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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 152, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/152\hfil Existence of almost periodic solutions]
{Existence of almost periodic solutions for
 Hopfield neural networks with continuously distributed delays
 and impulses}

\author[Y. Li, T. Zhang\hfil EJDE-2009/152\hfilneg]
{Yongkun Li, Tianwei Zhang}  % in alphabetical order

\address{Yongkun Li \newline
Department of Mathematics, Yunnan University\\
Kunming, Yunnan 650091, China}
\email{yklie@ynu.edu.cn}

\address{Tianwei Zhang \newline
Department of Mathematics, Yunnan University\\
Kunming, Yunnan 650091, China} 
\email{1200801347@stu.ynu.edu.cn}

\thanks{Submitted October 28, 2009. Published November 25, 2009.}
\thanks{Supported by grant 10971183 from the National Natural Sciences
Foundation of China} 
\subjclass[2000]{34K14; 34K45; 92B20}
\keywords{Almost periodic solution;  Hopfield neural networks;
 impulses; \hfill\break\indent Cauchy matrix}

\begin{abstract}
 By means of a Cauchy matrix, we prove the existence
 of almost periodic solutions for Hopfield neural networks with
 continuously distributed delays and impulses.
 An example is employed to illustrate our results.
\end{abstract}

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

\section{Introduction}

Let $\mathbb{R}$ be the set of  real numbers
$\mathbb{R^{+}}=[0,\infty)$, $\Omega\subset\mathbb{R}$,
$\Omega\neq\emptyset$.
The set of sequences that are unbounded and strictly increasing
is denoted by
$\mathbb{B}=\{\{\tau_{k}\}\in\mathbb{R}: \tau_{k}<\tau_{k+1},
k\in\mathbb{Z}, \lim_{k\to\pm\infty}\tau_{k}=\pm\infty\}$.

Recently,  Stamov \cite{s2} investigated the  generalized impulsive
Lasota-Wazewska model
\begin{equation} \label{e1.1}
\begin{gathered}
x'(t)=-a(t)x(t)+\sum_{i=1}^{n}\beta_{i}(t)
e^{-\gamma_{i}(t)x(t-\xi)},\quad t\neq \tau_{k},\\
\Delta x(\tau_{k})=\alpha_{k}x(\tau_{k})+\nu_{k},
\end{gathered}
\end{equation}
where $t\in \mathbb{R}$,
$\alpha(t), \beta_{i}(t), \gamma_{i}(t)
\in C(\mathbb{R},\mathbb{R}^{+})$, $i=1,2,\dots,n$,
$\xi$ is a positive constant,
$\{\tau_{k}\}\in \mathbb{B}$, with
 $\alpha_{k}, \nu_{k}\in \mathbb{R}$ for
$k\in\mathbb{Z}$.  By means of the Cauchy matrix he  obtained
sufficient conditions for the existence and exponential stability of
almost periodic solutions for \eqref{e1.1}.
In this paper, we  consider a more general model;
that is,  the following impulsive Hopfield neural networks with
continuously distributed delays
\begin{equation} \label{e1.2}
\begin{gathered}
\begin{aligned}
x'_{i}(t)&=-c_{i}(t)x_{i}(t)+\sum_{j=1}^{n}a_{ij}(t)f_{j}(x_{j}(t-\xi))\\
&\quad+\sum_{j=1}^{n}b_{ij}(t)
\int_{0}^{\infty}K_{ij}(u)g_{j}(x_{j}(t-u))\,{\rm d}u
+I_{i}(t),\quad t\neq \tau_{k},
\end{aligned}\\
\Delta x_{i}(\tau_{k})=\alpha_{ik}x_{i}(\tau_{k})+\nu_{ik},
\end{gathered}
\end{equation}
where $i=1,2,\dots,n$, $k\in\mathbb{Z}, x_{i}(t)$ denotes the potential
(or voltage) of cell $i$ at 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 time $t$; $a_{ij}(t)$ and $b_{ij}(t)$ are the connection
weights between cell $i$ and $j$ at time $t$; $\xi$ is a constant
and denotes the time delay; $K_{ij}(t)$ corresponds to the
transmission delay kernels; $f_{j}$ and $g_{j}$ are the activation
functions; $I_{i}(t)$ is an external input on the $i$th unit at time
$t$. Furthermore, $\{\tau_{k}\}\in \mathbb{B}$, with the constants
$\alpha_{ik}\in\mathbb{R}$, $\gamma_{ik}\in\mathbb{R}$, $k\in
\mathbb{Z}$, $i=1,2,\dots,n$.

