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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 23, 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 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/23\hfil Existence and stability for SICNNs]
{Existence and stability of almost periodic solutions
for SICNNs with neutral type delays}

\author[Q.-L. Liu, H.-S. Ding \hfil EJDE-2014/23\hfilneg]
{Qing-Long Liu, Hui-Sheng Ding}  % in alphabetical order

\address{Qing-Long Liu \newline
College of Mathematics and Information Science,
Jiangxi Normal University, Nanchang, Jiangxi 330022, China}
\email{758155543@qq.com}

\address{Hui-Sheng Ding (corresponding author)\newline
College of Mathematics and Information Science,
Jiangxi Normal University, Nanchang, Jiangxi 330022, China}
\email{dinghs@mail.ustc.edu.cn}

\thanks{Submitted November 4, 2013. Published January 10, 2014.}
\subjclass[2000]{34C25, 34K13, 34K25}
\keywords{Almost periodic; shunting inhibitory cellular neural networks;
 \hfill\break\indent neutral type delays; stability}

\begin{abstract}
 This article concerns the shunting inhibitory cellular neural
 networks with neutral type delays. Under a weaker condition than the
 usual Lipschitz condition, we establish the  existence and stability
 of almost periodic solutions for SICNNs with neutral type delays.
 An example is given to illustrate our main results.
\end{abstract}

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

\section{Introduction}


Since Bouzerdoum and Pinter
\cite{Bouzerdoum91,Bouzerdoum92,Bouzerdoum} introduced and analyzed the
shunting inhibitory cellular neural networks (SICNNs),
they have been extensively
applied in psychophysics, speech, perception, robotics, adaptive
pattern recognition, vision, and image processing (cf.
\cite{Bouzerdoum1,Bouzerdoum2} and references therein).

It is well known that studies on neural networks not only involve a
discussion of stability properties, but also involve many dynamic
behaviors such as periodic oscillatory behavior and almost periodic
oscillatory properties. In applications, if the various constituent
components of the temporally nonuniform environment is with
incommensurable (nonintegral multiples) periods, then one has to
consider the environment to be almost periodic since there is no a
priori reason to expect the existence of periodic solutions.
Therefore, if we consider the effects of the environmental factors,
almost periodicity is sometimes more realistic and more general than
periodicity. Also, as pointed out in \cite{Fink,corduneanu},
compared with periodic effects, almost periodic effects are more
frequent in many real world applications. In fact, this point of
view is partially verified by a recent work
\cite{zheng-ding-gaston}, where the authors proved that the ``amount''
of almost periodic functions (not periodic) is far more than the
``amount'' of continuous periodic functions in the sense of category.
Thus, studying the existence of almost periodic solutions for
differential equations is natural and necessary.

Recently, many authors have studied the existence and stability of
periodic solutions and almost periodic solutions for the following
SICNNs:
\begin{equation*}
x'_{ij}(t)=-a_{ij}x_{ij}(t)-\sum_{C_{kl}\in
N_r(i,j)}C^{kl}_{ij}f[x_{kl}(t-\tau(t))]x_{ij}(t)+L_{ij}(t),
\end{equation*}
and its variants. We refer the reader to
\cite{n6,n4,n5, n3,n1, farouk,ding,fang1,n2,fang2,liu2,liu1} and
reference therein for some of recent developments on this topic.


Especially, in a very recent work, the authors in \cite{fang2} investigated the
existence and stability of almost periodic solutions for the
following SICNNs with neutral type delays:
\begin{equation}\label{moxing}
\begin{aligned}
x'_{ij}(t)
&= -a_{ij}(t)x_{ij}(t)-\sum_{C_{kl}\in N_r(i,j)}C^{kl}_{ij}(t)
\int_0^\infty K_{ij}(u)f(x_{kl}(t-u))dux_{ij}(t)  \\
&\quad -\sum_{D_{kl}\in N_s(i,j)}D^{kl}_{ij}(t)\int_0^\infty
J_{ij}(u)g(x'_{kl}(t-u))dux_{ij}(t)+L_{ij}(t),
\end{aligned}
\end{equation}
$i=1,2,\dots ,m$, $j=1,2,\dots n$, where $m,n$ are two fixed positive integers,
$C_{ij}$ is the cell at the $(i,j)$ position of the lattice, the
 $r$-neighborhood $N_r(i,j)$ of $C_{ij}$ is defined as follows:
$$
N_r(i,j)=\{C_{kl}: \max{|k-i|,|l-j|}\leq r, 1\leq k \leq m,1\leq l\leq n \},
$$
and $N_s(i,j)$ is defined similarly. Here $x_{ij}(t)$ is the activity
of cell $C_{ij}$, $L_{ij}(t)$ is the external input to $C_{ij}$, the
coefficient $a_{ij}(t)$ is the passive decay rate of the cell
activity, $f,g$ are continuous activity functions of signal transmission, $C^{kl}_{ij}(t)$ represents the
connection or coupling strength of postsynaptic activity of the cell
transmitted to the cell $C_{ij}$, and $D^{kl}_{ij}(t)$ has a similar meaning.

In \cite{fang2}, the activation functions $f$ and $g$ satisfy the
global Lipschitz conditions. In this paper, as one will see, we
allow for more general activity functions, i.e., we will discuss the
existence and stability of almost periodic solutions for the
SICNNs \eqref{moxing} under a weaker Lipschitz conditions on $f$ and $g$.

Next, let us recall some basic notation and results about almost
periodic functions. For more details, we refer the reader to
\cite{corduneanu,Fink,gaston}.

\begin{definition} \rm
A continuous function $u$: $\mathbb{R}\to \mathbb{R}$ is called almost
periodic if for each $\varepsilon > 0$ there exists $l(\varepsilon) > 0$
such that every interval $I$ of length $l(\varepsilon)$ contains a number
$\tau$ with the property that
$$
|u(t+\tau)-u(t)|<\varepsilon.
$$
\end{definition}

We denote by $AP(\mathbb{R})$ the set of all almost periodic functions
from $\mathbb{R}$ to $\mathbb{R}$, and by $AP^1(\mathbb{R})$ the set of all continuously
differentiable functions $u:\mathbb{R}\to \mathbb{R}$ satisfying
$u,u'\in AP(\mathbb{R})$.


\begin{lemma}\label{ap-lemma}
Let $f,g\in AP(\mathbb{R})$ and $k\in L^1(\mathbb{R}^+)$. Then the following assertions
hold:
\begin{itemize}
\item[(a)] $f+g\in AP(\mathbb{R})$ and $f\cdot g\in AP(\mathbb{R})$;
\item[(b)] the function $t\mapsto f(t-\tau)$ belongs to $AP(\mathbb{R})$ for every $\tau\in\mathbb{R}$;
\item[(c)] $F\in AP(\mathbb{R})$, where
$$
F(t)=\int_0^{+\infty}k(u)f(t-u)du,\quad t\in\mathbb{R}.
$$
\item[(d)] $AP(\mathbb{R})$ is a Banach space under the norm
$\|f\|=\sup_{t\in\mathbb{R}}|f(t)|$.
\end{itemize}
\end{lemma}

