ G_{1}(x^{*})$.

\begin{remark} \label{rmk4.1} \rm
System \eqref{e4.1} is a very simple form
of recurrent neural networks with mixed  delays.
 Since $h_{i}(u)\neq u$, $i=1, 2$,
$\tilde{g}_{1}(x)=\tilde{g}_{2}(x) =|\sin x|x$,
$g_{1}(x)=g_{2}(x)=x^{2}=x\times x $. One can observe that
condition (T0) is  not satisfied. Therefore,  the results in
in this article and their references can not be applied to \eqref{e4.1}.
\end{remark}

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\end{document}