\begin{remark} \label{rmk1.1} \rm
If $i=1$, $f_{j}(x_{j}(t-\xi))=e^{-\gamma_{j}(t)x(t-\xi)}$,
$b_{ij}(t)=I_{i}(t)=0$, $j=1,2,\dots,n$, then \eqref{e1.2} reduces to
\eqref{e1.1}.
\end{remark}

Our main am of this paper is to investigate the existence of almost
periodic solutions of system \eqref{e1.2}.
Let $t_{0}\in \mathbb{R}$.
Introduce the following notation:

$PC(t_{0})$ is the space of all functions $\phi:[-\infty,t_{0}]\to \Omega$
having points of discontinuity at
$\theta_{1},\theta_{2},\dots\in(-\infty,t_{0})$
of the first kind and left continuous at these points.

For $J\subset\mathbb{R}$, $PC(J, \mathbb{R})$ is the space of all
piecewise continuous functions from $J$ to $\mathbb{R}$
with points of discontinuity of the first kind $\tau_{k}$,
at which it is left continuous.

The initial conditions associated with system \eqref{e1.2}
 are of the form
\begin{eqnarray*}
x_{i}(s)=\phi_{i}(s),\,\,s\in(-\infty,t_{0}],
\end{eqnarray*}
where $\phi_{i}\in PC(t_{0})$, $i=1,2,\dots,n$.

The remainder of this article is organized as follows:
In Section 2, we will introduce some necessary notations, definitions
and lemmas which will be used in the paper.
In Section 3, some sufficient conditions are derived ensuring the existence of
the almost periodic solution. At last, an illustrative example is given.


\section{Preliminaries}


In this section, we  introduce  necessary notations,
definitions and lemmas which will be used later.

\begin{definition}[\cite{s1}]\label{def1} \rm
The set of sequences $\{\tau_{k}^{j}\}$,
$\tau_{k}^{j}=\tau_{k+j}-\tau_{k}$, $k, j\in \mathbb{Z}$,
$\{\tau_{k}\}\in \mathbb{B}$ is
said to be uniformly almost periodic if for arbitrary $\epsilon>0$
there exists a relatively dense set of $\epsilon$-almost periods common
 for any sequences.
\end{definition}

\begin{definition}[\cite{s1}] \label{def2} \rm
A function $x(t)\in PC(\mathbb{R},\mathbb{R})$ is
said to be almost periodic, if the following hold:
\begin{itemize}
 \item[(a)] The  set of sequences
$\{\tau_{k}^{j}\}, \tau_{k}^{j}=\tau_{k+j}-
\tau_{k}, k, j\in \mathbb{Z}$,  $\{\tau_{k}\}\in \mathbb{B}$ is
uniformly almost periodic.

\item[(b)] For any $\epsilon>0$ there exists a real number
$\delta>0$ such that if the points $t'$
and $t''$ belong to one and the same interval of continuity of
$x(t)$ and satisfy the inequality $|t'-t''|<\delta$, then
$|x(t')-x(t'')|<\epsilon$.

\item[(c)] For any $\epsilon>0$ there exists a relatively
dense set $T$ such that if $\tau\in T$,
then $|x(t+\tau)-x(t)|<\epsilon$ for all $t\in \mathbb{R}$ satisfying
the condition $|t-\tau_{k}|>\epsilon$, $k\in \mathbb{Z}$.
\end{itemize}
The elements of $T$ are called $\epsilon$-almost periods.
\end{definition}

Together with the system \eqref{e1.2} we consider the linear system
\begin{equation} \label{e2.1}
\begin{gathered}
x'_{i}(t)=-c_{i}(t)x_{i}(t), \quad t\neq\tau_{k}, \\
\Delta x_{i}(\tau_{k})=\alpha_{ik}x_{i}(\tau_{k}), \quad k\in\mathbb{Z},
\end{gathered}
\end{equation}
where $t\in\mathbb{R}$, $i=1,2,\dots,n$.
Now let us consider the equations
\[
x'_{i}(t)=-c_{i}(t)x_{i}(t),\quad \tau_{k-1}< t\leq\tau_{k},
\quad\{\tau_{k}\}\in \mathbb{B}
\]
and their solutions
\[
x_{i}(t)=x_{i}(s)\exp\big\{-\int_{s}^{t}c_{i}(\sigma)\,{\rm d}\sigma\big\}
\]
for $\tau_{k-1}<s<t\leq\tau_{k}$, $i=1,2,\dots,n$.