\section{Existence of almost periodic solution}

For the rest of this article, we denote
\begin{gather*}
J=\{11,\dots ,1n,,\dots ,m1,\dots ,mn\},\\
x(t)=\{x_{ij}(t)\}=(x_{11}(t),\dots ,x_{1n}(t),..,x_{m1}(t),\dots ,x_{mn}(t)),\\
X=\{\varphi:\varphi=\{\varphi_{ij}\},\ \varphi_{ij},\varphi'_{ij}\in AP(\mathbb{R})\}.
\end{gather*}
For every $\varphi\in X$,
we denote
\begin{gather*}
\|\varphi\|=\sup_{t\in \mathbb{R} }\max_{ij\in J}\{|\varphi_{ij}(t)|\},\\
\|\varphi\|_X=\max\{\|\varphi\|,\|\varphi'\|\}
=\max\{\sup_{t\in \mathbb{R} }\max_{ij\in J}|\varphi_{ij}(t)|,
\sup_{t\in \mathbb{R} }\max_{ij\in J}|\varphi'_{ij}(t)|\}.
\end{gather*}
It is not difficult to verify that $X$ is a Banach space under the
norm $\|\cdot\|_X$.
For  every $ij \in J$, we denote
\begin{gather*}
a^+_{ij}:=\sup_{t\in \mathbb{R}}a_{ij}(t),\quad
a^-_{ij}:=\inf_{t\in \mathbb{R}}a_{ij}(t),\quad
L^{+}_{ij}:=\sup_{t\in \mathbb{R}}|L_{ij}(t)|,\\
\overline{C_{ij}^{kl}}:=\sup_{t\in \mathbb{R}}|C_{ij}^{kl}(t)|,\quad
\overline{D_{ij}^{kl}}:=\sup_{t\in \mathbb{R}}|D_{ij}^{kl}(t)|.
\end{gather*}

We will use the following assumptions:
\begin{itemize}
\item[(H1)] For every $ij\in J$, $a_{ij}$, $C_{ij}^{kl}$, $D_{ij}^{kl}$
and $L_{ij}$ are both almost periodic functions, and $a^-_{ij}>0$.

\item[(H2)] There exist four functions $f_1, f_2,g_1,g_2:\mathbb{R}\to\mathbb{R}$
and four positive constants $L_{f_1},L_{f_2},L_{g_1},L_{g_2}$ such that
$f=f_1f_2$, $g=g_1g_2$ and for all $u,v \in \mathbb{R}$, there holds
$$
|f_i(u)-f_i(v)|\leq L_{f_i}|u-v|,\quad |g_i(u)-g_i(v)|\leq L_{g_i}|u-v|,
\quad i=1,2.
$$

\item[(H3)] There exists a constant $\lambda_0>0$ such that
$$
\int_0^{\infty}|K_{ij}(u)|e^{\lambda_0 u}du<+\infty,
\quad\int_0^{\infty}|J_{ij}(u)|e^{\lambda_0 u}du<+\infty,\quad ij\in J.
$$

\item[(H4)] There exists a constant $d>0$ such that
\begin{gather*}
\max\Big\{\max_{ij\in J}\big\{\frac{A_{ij}+L_{ij}^+}{a_{ij}^-}\big\} ,
\max_{ij\in J}\big\{\frac{A_{ij}+L_{ij}^+}{a_{ij}^-}(a_{ij}^++a_{ij}^-)\big\}
\Big\}\leq d ,
\\
\max\Big\{\max_{ij\in J}\big\{\frac{B_{ij}(0)}{a_{ij}^-}\big\},
{\max _{ij\in J}}\big\{\frac{B_{ij}(0)}{a_{ij}^-}(a_{ij}^++a_{ij}^-)\big\}
\Big\}<1,
\end{gather*}
where
$M_{f_i}=\sup_{|x|\leq d}|f_i(x)|$,
$M_{g_i}=\sup_{|x|\leq d}|g_i(x)|$, $i=1,2$,
\begin{align*}
A_{ij}&=\sum_{C_{kl}\in N_r(i,j)}\overline{C^{kl}_{ij}}
 (dM_{f_1}M_{f_2})\int_0^\infty |K_{ij}(u)|du\\
&\quad +\sum_{D_{kl}\in N_s(i,j)}\overline{D^{kl}_{ij}}
 (dM_{g_1}M_{g_2})\int_0^\infty |J_{ij}(u)|du,
\end{align*}
and
\begin{align*}
B_{ij}(0)=&\sum_{C_{kl}\in N_r(i,j)}  \overline{C^{kl}_{ij}}
 \Big[(M_{f_1}M_{f_2})\int_0^\infty|K_{ij}(u)|du \\
&\quad + d(M_{f_1}L_{f_2}+M_{f_2}L_{f_1})\int_0^\infty|K_{ij}(u)|du\Big]\\
&\quad +\sum_{D_{kl}\in N_s(i,j)} \overline{D^{kl}_{ij}}
 \Big[(M_{g_1}M_{g_2})\int_0^\infty|J_{ij}(u)|du\\
&\quad +d(M_{g_1}L_{g_2}+M_{g_2}L_{g_1})\int_0^\infty|J_{ij}(u)|du\Big],
\end{align*}
\end{itemize}

\begin{theorem}\label{dingli2.1}
Under assumptions {\rm H1)--(H4)}, there exists a unique continuously
differentiable almost periodic
solution of  \eqref{moxing} in the region
$ \Omega=\{ \varphi \in X:\|
\varphi \|_{X} \leq d \}$.
\end{theorem}

\begin{proof}
For $\omega \in (0, \lambda_0]$, we denote
\begin{align*}
B_{ij}(\omega)
&=\sum_{C_{kl}\in N_r(i,j)} \overline{C^{kl}_{ij}}
\Big[(M_{f_1}M_{f_2})\int_0^\infty|K_{ij}(u)|du\\
&\quad + d(M_{f_1}L_{f_2}+M_{f_2}L_{f_1})\int_0^\infty|K_{ij}(u)|
 e^{\omega u}du\Big]\\
&\quad +\sum_{D_{kl}\in N_s(i,j)} \overline{D^{kl}_{ij}}
\Big[(M_{g_1}M_{g_2})\int_0^\infty|J_{ij}(u)|du\\
&\quad +d(M_{g_1}L_{g_2}+M_{g_2}L_{g_1})\int_0^\infty|J_{ij}(u)|
e^{\omega u}du\Big].
\end{align*}
For each $\varphi \in X $, we consider the almost periodic
differential equations
\begin{equation}\label{moxing2}
\begin{aligned}
x'_{ij}(t)&=  -a_{ij}(t)x_{ij}(t)
 -\sum_{C_{kl}\in N_r(i,j)}C^{kl}_{ij}(t)\int_0^\infty K_{ij}(u)
 f(\varphi_{kl}(t-u))du\varphi_{ij}(t)  \\
&\quad -\sum_{D_{kl}\in N_s(i,j)}D^{kl}_{ij}(t)\int_0^\infty
J_{ij}(u)g(\varphi'_{kl}(t-u))du\varphi_{ij}(t)+L_{ij}(t),\quad  ij\in J.
\end{aligned}
\end{equation}
Combining (H1) and Lemma \ref{ap-lemma}, we know that the
inhomogeneous part of equation \eqref{moxing2} is an almost periodic function.
Noting that $a^-_{ij}>0$, by \cite[Theorem 7.7]{Fink}, we conclude that
\eqref{moxing2} has a unique almost periodic solution $x^{\varphi}$ satisfying
\begin{align*}
x^{\varphi}(t)
&= \Big\{\int^t_{-\infty}e^{-\int^{t}_{s}a_{ij}(u)du}
\Big[- \sum_{C_{kl}\in N_r(i,j)}C^{kl}_{ij}(s)
 \int_0^\infty K_{ij}(u)f(\varphi_{kl}(s-u))du\varphi_{ij}(s) \\
&\quad - \sum_{D_{kl}\in N_s(i,j)}D^{kl}_{ij}(s)
\int_0^\infty J_{ij}(u)g(\varphi'_{kl}(s-u))du\varphi_{ij}(s)
+L_{ij}(s)\Big] ds\Big\}_{ij\in J}.
\end{align*}
Now, define a mapping $T$ on $ \Omega=\{ \varphi \in X:\|\varphi \|_{X} \leq d \}$
 by
$$
(T \varphi)(t)=x^{\varphi}(t),\quad \forall \varphi \in \Omega.
$$
It is easy to show that $T(\Omega)\subset X$.