As in \cite{l1}, the Cauchy matrix of the linear system \eqref{e2.1} is
\begin{align*}
&W_{i}(t,s)\\
&= \begin{cases}
\exp\big\{-\int_{s}^{t}c_{i}(\sigma)\,{\rm d}\sigma \big\},
 &\tau_{k-1}<s<t<\tau_{k}; \\[4pt]
\prod_{j=m}^{k+1}(1+\alpha_{ij})
\exp\big\{-\int_{s}^{t}c_{i}(\sigma)\,{\rm d}
\sigma \big\}, & \tau_{m-1}<s\leq\tau_{m}<\tau_{k}<t\leq\tau_{k+1}.
\end{cases}
\end{align*}
The solutions of system \eqref{e2.1} are of the form
\[
x_{i}(t;t_{0};x_{i}(t_{0}))=W_{i}(t,t_{0})x_{i}(t_{0}),\quad
t_{0}\in \mathbb{R}, \; i=1,2,\dots,n.
\]
For convenience, we introduce the notation
\[
\overline{f}=\sup_{t\in \mathbb{R}}|f(t)|,\quad
\underline{f}=\inf_{t\in \mathbb{R}}|f(t)|.
\]
In this article, we use the following hypotheses:
\begin{itemize}
\item[(H1)]
$c_{i}(t)\in C(\mathbb{R},\mathbb{R}^{+})$ is almost periodic
and there exists a positive constant
$c$ such that $c<c_{i}(t)$, $t\in \mathbb{R}$, $i=1,2,\dots,n$.

 \item[(H2)] The set of sequences $\{\tau_{k}^{j}\}$,
$\tau_{k}^{j}=\tau_{k+j}-\tau_{k}$,
$k\in\mathbb{Z}$, $j\in\mathbb{Z}$, $\{\tau_{k}\}\in \mathbb{B}$
is uniformly almost
periodic and there exists $\theta>0$ such that
$\inf_{k\in\mathbb{Z}}\tau_{k}^{1}=\theta>0$.

\item[(H3)] The sequence $\{\alpha_{ik}\}$ is almost
periodic and $1-e^{2}\leq\alpha_{ik}
\leq e^{2}-1$,
$k\in\mathbb{Z}$, $i=1,2,\dots,n$.

\item[(H4)] The sequence $\{\nu_{ik}\}$ is almost
periodic and $\gamma=\sup_{k\in\mathbb{Z}}|\nu_{ik}|$,
$k\in\mathbb{Z}$, $i=1,2,\dots,n$.

\item[(H5)] The functions $a_{ij}(t)$, $b_{ij}(t)$ and $I_{i}(t)$
are almost periodic in the sense of Bohr
and $|I_{i}(t)|<\infty$, $t\in\mathbb{R}$, $i,j=1,2,\dots,n.$

\item[(H6)] The functions $f_{j}(t)$ and $g_{j}(t)$
 are almost periodic in the sense of Bohr and
 $f_{j}(0)=g_{j}(0)=0$, $j=1,2,\dots,n$.
There exist positive bounded functions
$L_{f}(t)$ and $L_{g}(t)$ such that for $u$, $v\in\mathbb{R}$
\[
\max_{1\leq j\leq n}|f_{j}(u)-f_{j}(v)|\leq L_{f}(t)|u-v|,\quad
\max_{1\leq j\leq n}|g_{j}(u)-g_{j}(v)|\leq L_{g}(t)|u-v|.
\]

\item[(H7)] The delay kernels $K_{ij}\in C(\mathbb{R},\mathbb{R})$ and
there exists a positive constant
$K$ such that
\[
\int_{0}^{+\infty}|K_{ij}(s)|\,{\rm d}s\leq K,\quad i,j=1,2,\dots,n.
\]
\end{itemize}


\begin{lemma}[\cite{s1}] \label{lem2.1}
Assume {\rm (H1)-(H6)}.
Then for each $\epsilon>0$, there exist $\epsilon_{1}$,
$0<\epsilon_{1}<\epsilon$, relatively dense sets $T$ of real
numbers and $Q$ of whole numbers, such that the following relations
are fulfilled:
\begin{itemize}
\item[(a)] $|c_{i}(t+\tau)-c_{i}(t)|<\epsilon$,
 $t\in\mathbb{R}$, $\tau\in T$, $i=1,2,\dots,n$;

\item[(b)] $|a_{ij}(t+\tau)-a_{ij}(t)|<\epsilon$,
$t\in\mathbb{R}$, $\tau\in T$,
 $|t-\tau_{k}|>\epsilon$, $k\in\mathbb{Z}$, $i,j=1,2,\dots,n$;

\item[(c)] $|b_{ij}(t+\tau)-b_{ij}(t)|<\epsilon$,
$t\in\mathbb{R}$, $\tau\in T$,
$|t-\tau_{k}|>\epsilon$, $k\in\mathbb{Z}$, $i,j=1,2,\dots,n$;

\item[(d)] $|I_{i}(t+\tau)-I_{i}(t)|<\epsilon$,
$t\in\mathbb{R}$, $\tau\in T$,
 $|t-\tau_{k}|>\epsilon$, $k\in\mathbb{Z}$, $i=1,2,\dots,n$;

\item[(e)] $|f_{j}(t+\tau)-f_{j}(t)|<\epsilon$,
$t\in\mathbb{R}$, $\tau\in T$,  $|t-\tau_{k}|>\epsilon$,
$k\in\mathbb{Z}$, $j=1,2,\dots,n$;

\item[(f)] $|g_{j}(t+\tau)-g_{j}(t)|<\epsilon$,
$t\in\mathbb{R}$, $\tau\in T$,
 $|t-\tau_{k}|>\epsilon$, $k\in\mathbb{Z}$, $j=1,2,\dots,n$;

\item[(g)] $|\alpha_{i(k+q)}-\alpha_{ik}|<\epsilon$,
  $q\in Q$,  $k\in\mathbb{Z}$, $i=1,2,\dots,n$;

\item[(h)] $|\nu_{i(k+q)}-\nu_{ik}|<\epsilon$,
 $q\in Q$,  $k\in\mathbb{Z}$, $i=1,2,\dots,n$;

\item[(i)] $|\tau_{k}^{q}-\tau|<\epsilon_{1}$,
$q\in Q$, $\tau\in T$, $k\in\mathbb{Z}$, $i=1,2,\dots,n$.
\end{itemize}
\end{lemma}

\begin{lemma}[\cite{s1}] \label{lem2.2}
Let $\{\tau_{k}\}\in \mathbb{B}$ and the condition {\rm (H2)} hold.
Then for $1>0$ there exists a positive integer $A$ such that on
each interval of length $1$, we have
no more than $A$ elements of the sequence $\{\tau_{k}\}$, i.e.,
\[
i(s,t)\leq A(t-s)+A,
\]
where $i(s,t)$ is the number of the points $\tau_{k}$ in the
interval $(s,t)$.
\end{lemma}

\begin{lemma} \label{lem2.3}
Assume {\rm (H1)-(H3)}. Then for the Cauchy matrix $W_{i}(t,s)$
of system \eqref{e2.1}, we have
\[
|W_{i}(t,s)|\leq e^{2A}e^{-\alpha(t-s)},\quad t\geq s, \;
t,s\in \mathbb{R}, \; i=1,2,\dots,n,
\]
where $\alpha=c-2A$, $A$ is determined in Lemma \ref{lem2.2}.
\end{lemma}

\begin{proof}
Since the sequence $\{\alpha_{ik}\}$ is almost periodic, then it
is bounded and from
(H3) it follows that $|1+\alpha_{ik}|\leq e^{2}$, $k\in\mathbb{Z}$,
$i=1,2,\dots,n$.
From the expression of $W_{i}(t,s)$ and the above inequality it
follows that
\begin{align*}
|W_{i}(t,s)|&= |1+\alpha_{ik}|^{i(s,t)}e^{-\int_{s}^{t}c_{i}(\theta)\,{\rm d}\theta}\\
            &\leq |1+\alpha_{ik}|^{A(t-s)+A}e^{-c(t-s)}\\
            &\leq e^{2A}e^{-(c-2A)(t-s)}\\
            &= e^{2A}e^{-\alpha(t-s)},
\end{align*}
where $t\geq s$, $t,s\in\mathbb{R}$, $i=1,2,\dots,n$. The proof  is
complete.
\end{proof}


 From \cite[Lemma 3]{l1}, we  obtain the following lemma.

\begin{lemma} \label{lem2.4}
Assume {\rm (H1)-(H3)} and the  condition
\begin{itemize}
\item[(H8)] $\alpha=c-2A>0$.
\end{itemize}
Then for any $\epsilon>0$, $t\geq s$, $t,s\in \mathbb{R}$,
$|t-\tau_{k}|>\epsilon$,
$|s-\tau_{k}|>\epsilon$, $k\in \mathbb{Z}$ there
exists a relatively dense set $T$ of the function $c_{i}(t)$ and a
positive constant $\Gamma$ such that for $\tau\in T$ it follows that
\[
|W_{i}(t+\tau,s+\tau)-W_{i}(t,s)|\leq \epsilon
\Gamma e^{-\frac{\alpha}{2}(t-s)}, \quad
 t\geq s, \; t,s\in \mathbb{R}, \; i=1,2,\dots,n.
\]
\end{lemma}