Next, let us check that $T(\Omega)\subset \Omega$. It suffices to prove that
$\|T\varphi\|_X\leq d$ for all $\varphi\in \Omega$.
By (H2) and (H3), we have
\begin{align*}
&\|T\varphi\|\\
&=  \sup_{t\in \mathbb{R}}\max_{ij\in J}
\Big\{ \Big|\int^t_{-\infty}e^{-\int^{t}_{s}a_{ij}(u)du}\\
&\quad\times \Big[ -  \sum_{C_{kl}\in N_r(i,j)}C^{kl}_{ij}(s)
 \int_0^\infty K_{ij}(u)f(\varphi_{kl}(s-u))du\varphi_{ij}(s) \\
&\quad -\sum_{D_{kl}\in N_s(i,j)}D^{kl}_{ij}(s)
 \int_0^\infty J_{ij}(u)g(\varphi'_{kl}(s-u))du\varphi_{ij}(s)
 +L_{ij}(s)\Big]ds \Big| \Big\}
\\
& \leq  \sup_{t\in \mathbb{R}}  \max_{ij\in J}
 \Big\{\int^t_{-\infty}e^{-\int^{t}_{s}a_{ij}(u)du}\\
&\quad\times \Big[  \sum_{C_{kl}\in N_r(i,j)}\overline{C^{kl}_{ij}}
 \int_0^\infty |K_{ij}(u)||f(\varphi_{kl}(s-u))|du|\varphi_{ij}(s)| \\
&\quad +\sum_{D_{kl}\in N_s(i,j)}\overline{D^{kl}_{ij}}\int_0^\infty
 |J_{ij}(u)||g(\varphi'_{kl}(s-u))|du|\varphi_{ij}(s)|+|L_{ij}(s)|\Big]ds\Big\}
\\
& \leq  \sup_{t\in \mathbb{R}}  \max_{ij\in J}
 \Big\{\int^t_{-\infty}e^{a_{ij}^-(s-t)}
 \Big[\sum_{C_{kl}\in N_r(i,j)}\overline{C^{kl}_{ij}}(dM_{f_1}M_{f_2})
 \int_0^\infty |K_{ij}(u)|du  \\
&\quad +\sum_{D_{kl}\in N_s(i,j)}\overline{D^{kl}_{ij}}(dM_{g_1}M_{g_2})
 \int_0^\infty |J_{ij}(u)|du+L_{ij}^+\Big]ds\Big\}
\\
& \leq  \max_{ij\in J}\Big\{
 \Big[\sum_{C_{kl}\in N_r(i,j)}\overline{C^{kl}_{ij}}
 (dM_{f_1}M_{f_2})\int_0^\infty |K_{ij}(u)|du\\
&\quad  +\sum_{D_{kl}\in N_s(i,j)}\overline{D^{kl}_{ij}}(dM_{g_1}M_{g_2})
 \int_0^\infty |J_{ij}(u)|du+L_{ij}^+\Big]/ a_{ij}^- \Big\}\\
& =  \max_{ij\in J}\bigg\{\frac{A_{ij}+L_{ij}^+}{a_{ij}^-} \bigg\},
\end{align*}
and
\begin{align*}
\|(T\varphi)'\|
& =  \sup_{t\in \mathbb{R}}\max_{ij\in J}
\Big\{ \Big|-a_{ij}(t)\int^t_{-\infty}e^{-\int^{t}_{s}a_{ij}(u)du}\\
&\quad\times \Big[ -  \sum_{C_{kl}\in N_r(i,j)}C^{kl}_{ij}(t)
  \int_0^\infty K_{ij}(u)f(\varphi_{kl}(s-u))du\varphi_{ij}(s)\\
&\quad  -\sum_{D_{kl}\in N_s(i,j)}D^{kl}_{ij}(t)
  \int_0^\infty J_{ij}(u)g(\varphi'_{kl}(s-u))du\varphi_{ij}(s)
 +L_{ij}(s)\Big]ds \\
&\quad  +\Big[-  \sum_{C_{kl}\in N_r(i,j)}C^{kl}_{ij}(t)
 \int_0^\infty K_{ij}(u)f(\varphi_{kl}(t-u))du\varphi_{ij}(t)\\
&\quad  -\sum_{D_{kl}\in N_s(i,j)}D^{kl}_{ij}(t)
 \int_0^\infty J_{ij}(u)g(\varphi'_{kl}(t-u))du\varphi_{ij}(t)+L_{ij}(t)\Big]
\Big| \Big\}\\
& \leq \sup_{t\in \mathbb{R}}\max_{ij\in J}
 \Big\{a_{ij}^+\cdot \frac{A_{ij}+L_{ij}^+}{a_{ij}^-}+ A_{ij}+L_{ij}^+\Big\}\\
& = \max_{ij\in J}\Big\{\frac{A_{ij}+L_{ij}^+}{a_{ij}^-}(a_{ij}^+
 +a_{ij}^-)\Big\}.
\end{align*}
Then,  from (H4) it follows that
$$
\|(T\varphi)\|_X\leq \max\Big\{\max_{ij\in J}
\big\{\frac{A_{ij}+L_{ij}^+}{a_{ij}^-}\big\} ,\max_{ij\in J}
\big\{\frac{A_{ij}+L_{ij}^+}{a_{ij}^-}(a_{ij}^++a_{ij}^-)\big\}\Big\}\leq d,
$$
which implies that $T( \Omega)\subset  \Omega$.