\section{Main results}

Let
\[
P=\max_{1\leq i\leq n}
   \Big\{\frac{\overline{I_{i}}e^{2A}}{\alpha}+\frac{\gamma e^{2A}}
        {1-e^{-\alpha}}\Big\}.
\]

\begin{theorem} \label{thm3.1}
Assume {\rm (H1)-(H8)} and
\begin{itemize}
\item[(H9)]    $r=\max_{1\leq i\leq n}\Big\{ \frac{e^{2A}}{\alpha}
\Big(\sum_{j=1}^{n}\overline{a}_{ij}\overline{L}_{f}
+\sum_{j=1}^{n}\overline{b}_{ij}\overline{L}_{g}K\Big)
                   \Big\}<1$.
\end{itemize}
Then \eqref{e1.2} has a unique almost periodic solution.
\end{theorem}

\begin{proof}
Set $\mathbb{X}=\{\varphi(t)\in PC(\mathbb{R},\mathbb{R}^{n}):\varphi(t)=
(\varphi_{1}(t),\varphi_{2}(t),\dots ,\varphi_{n}(t))^{T}$, where
$\varphi_{i}(t)$ is a almost
periodic function satisfying
$\|\varphi\|=\max_{1\leq i\leq n}\{\sup_{t\in\mathbb{R}}
|\varphi_{i}(t)|\}\leq N=\frac{P}{1-r}$,
$i=1,2,\dots,n\}$
with the norm
$\|\varphi\|=\max_{1\leq i\leq n}
\{\sup_{t\in\mathbb{R}}|\varphi_{i}(t)|\}$.
We define an map $\Phi$ on $\mathbb{X}$ by
\[
(\Phi \varphi)(t)=((\Phi_{1} \varphi)(t),(\Phi_{2} \varphi)(t),
\dots,(\Phi_{n} \varphi)(t))^{T},
\]
where $t\in \mathbb{R}$,
\begin{equation} \label{e3.1}
\begin{aligned}
&(\Phi_{i} \varphi)(t)\\
&=\int_{-\infty}^{t}W_{i}(t,s)
\Big(\sum_{j=1}^{n}a_{ij}(s)f_{j}(\varphi_{j}(s-\xi)) \\
&\quad +\sum_{j=1}^{n}b_{ij}(s)\int_{0}^{\infty}K_{ij}(u)
 g_{j}(\varphi_{j}(s-u))\,{\rm d}u
+I_{i}(s)\Big)\,{\rm d} s +\sum_{\tau_{k}<t}W_{i}(t,\tau_{k})\nu_{ik},
\end{aligned}
\end{equation}
where $k\in\mathbb{Z}$, $i=1,2,\dots,n$.
And let $\mathbb{X}^{*}$ be a subset of $\mathbb{X}$ defined by
\[
\mathbb{X}^{*}=\Big\{\varphi\in \mathbb{X}:\|\varphi-\phi\|\leq
\frac{r P}{1-r}\Big\},
\]
where
$\phi=(\phi_{1},\phi_{2},\dots,\phi_{n})^{T}$
and
\[
\phi_{i}=\int_{-\infty}^{t}W_{i}(t,s)I_{i}(s)\,{\rm d} s
+\sum_{\tau_{k}<t}W_{i}(t,\tau_{k})\nu_{ik}, \quad
k\in\mathbb{Z},\,\, i=1,2,\dots,n.
\]
We have
\begin{equation} \label{e3.2}
\begin{aligned}
\|\phi\|&= \max_{1\leq i\leq n}
\Big\{\sup_{t\in\mathbb{R}}\Big|\int_{-\infty}^{t}W_{i}(t,s)I_{i}(s)\,{\rm d} s
+\sum_{\tau_{k}<t}W_{i}(t,\tau_{k})\nu_{ik}\Big|\Big\} \\
&\leq \max_{1\leq i\leq n}\Big\{\sup_{t\in\mathbb{R}}
\Big(\int_{-\infty}^{t}|W_{i}(t,s)||I_{i}(s)|\,{\rm d} s
 +\sum_{\tau_{k}<t}|W_{i}(t,\tau_{k})||\nu_{ik}|\Big)\Big\} \\
&\leq \max_{1\leq i\leq n}\Big\{\sup_{t\in\mathbb{R}}
   \Big(\int_{-\infty}^{t}e^{2A}e^{-\alpha(t-s)}\overline{I}_{i}\,{\rm d} s
   +\sum_{\tau_{k}<t}e^{2A}e^{-\alpha(t-\tau_{k})}\nu_{ik}\Big)\Big\} \\
&\leq \max_{1\leq i\leq n}
     \Big\{\frac{\overline{I_{i}}e^{2A}}{\alpha}+\frac{\gamma e^{2A}}
        {1-e^{-\alpha}}\Big\}
=P.
\end{aligned}
\end{equation}
Then for arbitrary $\varphi\in \mathbb{X}^{*}$ from \eqref{e3.1}
and \eqref{e3.2} we have
\[
\|\varphi\|\leq\|\varphi-\phi\|+\|\phi\|\leq \frac{r P}{1-r}+P=\frac{P}{1-r}.
\]
Now we prove that $\Phi$ is self-mapping from $\mathbb{X}^{*}$ to
$\mathbb{X}^{*}$.
For arbitrary $\varphi\in \mathbb{X}^{*}$ it follows that
\begin{equation} \label{e3.3}
\begin{aligned}
\|\Phi\varphi-\phi\|
&= \max_{1\leq i\leq n}\Big\{
\sup_{t\in\mathbb{R}}\Big|\int_{-\infty}^{t}W_{i}(t,s)
\Big(\sum_{j=1}^{n}a_{ij}(s)f_{j}(\varphi_{j}(s-\xi)) \\
&\quad +\sum_{j=1}^{n}b_{ij}(s)\int_{0}^{\infty}K_{ij}(u)g_{j}(\varphi_{j}(s-u))\,{\rm d}u
\Big)\,{\rm d} s\Big|\Big\} \\
&\leq \max_{1\leq i\leq n}\Big\{
                     \sup_{t\in\mathbb{R}}\Big(\int_{-\infty}^{t}
                     e^{2A}e^{-\alpha(t-s)}
\Big(\sum_{j=1}^{n}\overline{a}_{ij}\overline{L}_{f}
+\sum_{j=1}^{n}\overline{b}_{ij}\overline{L}_{g}K\Big)\,{\rm d} s\Big)
                   \Big\}\|\varphi\| \\
&\leq \max_{1\leq i\leq n}\Big\{
 \frac{e^{2A}}{\alpha}
\Big(\sum_{j=1}^{n}\overline{a}_{ij}\overline{L}_{f}
+\sum_{j=1}^{n}\overline{b}_{ij}\overline{L}_{g}K\Big)
                   \Big\}\|\varphi\| \\
&= r\|\varphi\|\leq\frac{r P}{1-r}.
\end{aligned}
\end{equation}
Moreover, we get
\begin{equation} \label{e3.4}
\begin{aligned}
\|\Phi\varphi\|
&= \max_{1\leq i\leq n}\Big\{
\sup_{t\in\mathbb{R}}\Big|\int_{-\infty}^{t}W_{i}(t,s)
\Big(\sum_{j=1}^{n}a_{ij}(s)f_{j}(\varphi_{j}(s-\xi)) \\
&\quad +\sum_{j=1}^{n}b_{ij}(s)\int_{0}^{\infty}K_{ij}(u)g_{j}(\varphi_{j}(s-u))\,{\rm d}u
+I_{i}(s)\Big)\,{\rm d} s +\sum_{\tau_{k}<t}W_{i}(t,\tau_{k})\nu_{ik}\Big|\Big\} \\
&\leq \frac{r P}{1-r}+P \\
&= \frac{P}{1-r}=N.
\end{aligned}
\end{equation}
On the other hand, let $\tau\in T$, $q\in Q$, where the sets $T$
and $Q$ are determined in Lemma \ref{lem2.1}.
Then
\begin{align}
&|(\Phi_{i}\varphi)(t+\tau)-(\Phi_{i}\varphi)(t)| \nonumber\\
&= \Big|\int_{-\infty}^{t+\tau}W_{i}(t+\tau,s)
\Big(\sum_{j=1}^{n}a_{ij}(s)f_{j}(\varphi_{j}(s-\xi)) \nonumber \\
&\quad +\sum_{j=1}^{n}b_{ij}(s)\int_{0}^{\infty}K_{ij}(u)g_{j}(\varphi_{j}(s-u))\,{\rm d}u
+I_{i}(s)\Big)\,{\rm d} s \nonumber \\
&\quad +\sum_{\tau_{k}<t+\tau}W_{i}(t+\tau,\tau_{k})\nu_{ik}
-\sum_{\tau_{k}<t}W_{i}(t,\tau_{k})\nu_{ik} \nonumber \\
&\quad -\int_{-\infty}^{t}W_{i}(t,s)
\Big(\sum_{j=1}^{n}a_{ij}(s)f_{j}(\varphi_{j}(s-\xi)) \nonumber\\
&\quad +\sum_{j=1}^{n}b_{ij}(s)\int_{0}^{\infty}K_{ij}(u)g_{j}(\varphi_{j}(s-u))\,{\rm d}u
+I_{i}(s)\Big)\,{\rm d} s\Big|  \nonumber\\
&\leq \Big|\int_{-\infty}^{t}W_{i}(t+\tau,s+\tau)
\Big(\sum_{j=1}^{n}a_{ij}(s+\tau)f_{j}(\varphi_{j}(s+\tau-\xi)) \nonumber\\
&\quad +\sum_{j=1}^{n}b_{ij}(s+\tau)\int_{0}^{\infty}K_{ij}(u)g_{j}(\varphi_{j}(s+\tau-u))\,{\rm d}u
+I_{i}(s+\tau)\Big)\,{\rm d} s  \nonumber \\
&\quad +\sum_{\tau_{k}<t}W_{i}(t+\tau,\tau_{k+q})\nu_{i(k+q)}
-\sum_{\tau_{k}<t}W_{i}(t,\tau_{k})\nu_{ik} \nonumber\\