Let $\varphi,\psi\in  \Omega$, and for $ij\in J$ denote
\begin{align*}
\alpha_{ij}(s)
&=\sum_{C_{kl}\in N_r(i,j)}C^{kl}_{ij}(s)
\Big(\int_0^\infty K_{ij}(u)f(\varphi_{kl}(s-u))du\varphi_{ij}(s)\\
&\quad -\int_0^\infty K_{ij}(u)f(\psi_{kl}(s-u))du\psi_{ij}(s)\Big),
\end{align*}
and
\begin{align*}
\beta_{ij}(s)
&=\sum_{D_{kl}\in N_s(i,j)}D^{kl}_{ij}(s)
 \Big(\int_0^\infty J_{ij}(u)g(\varphi'_{kl}(s-u))du\varphi_{ij}(s)\\
&\quad -\int_0^\infty J_{ij}(u)g(\psi'_{kl}(s-u))du\psi_{ij}(s)\Big).
\end{align*}
By (H2), for each $ij\in J$, we obtain
\begin{align*}
& |\alpha_{ij}(s)| \\
& \leq \sum_{C_{kl}\in N_r(i,j)}\overline{C^{kl}_{ij}}
\Big\{\Big|\int_0^\infty K_{ij}(u)f(\varphi_{kl}(s-u))du
 \varphi_{ij}(s)\\
&\quad -\int_0^\infty K_{ij}(u)f(\varphi_{kl}(s-u))du\psi_{ij}(s)\Big|\\
&\quad  +\Big|\int_0^\infty K_{ij}(u)f(\varphi_{kl}(s-u))du\psi_{ij}(s)
 -\int_0^\infty K_{ij}(u)f(\psi_{kl}(s-u))du\psi_{ij}(s)\Big|\Big\}\\
& \leq \sum_{C_{kl}\in N_r(i,j)}\overline{C^{kl}_{ij}}
 \Big|\Big[\int_0^\infty K_{ij}(u)f_1(\varphi_{kl}(s-u))
 f_2(\varphi_{kl}(s-u))du\varphi_{ij}(s) \\
&\quad -\int_0^\infty K_{ij}(u)f_1(\varphi_{kl}(s-u))
 f_2(\varphi_{kl}(s-u))du\psi_{ij}(s)\Big] \\
&\quad  +\Big[\int_0^\infty K_{ij}(u)f_1(\varphi_{kl}(s-u))
 f_2(\varphi_{kl}(s-u))du\psi_{ij}(s)\\
&\quad  -\int_0^\infty K_{ij}(u)f_1(\varphi_{kl}(s-u))
 f_2(\psi_{kl}(s-u))du\psi_{ij}(s)\Big]\\
&\quad  +\Big[\int_0^\infty K_{ij}(u)f_1(\varphi_{kl}(s-u))
 f_2(\psi_{kl}(s-u))du\psi_{ij}(s)\\
&\quad  -\int_0^\infty K_{ij}(u)f_1(\psi_{kl}(s-u))
 f_2(\psi_{kl}(s-u))du\psi_{ij}(s)\Big]\Big|\\
& \leq \sum_{C_{kl}\in N_r(i,j)}\overline{C^{kl}_{ij}}
 [M_{f_1}M_{f_2}+d(M_{f_1}L_{f_2}+M_{f_2}L_{f_1})]
 \int_0^\infty |K_{ij}(u)|du\|\varphi-\psi\|_X.
\end{align*}
Similarly, for each $ij \in J$, we have
$$
|\beta_{ij}(s)|\leq \sum_{D_{kl}\in N_s(i,j)}\overline{D^{kl}_{ij}}
[M_{g_1}M_{g_2}+d(M_{g_1}L_{g_2}+M_{g_2}L_{g_1})]
\int_0^\infty |J_{ij}(u)|du\|\varphi-\psi\|_X.
$$
Thus,
$$
|\alpha_{ij}(s)|+|\beta_{ij}(s)|\leq B_{ij}(0)||\varphi-\psi||_X.
$$
It follows that
\begin{align*}
\|T\varphi-T\psi\|
&=\sup_{t\in \mathbb{R}}\max_{ij\in J}
\Big\{ \Big|\int^t_{-\infty}e^{-\int^{t}_{s}a_{ij}(u)du}[\alpha_{ij}(s)
 +\beta_{ij}(s)]ds \Big| \Big\}\\
&\leq \sup_{t\in \mathbb{R}}\max_{ij\in J}
 \int^t_{-\infty}e^{-\int^{t}_{s}a_{ij}(u)du}
 (|\alpha_{ij}(s)|+|\beta_{ij}(s)|)ds\\
&\leq \sup_{t\in \mathbb{R}}\max_{ij\in J}
 \Big\{\int^t_{-\infty}e^{-\int^{t}_{s}a_{ij}(u)du}ds\Big\} B_{ij}(0)
 \|\varphi-\psi\|_X\\
&\leq \max_{ij\in J}\Big\{\frac{B_{ij}(0)}{a_{ij}^-}\Big\}\|\varphi-\psi\|_X,
\end{align*}
and
\begin{align*}
&\|(T\varphi-T\psi)'\|\\
&=\sup_{t\in \mathbb{R}}\max_{ij\in J}
 \Big\{ \Big|-a_{ij}(t)\int^t_{-\infty}e^{-\int^{t}_{s}a_{ij}(u)ds}
 (\alpha_{ij}(s)+\beta_{ij}(s))ds+(\alpha_{ij}(t)+\beta_{ij}(t)) \Big| \Big\}\\
&\leq \sup_{t\in \mathbb{R}}\max_{ij\in J}
\Big\{ a_{ij}^+\Big|\int^t_{-\infty}e^{-\int^{t}_{s}a_{ij}(u)du}
 (\alpha_{ij}(s)+\beta_{ij}(s))ds\Big|+(|\alpha_{ij}(t)|+|\beta_{ij}(t)|) \Big\}\\
&\leq \sup_{t\in \mathbb{R}}\max_{ij\in J}
 \Big\{a_{ij}^+\frac{B_{ij}(0)}{a_{ij}^-}\|\varphi-\psi\|_X+B_{ij}(0)
 \|\varphi-\psi\|_X\Big\}\\
&\leq \max_{ij\in J}\Big\{\frac{B_{ij}(0)}{a_{ij}^-}(a_{ij}^++a_{ij}^-)\Big\}
 \|\varphi-\psi\|_X.
\end{align*}
Combining the above two inequalities, we obtain
$$
\|T\varphi-T\psi\|_{X}\leq \max\Big\{\max_{ij\in J}
\big\{\frac{B_{ij}(0)}{a_{ij}^-}\big\},
{\max _{ij\in J}}\big\{\frac{B_{ij}(0)}{a_{ij}^-}(a_{ij}^++a_{ij}^-)\big\}\Big\}
\|\varphi-\psi\|_{X}.
$$
Noticing that
$$\
max\Big\{\max_{ij\in J}\big\{\frac{B_{ij}(0)}{a_{ij}^-}\big\},
{\max _{ij\in J}}\big\{\frac{B_{ij}(0)}{a_{ij}^-}(a_{ij}^++a_{ij}^-)\big\}\Big\}
<1,
$$
By the Banach contraction principle, $T$ has a unique fixed point
$x$ in $\Omega$, which is just a continuously differentiable almost periodic
solution of Equation \eqref{moxing}.
\end{proof}

\section{Stability of almost periodic solutions}

In this section, we will establish some results about the locally exponential
stability of the almost periodic solution for Equation \eqref{moxing}.