&\quad -\int_{-\infty}^{t}W_{i}(t,s)
\Big(\sum_{j=1}^{n}a_{ij}(s)f_{j}(\varphi_{j}(s-\xi)) \label{e3.5} \\
&\quad +\sum_{j=1}^{n}b_{ij}(s)\int_{0}^{\infty}K_{ij}(u)g_{j}(\varphi_{j}(s-u))\,{\rm d}u
+I_{i}(s)\Big)\,{\rm d} s \Big|  \nonumber\\
&\leq \int_{-\infty}^{t}|W_{i}(t+\tau,s+\tau)-W_{i}(t,s)|
\Big|\sum_{j=1}^{n}a_{ij}(s+\tau)f_{j}(\varphi_{j}(s+\tau-\xi)) \nonumber\\
&\quad +\sum_{j=1}^{n}b_{ij}(s+\tau)\int_{0}^{\infty}K_{ij}(u)g_{j}(\varphi_{j}(s+\tau-u))\,{\rm d}u
+I_{i}(s+\tau)\Big|\,{\rm d} s  \nonumber\\
&\quad +\int_{-\infty}^{t}|W_{i}(t,s)|
\Big|\sum_{j=1}^{n}a_{ij}(s+\tau)f_{j}(\varphi_{j}(s+\tau-\xi)) \nonumber\\
&\quad -\sum_{j=1}^{n}a_{ij}(s)f_{j}(\varphi_{j}(s-\xi))
-\sum_{j=1}^{n}b_{ij}(s)\int_{0}^{\infty}K_{ij}(u)g_{j}
 (\varphi_{j}(s-u))\,{\rm d}u  \nonumber\\
&\quad +\sum_{j=1}^{n}b_{ij}(s+\tau)\int_{0}^{\infty}K_{ij}(u)g_{j}
 (\varphi_{j}(s+\tau-u))\,{\rm d}u \nonumber\\
&\quad +I_{i}(s+\tau)-I_{i}(s)\Big|\,{\rm d} s+\sum_{\tau_{k}<t}
 |W_{i}(t,\tau_{k})||\nu_{i(k+q)}-\nu_{ik}| \nonumber\\
&\quad +\sum_{\tau_{k}<t}|W_{i}(t+\tau,\tau_{k+q})-W_{i}(t,\tau_{k})|
|\nu_{i(k+q)}| \nonumber\\
&\leq C\epsilon, \nonumber
\end{align}
where
\begin{align*}
C&= \max_{1 \leq i \leq n}\Big\{
\frac{1}{\alpha}\sum_{j=1}^{n}(2\Gamma\overline{a}_{ij}\overline{L}_{f}
+2\Gamma\overline{b}_{ij}\overline{L}_{g}K
+\overline{L}_{f}e^{2A}+\overline{L}_{g}e^{2A}K)N
+\frac{e^{2A}+2\Gamma \overline{I}_{i}}{\alpha} \\
&\quad +\frac{e^{2A}}{\alpha}\sum_{j=1}^{n}(\overline{a}_{ij}\overline{L}_{f}
+\overline{b}_{ij}\overline{L}_{g}K)
+\frac{\gamma\Gamma}{1-e^{-\frac{\alpha}{2}}}+\frac{e^{2A}}{1-e^{-\alpha}}\Big\}.
\end{align*}
 From \eqref{e3.3}-\eqref{e3.5}, we obtain that
$\Phi\varphi\in\mathbb{X}^{*}$.
Let $\varphi\in\mathbb{X}^{*}$, $\psi\in\mathbb{X}^{*}$. We have
\begin{align*}
\|\Phi\varphi-\Phi\psi\|
&= \max_{1\leq i\leq n}\Big\{\sup_{t\in\mathbb{R}}\Big|\int_{-\infty}^{t}W_{i}(t,s)
\Big(\sum_{j=1}^{n}a_{ij}(s)f_{j}(\varphi_{j}(s-\xi)) \\
&\quad +\sum_{j=1}^{n}b_{ij}(s)\int_{0}^{\infty}K_{ij}(u)g_{j}(\varphi_{j}(s-u))\,{\rm d}u
\Big)\,{\rm d} s \\
&\quad -\int_{-\infty}^{t}W_{i}(t,s)
\Big(\sum_{j=1}^{n}a_{ij}(s)f_{j}(\psi_{j}(s-\xi)) \\
&\quad +\sum_{j=1}^{n}b_{ij}(s)\int_{0}^{\infty}K_{ij}(u)g_{j}(\psi_{j}(s-u))\,{\rm d}u
\Big)\,{\rm d} s
\Big|\Big\} \\
&\leq \max_{1\leq i\leq n}\Big\{
 \frac{e^{2A}}{\alpha}
\Big(\sum_{j=1}^{n}\overline{a}_{ij}\overline{L}_{f}
+\sum_{j=1}^{n}\overline{b}_{ij}\overline{L}_{g}K\Big)
\Big\}\|\varphi-\psi\| \\
&= r\|\varphi-\psi\| \\
&< \|\varphi-\psi\|.
\end{align*}
 From this inequality, it follows that $\Phi$ is contracting operator
in $\mathbb{X}^{*}$. So \eqref{e1.2} has a unique almost periodic
solution. This completes the proof.
\end{proof}