\begin{theorem}\label{dingli3.1}
Assume {\rm (H1)--(H4)} hold. Let $x(t)=\{x_{ij}(t)\}$ be the unique
continuously differentiable almost periodic solution of  \eqref{moxing}
in $\Omega$, and $y(t)=\{y_{ij}(t)\}$ be an arbitrary
continuously differentiable solution of Equation \eqref{moxing} in the
region $\Omega$.
Then, there exist two constants $\lambda,M>0$ such that
$$
\|x(t)-y(t)\|_1\leq M e^{-\lambda t}, \quad \forall t\in \mathbb{R},
$$
where
$$
\|x(t)-y(t)\|_1 :=\max\{\max_{ij\in J}|x_{ij}(t)-y_{ij}(t)|,
\max_{ij\in J}|x'_{ij}(t)-y'_{ij}(t)|\}.
$$
\end{theorem}

\begin{proof}
For $\omega\in [0,\lambda_0]$, we denote
$$
T_{ij}(\omega)=a_{ij}^--\omega-B_{ij}(\omega), \quad
S_{ij}(\omega)=a_{ij}^--\omega-(a_{ij}^++a_{ij}^-)B_{ij}(\omega).
$$
By (H4), we have $T_{ij}(0)>0$  and   $S_{ij}(0)>0$ for all $ij\in J$.
Then, due to the continuity of $T_{ij}(\omega)$ and $S_{ij}(\omega)$,
there exists a sufficiently small positive constant
$\lambda<\min\big\{\min_{ij\in J}\{a_{ij}^-\},\lambda_0\big\}$ such that
$$
T_{ij}(\lambda)>0,\quad S_{ij}(\lambda)>0,\quad ij\in J,
$$
which means that
\begin{equation}\label{bds3.1}
\frac{B_{ij}(\lambda)}{a^{-}_{ij}-\lambda}<1, \quad
\frac{B_{ij}(\lambda)}{a^{-}_{ij}-\lambda}(a_{ij}^++a_{ij}^-)<1,\quad ij\in J.
\end{equation}
for all $ij\in J$. Setting
$M_0=\max_{ij\in J}\big\{\frac{a_{ij}^-}{B_{ij}(0)}\big\}$,
the following three inequalities hold:
\begin{equation}\label{three equations}
M_0>1,\quad \frac{1}{M_0}-\frac{B_{ij}(\lambda)}{a_{ij}^--\lambda}\leq0 ,\quad
B_{ij}(\lambda)(\frac{a_{ij}^+}{a^{-}_{ij}-\lambda}+1)<1,\quad ij\in J.
\end{equation}
Now, we denote
\begin{gather*}
z(t)=\big\{z_{ij}(t):z_{ij}(t)=x_{ij}(t)-y_{ij}(t)\big\},
\\
\begin{aligned}
R_{ij}(s)&=\sum_{C_{kl}\in N_r(i,j)}C^{kl}_{ij}(s)
\Big[\int_0^\infty K_{ij}(u)f(y_{kl}(s-u))duy_{ij}(s)\\
&\quad -\int_0^\infty K_{ij}(u)f(x_{kl}(s-u))dux_{ij}(s)\Big],
\end{aligned}\\
\begin{aligned}
Q_{ij}(s)&=\sum_{D_{kl}\in N_s(i,j)}D^{kl}_{ij}(s)
\Big[\int_0^\infty J_{ij}(u)g(y'_{kl}(s-u))duy_{ij}(s)\\
&\quad -\int_0^\infty J_{ij}(u)g(x'_{kl}(s-u))dux_{ij}(s)\Big].
\end{aligned}
\end{gather*}
Since $x(t)$ and $y(t)$ are both solutions to equation \eqref{moxing}, we have
\begin{equation} \label{3.1}
 z'_{ij}(s)+a_{ij}(s)z_{ij}(s)=R_{ij}(s)+Q_{ij}(s).
\end{equation}
Multiplying  by $e^{\int_0^sa_{ij}(u)du}$ and integrating on $[0,t]$, we obtain
\begin{equation}\label{3.2}
z_{ij}(t)=z_{ij}(0)e^{-\int_0^ta_{ij}(u)du}+\int_0^te^{-\int_s^ta_{ij}(u)du}(R_{ij}(s)+Q_{ij}(s))ds
\end{equation}
Let
$$
M:=M_0\cdot \max\big\{\sup_{t\leq 0}\max_{ij\in J}|x_{ij}(t)-y_{ij}(t)|,
\sup_{t\leq 0}\max_{ij\in J}|x'_{ij}(t)-y'_{ij}(t)|\big\}.
$$
Without loss for generality, we can assume that $M>0$.
Then, for all $t\leq 0$, noting that $M_0>1$, we have
$$
\|z(t)\|_1= \max\big\{\max_{ij\in
J}|x_{ij}(t)-y_{ij}(t)|,\max_{ij\in
J}|x'_{ij}(t)-y'_{ij}(t)|\big\}<Me^{-\lambda t} .
$$

Next, we prove the inequality
$$
\|z(t)\|_1\leq Me^{-\lambda t},\quad t>0,
$$
by contradiction. If the above inequality is not true, then
$$
 V:=\{t>0:  \|z(t)\|_1 > Me^{-\lambda t} \}\neq \emptyset.
$$
 Letting $t_1=\inf V$, then $t_1>0$ and
\begin{align}\label{maodun}
\|z(t)\|_1&\leq M e^{-\lambda t}, \quad \forall t\in(-\infty,t_{1}),\quad
\|z(t_1)\|_1= M e^{-\lambda t_1}.
\end{align}