\begin{remark} \label{rmk3.1} \rm
In \cite{s2}, $\alpha_{k}, k\in\mathbb{Z}$ are required to
take values in $[-1,0]$, which is a more strict requirement (H2)
in this article.
\end{remark}

\section{An example}

Consider the  impulsive Hopfield neural network
\begin{equation} \label{e4.1}
\begin{gathered}
x'(t)=-c(t)x(t)+f(x(t-\frac{1}{2}))
+\frac{1}{20}\int_{0}^{\infty}K(u)g(x(t-u))\,{\rm d}u
+I(t),\quad  t\neq \tau_{k},\\
\Delta x(\tau_{k})=\alpha_{k}x(\tau_{k})+\nu_{k},\quad
k\in \mathbb{Z},
\end{gathered}
\end{equation}
 where (H2) and (H4) hold with $A=2$, $c(t)=e^{8}+\cos t$,
$f(t)=\frac{1}{2}|t|$, $K(t)=e^{-4t}$, $g(t)=\frac{1}{4}\sin^{2}t$,
$I(t)=2+\sin t$, the sequence $\{\alpha_{k}\}$ is almost periodic
and $1-e^{2}\leq \alpha_{k}\leq e^{2}-1$, $k\in\mathbb{Z}$.
Obviously, $c=e^{8}-1$, $\overline{a}=1$,
$\overline{b}=\frac{1}{20}$,
$\overline{L}_{f}=\overline{L}_{g}=\frac{1}{2}$, $K=\frac{1}{4}$.
Then $\alpha=e^{8}-5>0$,
$r=\frac{e^{4}}{e^{8}-5}(1\times\frac{1}{2}+\frac{1}{20}\times
\frac{1}{4}\times 5)<1$, so (H8)-(H9) hold. It is easy to verify
that (H1)-(H7) is satisfied. According to Theorem \ref{thm3.1}, \eqref{e4.1} has
one unique almost periodic solution.


 \begin{thebibliography}{00}

\bibitem{s2} G. T. Stamov;
\emph{On the existence of almost periodic solutions for the impulsive
Lasota-Wazewska model}, Appl. Math. Lett. 22 (2009) 516-520.

\bibitem{s1} A. M. Samoilenko, N. A. Perestyuk;
\emph{Differential Equations with Impulse Effect},
World Scientific, Singapore, 1995.

\bibitem{l1} V. Lakshmikantham, D. D. Bainov, P. S. Simeonov;
\emph{Theory of Impulsive Differential Equations},
World Scientific, Singapore, New Jersey, London, 1989.

\end{thebibliography}
\end{document}