For $s\in [0, t_1]$ and $ij\in J$, by the assumptions and \eqref{maodun},
we have
\begin{align*}
&|R_{ij}(s)|\\
&=\sum_{C_{kl}\in N_r(i,j)}|C^{kl}_{ij}(s)|
 \Big|\int_0^\infty K_{ij}(u)f(y_{kl}(s-u))duy_{ij}(s)\\
&\quad -\int_0^\infty K_{ij}(u)f(x_{kl}(s-u))dux_{ij}(s)\Big|\\
&\leq \sum_{C_{kl}\in N_r(i,j)}\overline{C^{kl}_{ij}}
 \Big\{\Big|\int_0^\infty K_{ij}(u)f(y_{kl}(s-u))duy_{ij}(s)\\
&\quad -\int_0^\infty K_{ij}(u)f(y_{kl}(s-u))dux_{ij}(s)\Big|\\
&\quad +\Big|\int_0^\infty K_{ij}(u)f(y_{kl}(s-u))dux_{ij}(s)
-\int_0^\infty K_{ij}(u)f(x_{kl}(s-u))dux_{ij}(s)\Big|\Big\}\\
&\leq \sum_{C_{kl}\in N_r(i,j)}\overline{C^{kl}_{ij}}
 \Big\{\Big|\int_0^\infty K_{ij}(u)f_1(y_{kl}(s-u))f_2(y_{kl}(s-u))duy_{ij}(s)
 \Big|\\
&\quad -\int_0^\infty K_{ij}(u)f_1(y_{kl}(s-u))f_2(y_{kl}(s-u))dux_{ij}(s)\Big|\\
&\quad +\Big|\int_0^\infty K_{ij}(u)f_1(y_{kl}(s-u))f_2(y_{kl}(s-u))dux_{ij}(s)\\
&\quad -\int_0^\infty K_{ij}(u)f_1(x_{kl}(s-u))f_2(y_{kl}(s-u))dux_{ij}(s)\Big|\\
&\quad +\Big|\int_0^\infty K_{ij}(u)f_1(x_{kl}(s-u))f_2(y_{kl}(s-u))dux_{ij}(s)\\
&\quad -\int_0^\infty K_{ij}(u)f_1(x_{kl}(s-u))f_2(x_{kl}(s-u))dux_{ij}(s)\Big|\Big\}\\
&\leq \sum_{C_{kl}\in N_r(i,j)}\overline{C^{kl}_{ij}}
 \Big\{(M_{f_1}M_{f_2})\int_0^\infty|K_{ij}(u)|du|z_{ij}(s)|\\
&\quad + d(M_{f_1}L_{f_2}+M_{f_2}L_{f_1})\int_0^\infty|K_{ij}(u)||z_{kl}(s-u)|du\Big\}\\
&\leq \sum_{C_{kl}\in N_r(i,j)}\overline{C^{kl}_{ij}}
 \Big\{(M_{f_1}M_{f_2})\int_0^\infty|K_{ij}(u)|du\\
&\quad + d(M_{f_1}
 L_{f_2}+M_{f_2}L_{f_1})\int_0^\infty|K_{ij}(u)|e^{\lambda u}du\Big\}
 Me^{-\lambda s}.
\end{align*}
Similarly, for $s\in [0, t_1]$ and $ij\in J$, we  have
\begin{align*}
 |Q_{ij}(s)|
&\leq \sum_{D_{kl}\in N_s(i,j)}\overline{D^{kl}_{ij}}
 \Big\{(M_{g_1}M_{g_2})\int_0^\infty|J_{ij}(u)|du|z_{ij}(s)|\\
&\quad + d(M_{g_1}L_{g_2}+M_{g_2}L_{g_1})\int_0^\infty|J_{ij}(u)||z'_{kl}(s-u)|du\Big\}\\
&\leq \sum_{D_{kl}\in N_s(i,j)}\overline{D^{kl}_{ij}}
\Big\{(M_{g_1}M_{g_2})\int_0^\infty|J_{ij}(u)|du\\
&\quad + d(M_{g_1}L_{g_2}+M_{g_2}L_{g_1}) \int_0^\infty|J_{ij}(u)|e^{\lambda u}du\Big\}
 Me^{-\lambda s}.
\end{align*}
Then we have
\begin{equation}\label{er}
|R_{ij}(s)|+|Q_{ij}(s)|\leq Me^{-\lambda s}B_{ij}(\lambda),\quad 
 s\in [0,t_1],\ ij\in J.
\end{equation}
Combining \eqref{3.2} and \eqref{er}, we have
\begin{equation}\label{3.8}
\begin{aligned}
|z_{ij}(t_1)|
&= \big|z_{ij}(0)e^{-\int_0^{t_1}a_{ij}(u)du}
 +\int_0^{t_1}e^{-\int_s^{t_1}a_{ij}(u)du}(R_{ij}(s)+Q_{ij}(s))ds\big| \\
&\leq \frac{M}{M_0}e^{-a_{ij}^-t_1}+\int_0^{t_1}
 e^{(a_{ij}^{-}-\lambda)s-a_{ij}^{-}t_1}ds\cdot MB_{ij}(\lambda) \\
&= \frac{M}{M_0}e^{-a_{ij}^-t_1}+\frac{(e^{-\lambda t_1}
 -e^{-a_{ij}^- t_1})B_{ij}(\lambda)}{a_{ij}^--\lambda}\cdot M  \\
&\leq  Me^{-\lambda t_1}\Big\{\frac{e^{(\lambda-a_{ij}^-)t_1}}{M_0}
 +\frac{[1-e^{(\lambda-a_{ij}^-) t_1}]B_{ij}(\lambda))}{a_{ij}^--\lambda}\Big\}\\
&=  Me^{-\lambda t_1}\Big\{\Big(\frac{1}{M_0}-\frac{B_{ij}(\lambda)}{a_{ij}^-
-\lambda}\Big)e^{(\lambda-a_{ij}^-)t_1}
 +\frac{B_{ij}(\lambda)}{a_{ij}^--\lambda}\Big\}. 
\end{aligned}
\end{equation}
Then, by \eqref{bds3.1} and \eqref{three equations}, we deduce that
\begin{equation}\label{3.9}
 |z_{ij}(t_1)|< Me^{-\lambda t_1},\quad ij\in J.
\end{equation}
Recalling that
$$
z'_{ij}(t) = -a_{ij}(t)z_{ij}(t)+R_{ij}(t)+Q_{ij}(t),
$$
by \eqref{er} and \eqref{3.8}, we have
\begin{align*}
|z'_{ij}(t_1)|
& =  |-a_{ij}(t_1)z_{ij}(t_1)+R_{ij}(t_1)+Q_{ij}(t_1)|\\
&\leq  a_{ij}^+|z_{ij}(t_1)|+|R_{ij}(t_1)|+|Q_{ij}(t_1)|\\
&< a_{ij}^+ Me^{-\lambda t_1}\Big\{\Big(\frac{1}{M_0}
 -\frac{B_{ij}(\lambda)}{a_{ij}^--\lambda}\Big)e^{(\lambda-a_{ij}^-)t_1}
 +\frac{B_{ij}(\lambda)}{a_{ij}^--\lambda}\Big\}
+M e^{-\lambda t_1}B_{ij}(\lambda)\\
&=  Me^{-\lambda t_1}\Big\{a_{ij}^+\Big(\frac{1}{M_0}
-\frac{B_{ij}(\lambda)}{a_{ij}^--\lambda}\Big)e^{(\lambda-a_{ij}^-)t_1}
+B_{ij}(\lambda) (\frac{a_{ij}^
+}{a^{-}_{ij}-\lambda}+1)\Big\}
\end{align*}
Then, from \eqref{three equations} it follows  that
\begin{equation}\label{3.10}
|z'_{ij}(t_1)|< M e^{-\lambda t_1},\quad  ij\in J.
\end{equation}
Combining \eqref{3.9} and \eqref{3.10}, we obtain
\begin{align*}
\|z(t_1)\|_1 <Me^{-\lambda t_1},
\end{align*}
which contradicts with \eqref{maodun}. Thus, we obtain
$$
\|z(t)\|_1\leq Me^{-\lambda t} ,
$$
for all $t\in\mathbb{R}$. This completes the proof.
\end{proof}

\begin{remark}\label{remark3.2}  \rm
Compared with the results in \cite{fang2}, our Lipschitz conditions
are weaker, and thus our results may have a wider range of
applications.
\end{remark}


Next, we give an example to illustrate our main results.

\begin{example}\label{example}\rm
Consider the following SICNNs with neutral delays:
 \begin{equation}\label{liti4.1}
\begin{aligned}
x'_{ij}(t)
&=  -  a_{ij}(t)x_{ij}(t)-\sum_{C_{kl}\in N_1(i,j)}C^{kl}_{ij}(t)
\int_0^\infty K_{ij}(u)f(x_{kl}(t-u))dux_{ij}(t)  \\
&\quad - \sum_{D_{kl}\in N_1(i,j)}D^{kl}_{ij}(t)
\int_0^\infty J_{ij}(u)g(x'_{kl}(t-u))dux_{ij}(t)+L_{ij}(t),
\end{aligned}
\end{equation}
where $i=1,2,3$, $j=1,2,3$.
For $i,j=1,2,3$ and $t\in\mathbb{R}$, let
\begin{gather*}
f(t)=\frac{1}{4}|t|\cos t,\quad 
g(t)=\frac{1}{16}(t^2-1),\quad 
L_{ij}(t)=\frac{1}{8}|\sin\sqrt{2}t+\sin t|, \\
K_{ij}(t)=e^{-6t},\quad J_{ij}(t)=e^{-4t}.
\end{gather*}
Moreover,
\begin{gather*}   
\begin{pmatrix}
    a_{11}(t) & a_{12}(t) & a_{13}(t)\\
    a_{21}(t) & a_{22}(t) & a_{23}(t)\\
    a_{31}(t) & a_{32}(t) & a_{33}(t)
\end{pmatrix}
=\begin{pmatrix}
    5+|\sin t| & 5+|\sin\sqrt{2}t| & 9+|\sin\sqrt{3}t|\\
    6+|\cos t| & 6+\frac{|\cos t+\cos{\sqrt{2} t}|}{2} & 7+|\cos \sqrt{2}t|\\
    8+|\sin t| & 8+|\sin\sqrt{2}t| & 5+|\sin\sqrt{3}t|
\end{pmatrix},
\\
\begin{pmatrix}
    C_{11}(t) & C_{12}(t) & C_{13}(t)\\
    C_{21}(t) & C_{22}(t) & C_{23}(t)\\
    C_{31}(t) & C_{32}(t) & C_{33}(t)
\end{pmatrix}
= \begin{pmatrix}
    0.1|\sin t| & 0.3|\sin\sqrt{2}t| & 0.5|\sin\sqrt{3}t|\\
    0.2|\cos t| & 0.1|\cos \sqrt{2}t| & 0.2|\cos\sqrt{3}t|\\
    0.1|\sin t| & 0.2|\sin\sqrt{2}t| & 0.1|\cos\sqrt{3}t|
\end{pmatrix}
\\
\begin{pmatrix}
    D_{11}(t) & D_{12}(t) & D_{13}(t)\\
    D_{21}(t) & D_{22}(t) & D_{23}(t)\\
    D_{31}(t) & D_{32}(t) & D_{33}(t)
\end{pmatrix}
= 
\begin{pmatrix}
    0.1|\cos t| & 0.3|\cos\sqrt{2}t| & 0.5|\cos\sqrt{3}t|\\
    0.2|\sin t| & 0.1|\sin \sqrt{2}t| & 0.2|\sin\sqrt{3}t|\\
    0.1|\cos t| & 0.2|\cos\sqrt{2}t| & 0.1|\cos\sqrt{3}t|
\end{pmatrix}.
\end{gather*}

Let $d=1$, $\lambda_0=1,$ $f_1(t)=\frac{1}{2}|t|$, $f_2(x)=\frac{1}{2}\cos t$,
$g_1(t)=\frac{1}{4}(t-1)$, $g_2(t)=\frac{1}{4}(t+1)$. 
By a direct calculation, we obtain
$L_{f_1}=L_{f_2}=\frac{1}{2}$, $L_{g_1}=L_{g_2}=\frac{1}{4}$, 
$M_{f_1}=M_{f_2}=\frac{1}{2},$ $M_{g_1}=M_{g_2}=\frac{1}{2}$, 
$L_{ij}^+=\frac{1}{4}.$
Then, it is easy to see that (H1)--(H3) hold.

Next, let us verify (H4). We have
\begin{gather*}
\begin{pmatrix}
    a_{11}^- & a_{12}^-   & a_{13}^-\\
    a_{21}^- & a_{22}^-   & a_{23}^-\\
    a_{31}^- & a_{32}^-   & a_{33}^-
\end{pmatrix}
=\begin{pmatrix}
    5 & 5 & 9\\
    6 & 6 & 7\\
    8 & 8 & 5
\end{pmatrix}, \quad
\begin{pmatrix}
    a_{11}^+ & a_{12}^+ & a_{13}^+\\
    a_{21}^+ & a_{22}^+ & a_{23}^+\\
    a_{31}^+ & a_{32}^+ & a_{33}^+
\end{pmatrix}
= \begin{pmatrix}
    6 & 6 & 10\\
    7 & 7 & 8\\
    9 & 9 & 6
\end{pmatrix},
\\
\begin{pmatrix}
\sum_{C_{kl}\in N_1(1,1)}\overline{C_{11}^{kl}} 
 &\sum_{C_{kl}\in N_1(1,1)}\overline{C_{12}^{kl}}  
 & \sum_{C_{kl}\in N_1(1,1)}\overline{C_{13}^{kl}}\\
\sum_{C_{kl}\in N_1(1,1)}\overline{C_{21}^{kl}}
 &\sum_{C_{kl}\in N_1(1,1)}\overline{C_{22}^{kl}}
 & \sum_{C_{kl}\in N_1(1,1)}\overline{C_{23}^{kl}}\\
    \sum_{C_{kl}\in N_1(1,1)}\overline{C_{31}^{kl}}
 &\sum_{C_{kl}\in N_1(1,1)}\overline{C_{32}^{kl}}
 & \sum_{C_{kl}\in N_1(1,1)}\overline{C_{33}^{kl}}
\end{pmatrix}
= \begin{pmatrix}
0.7 & 1.4 & 1.1 \\
1 & 1.8 & 1.4 \\
0.6 & 0.9 &0.6 
\end{pmatrix},
\\
\begin{pmatrix}
   \sum_{D_{kl}\in N_1(1,1)}\overline{D_{11}^{kl}} 
 &\sum_{D_{kl}\in N_1(1,1)}\overline{D_{12}^{kl}}  
 & \sum_{D_{kl}\in N_1(1,1)}\overline{D_{13}^{kl}}\\
   \sum_{D_{kl}\in N_1(1,1)}\overline{D_{21}^{kl}} 
 &\sum_{D_{kl}\in N_1(1,1)}\overline{D_{22}^{kl}}  
 & \sum_{D_{kl}\in N_1(1,1)}\overline{D_{23}^{kl}}\\
    \sum_{D_{kl}\in N_1(1,1)}\overline{D_{31}^{kl}} 
 &\sum_{D_{kl}\in N_1(1,1)}\overline{D_{32}^{kl}}  
 & \sum_{D_{kl}\in N_1(1,1)}\overline{D_{33}^{kl}}
\end{pmatrix}
= \begin{pmatrix}
0.7 & 1.4 & 1.1 \\
1 & 1.8 & 1.4 \\
0.6 & 0.9 &0.6 
\end{pmatrix},
\\
\begin{aligned}
A_{ij}
&= \sum_{C_{kl}\in N_1(i,j)}\overline{C^{kl}_{ij}}(dM_{f_1}M_{f_2})
 \int_0^\infty |K_{ij}(u)|du\\
&\quad +\sum_{D_{kl}\in N_1(i,j)}\overline{D^{kl}_{ij}}(dM_{g_1}M_{g_2})
 \int_0^\infty |J_{ij}(u)|du\\
&= \frac{5}{48}\sum_{C_{kl}\in N_1(i,j)}\overline{C^{kl}_{ij}},
\end{aligned}
\\
\begin{aligned}
B_{ij}(0)
&= \sum_{C_{kl}\in N_1(i,j)} \overline{C^{kl}_{ij}}
\Big[(M_{f_1}M_{f_2})\int_0^\infty|K_{ij}(u)|du\\
&\quad + d(M_{f_1}L_{f_2}+M_{f_2}L_{f_1})\int_0^\infty|K_{ij}(u)|du\Big]\\
&\quad +\sum_{D_{kl}\in N_1(i,j)}\overline{D^{kl}_{ij}}
\Big[(M_{g_1}M_{g_2})\int_0^\infty|J_{ij}(u)|du\\
&\quad +d(M_{g_1}L_{g_2}+M_{g_2}L_{g_1})\int_0^\infty|J_{ij}(u)|du\Big]\\
& =\frac{1}{4}\sum_{C_{kl}\in N_1(i,j)}\overline{C^{kl}_{ij}}.
\end{aligned}
\end{gather*}
Then, we obtain
\begin{gather*}
\begin{pmatrix}
A_{11} & A_{12} &A_{13} \\
A_{21} & A_{22} &A_{23} \\
A_{31} & A_{32} &A_{33} 
\end{pmatrix}
= \begin{pmatrix}
\frac{3.5}{48} & \frac{7}{48}& \frac{5.5}{48}\\
\frac{5}{48} & \frac{9}{48}& \frac{7}{48}\\
\frac{3}{48} & \frac{4.5}{48}& \frac{3}{48}
\end{pmatrix},
\\
\begin{pmatrix}
B_{11}(0) & B_{12}(0) &B_{13}(0) \\
B_{21}(0) & B_{22}(0) &B_{23}(0) \\
B_{31}(0) & B_{32}(0) &B_{33}(0)
\end{pmatrix}
= \begin{pmatrix}
\frac{0.7}{4} & \frac{1.4}{4}& \frac{1.1}{4}\\
\frac{1}{4} & \frac{1.8}{4}& \frac{1.4}{4}\\
\frac{0.6}{4} & \frac{0.9}{4}& \frac{0.6}{4}
\end{pmatrix}
\end{gather*}
It follows that
\begin{align*}
& \max\Big\{\max_{ij\in J}\big\{\frac{A_{ij}+L_{ij}^+}{a_{ij}^-}\big\} ,
\max_{ij\in J}\big\{\frac{A_{ij}+L_{ij}^+}{a_{ij}^-}(a_{ij}^++a_{ij}^-)\big\}\Big\}
\\
&= \max_{ij\in J}\Big\{\frac{A_{ij}+L_{ij}^+}{a_{ij}^-}(a_{ij}^+
 +a_{ij}^-)\Big\}\\
&\leq \big(\max_{ij\in J}A_{ij}^+
 +\max_{ij\in J}L_{ij}^+\big) \max_{ij\in J}
\big\{\frac{a_{ij}^++a_{ij}^-}{a_{ij}^-}\big\}\\
&= \big(A_{22}+\frac{1}{4}\big)\frac{a_{11}^++a_{11}^-}{a_{11}^-}\\
&= \big(\frac{9}{48}+\frac{1}{4}\big)\cdot\frac{11}{5}<1=d,
\end{align*}
and
\begin{align*}
& \max\Big\{\max_{ij\in J}\big\{\frac{B_{ij}(0)}{a_{ij}^-}\big\},
{\max _{ij\in J}}\big\{\frac{B_{ij}(0)}{a_{ij}^-}(a_{ij}^++a_{ij}^-)\big\}\Big\}\\
&= \max_{ij\in J}\big\{\frac{B_{ij}(0)}{a_{ij}-}(a_{ij}^++a_{ij}^-)\big\}\\
&\leq \max_{ij\in J}\{B_{ij}(0)\}\max_{ij\in J}
\big\{\frac{a_{ij}^++a_{ij}^-}{a_{ij}^-}\big\}\\
&= B_{22}(0)\frac{a_{11}^++a_{11}^-}{a_{11}^-}\\
&=  \frac{1.8}{4}\cdot\frac{11}{5}<1.
\end{align*}
Thus,  condition (H4) holds. By Theorem \ref{dingli2.1} and 
Theorem \ref{dingli3.1}, Equation \eqref{liti4.1} admits a unique 
differentiable almost periodic solution $x^*(t)$ in the  region
$\Omega=\{\varphi\in X:\|\varphi\|_X\leq 1\}$, and $x^*(t)$ 
is locally exponentially stable in $\Omega$.
\end{example}


\begin{remark}\rm
In Example \ref{example}, let the iterative sequences
$x_n(t)=\{x_{n_{ij}}(t)\}$ be
 $$
x_{0_{ij}}(t)=0,\quad \forall t\in\mathbb{R},\;  i=1,2,3,\;
j=1,2,3,
$$ 
and
\begin{align*}
x_{n_{ij}}(t)
&= \int^t_{-\infty}e^{-\int^{t}_{s}a_{ij}(u)du}\\
&\quad\times \Big[- \sum_{C_{kl}\in N_1(i,j)}C^{kl}_{ij}(s)
\int_0^\infty K_{ij}(u)f(x_{{n-1}_{kl}}(s-u))du\cdot x_{{n-1}_{ij}}(s) \\
&\quad  - \sum_{D_{kl}\in N_1(i,j)}D^{kl}_{ij}(s)\int_0^\infty
J_{ij}(u)g(x'_{{n-1}_{kl}}(s-u))du\cdot
x_{{n-1}_{ij}}(s)+L_{ij}(s)\Big]ds,
\end{align*}
for all $ t\in\mathbb{R}$,  $i=1,2,3$, $j=1,2,3,$ and $n=1,2,3,\ldots$.
From the proof of Theorem \ref{dingli2.1}, it follows 
$$
\|x_n-x^*\|_{X}\to 0,\quad n\to\infty,
$$
where $x^*$ is the unique almost periodic solution of 
\eqref{liti4.1} in the region 
$\Omega=\{\varphi\in X:\|\varphi\|_X\leq 1\}$. So one can use this 
method to  compute numerically the almost periodic solution $x^*$.
\end{remark}


\begin{remark}\label{remark4.2}\rm
In the above example, $f$ and $g$ do not satisfy the global
Lipschitz condition. So the results in \cite{fang2} can not be applied 
to this example.
\end{remark}

\subsection*{Acknowledgements}
Qing-Long Liu acknowledges support from the Graduate Innovation Fund of
Jiangxi Province (YC2013-S100).
Hui-Sheng Ding acknowledges support from the NSF of China (11101192),
the Program for Cultivating Young
Scientist of Jiangxi Province (20133BCB23009),
and the NSF of Jiangxi Province.
The authors are grateful to the anonymous reviewers for their
valuable  suggestions, which improve the quality of this article.


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