\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 99, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/99\hfil Almost periodic solutions] {Almost periodic solutions for higher-order Hopfield neural networks without bounded activation functions} \author[F. Zhang, Y. Li \hfil EJDE-2007/99\hfilneg] {Fuxing Zhang, Ya Li} % in alphabetical order \address{Fuxing Zhang \newline Department of Mathematics, Shaoyang University, Shaoyang, Hunan, 422000, China} \email{fuxingzhang2006@163.com} \address{Ya Li \newline Editorial Department of Journal of Hunan University, Changsha 410082, China } \email{yali88888@sohu.com} \thanks{Submitted March 11, 2007. Published July 13, 2007.} \thanks{Supported by grant 10371034 from NNSF of China} \subjclass[2000]{34C25, 34K13} \keywords{High-order Hopfield neural networks; almost periodic solution; \hfill\break\indent exponential stability; time-varying delays} \begin{abstract} In this paper, we consider higher-order Hopfield neural networks (HHNNs) with time-varying delays. Based on the fixed point theorem, Lyapunov functional method, differential inequality techniques, and without assuming the boundedness on the activation functions, we establish sufficient conditions for the existence and local exponential stability of the almost periodic solutions. The results of this paper are new and they complement previously known results. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \allowdisplaybreaks \section{Introduction} Consider the following higher-order Hopfield neural networks (HHNNs), with time-varying delays, \begin{equation} \begin{aligned} x_i'(t)&=-c_ix_i(t)+\sum_{j=1}^na_{ij}(t)g_j(x_j(t-\tau_{ij}(t)))\\ &\quad +\sum_{j=1}^n\sum_{l=1}^nb_{ijl}(t)g_j(x_j(t-\sigma _{ijl}(t)))g_l(x_l(t-\nu _{ijl}(t))) +I_i(t), \end{aligned} \label{e1.1} \end{equation} for $i=1,2,\dots ,n$, where $n$ 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 $t$, $c_i>0$ 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, $a_{ij}(t)$ and $b_{ijl}(t)$ are the first- and second-order connection weights of the neural network, $\tau _{ij}(t)\geq 0$, $\sigma _{ijl}(t)\geq 0$ and $\upsilon _{ijl}(t)\geq 0$ correspond to the transmission delays, $I_i(t)$ denote the external inputs at time $t$, and $g_j$ is the activation function of signal transmission. Due to the fact that high-order neural networks have stronger approximation property, faster convergence rate, greater storage capacity, and higher fault tolerance than lower-order neural networks, high-order neural networks 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 equilibrium points and periodic solutions of HHNNs \eqref{e1.1} in the literature. We refer readers to \cite{c1,d1,w1,x1} and the references cited therein. The assumption \begin{itemize} \item[(T0)] for each $j\in \{1, 2, \dots ,n\}$, $g_j:\mathbb{R}\to\mathbb{R}$ is bounded, i.e., there exists a constant $L_j$ such that \begin{equation} |g_j(u)|\leq L_j,\quad \mbox{for all } u\in \mathbb{R}\label{e1.2} \end{equation} \end{itemize} has been considered as a fundamental condition for the existence and stability of equilibrium points and periodic solutions solutions of HHNNs \eqref{e1.1}. To the best of our knowledge, few authors have considered the problems of periodic and almost periodic solutions of HHNNs \eqref{e1.1} without the assumptions (T0). Thus, it is worth while to investigate the existence and stability of almost periodic solutions of HHNNs \eqref{e1.1} in this case. In this paper we shall study the existence and stability of almost periodic solutions for \eqref{e1.1}. By applying the fixed point theorem, Lyapunov functional method and differential inequality techniques, we derive some new sufficient conditions ensuring the existence and local exponential stability of the almost periodic solution of \eqref{e1.1}. These results are new and they complement previously known results. In particular, an example is also provided to illustrate the effectiveness of the new results. Throughout this paper, for $i, j, l=1, 2, \dots, n$, it will be assumed that $I_{i}$, $a_{ij}$, $ b_{ijl}$, $\tau_{ij}$, $\sigma_{ijl}$, $\nu_{ijl}:\mathbb{R}\to\mathbb{R}$ are almost periodic functions, and there exist constants $\tau$, $\overline{a_{ij}}$, $\overline{b_{ijl}}$ and $\overline{I_{i}}$ such that \begin{equation} \begin{gathered} \tau=\max\big\{\max_{1\leq i,j \leq n}\sup_{t\in \mathbb{R}}\tau_{ij}(t),\max_{1\leq i,j,l \leq n}\sup_{t\in \mathbb{R}}\sigma_{ijl}(t), \max_{1\leq i,j,l \leq n}\sup_{t\in \mathbb{R}}\nu_{ijl}(t)\big\}, \\ \sup_{t\in \mathbb{R}}|b_{ijl}(t)|= \overline{b_{ijl}}, \quad \sup_{t\in \mathbb{R}}|a_{ij}(t)|= \overline{a_{ij}}, \quad \sup_{t\in \mathbb{R}}|I_{i}(t)|=\overline{I_{i}}. \end{gathered}\label{e1.3} \end{equation} We also assume that the following conditions hold: \begin{itemize} \item[(H1)] For each $j\in\{1, 2, \dots, n \}$, there exists a nonnegative constant $L^{g}_{j} $ such that $\ g_{j}(0)=0$, $|g_{j}(u)-g_{j}(v)|\leq L^{g}_{j}|u-v|$, for all $u, \ v\in \mathbb{R}$. \item[(H2)] Assume that there exist nonnegative constants $L , q $ and $\delta$ such that \begin{gather*} L=\max_{1\leq i \leq n}\{\frac{\overline{I_{i}}}{c_{i}}\}, \quad \delta=\max_{ 1\leq i \leq n }\{ c_{i}^{-1} [ \sum^n_{j=1} \overline{a_{ij}}L^{g}_{j} +\sum^n_{j=1}\sum^n_{l=1}\overline{b_{ijl}} L^{g}_{j}L^{g}_{l} ] \} <1,\\ \frac { L}{1-\delta}\leq 1, \quad q=\max_{1\leq i\leq n} \big\{ c^{-1}_{i} \Big( \sum^n_{j=1} \overline{a_{ij}}L_{j}^{g} + \frac{2L}{1-\delta}\sum ^n_{j=1}\sum^n_{l=1}\overline{b_{ijl}} L_{j}^{g}L_{l}^{g}\Big)\big\}<1. \end{gather*} \end{itemize} For convenience, we introduce the following notation. We use $x=(x_1,x_2, \dots ,x_n)^T$ 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 $\Vert x\Vert =\max_{1\leq i\leq n}|x_i|$. A vector $x\geq 0$ means that all entries of $x$ are greater than or equal to zero. $x>0$ is defined similarly. For vectors $x$ and $y$, $x\geq y$ (resp. $x>y$) means that $x-y\geq 0$ (resp. $x-y>0$). For the rest of this paper, we set \begin{gather*} \{x_j(t)\}=(x_1(t),x_2(t),\dots ,x_n(t))^T,\\ B=\{\varphi|\varphi =\{\varphi _j(t)\} =(\varphi _1(t),\varphi _2(t),\dots ,\varphi _n(t))^T\}, \end{gather*} where $\varphi $ is an almost periodic function on $R$. For all $\varphi \in B$, we define the induced module $\Vert \varphi \Vert _B$ by $\Vert \varphi \Vert _B=\sup_{t\in \mathbb{R}}\Vert \varphi (t)\Vert $. Therefore $B$ is a Banach space. The initial conditions associated with system \eqref{e1.1} are of the form \begin{equation} x_{i}(s)=\varphi_{i}(s),s\in [-\tau, \ 0], \ i=1,2,\dots,n, \label{e1.4} \end{equation} where $\varphi=(\varphi_{1}(t), \varphi_{2}(t), \dots, \varphi_{n}(t))^{T}\in C([-\tau, 0]; R^{n} )$. \subsection*{Definition} \cite{f1,h1} Let $u(t):\mathbb{R}\to \mathbb{R}^{n}$ be continuous in $t$. $u(t)$ is said to be almost periodic on $\mathbb{R}$ if, for any $\varepsilon>0$, the set $T(u,\varepsilon)=\{\delta:|u(t+\delta)-u(t)|<\varepsilon$, for all $t\in \mathbb{R}\}$ is relatively dense, i.e., for $\forall\varepsilon>0$, it is possible to find a real number $l=l(\varepsilon)>0$, for any interval with length $l(\varepsilon)$, there exists a number $\delta=\delta(\varepsilon)$ in this interval such that $|u(t+\delta)-u(t)|<\varepsilon$, for for all $t\in \mathbb{R}$. The remaining part of this paper is organized as follows. In Section 2, we shall derive new sufficient conditions for the existence of almost periodic solutions of \eqref{e1.1}. In Section 3, we present some new sufficient conditions for the local exponential stability of the almost periodic solution of \eqref{e1.1}. In Section 4, we shall give some examples and remarks to illustrate our results obtained in the previous sections. \section{Existence of Almost Periodic Solutions} \begin{theorem} \label{thm2.1} Let conditions (H1) and (H2) hold. Then, there exists a unique almost periodic solution to \eqref{e1.1} in the region $B^{*}=\{\varphi |\varphi \in B,\Vert \varphi -\varphi _0\Vert _B\le \frac{\delta L}{1-\delta }\}$, where \begin{align*} \varphi _0(t)&=\big\{\int_{-\infty }^t\exp (-c_j(t-s))I_j(s)ds\big\}\\ &=\Big(\int_{-\infty }^t\exp (-c_1(t-s))I_1(s)ds, \int_{-\infty }^t\exp (-c_2(t-s))I_2(s)ds,\\ &\quad \dots,\int_{-\infty }^t\exp (-c_n(t-s))I_n(s)ds\Big)^T. \end{align*} \end{theorem} \begin{proof} For each $\varphi \in B$, we consider the almost periodic solution $x^\varphi (t)$ to the nonlinear almost periodic differential equations \begin{equation} \begin{aligned} x_i'(t)&=-c_ix_i(t)+\sum_{j=1}^na_{ij}(t)g_j(\varphi _j(t-\tau _{ij}(t)))\\ &\quad +\sum_{j=1}^n\sum_{l=1}^nb_{ijl}(t)g_j(\varphi _j(t-\sigma _{ijl}(t))) g_l(\varphi _l(t-\nu _{ijl}(t)))+I_i(t), \end{aligned}\label{e2.1} \end{equation} for $i=1,2,\dots ,n$. Then $\tau _{ij}(t)$, $b_{ij}(t)$ and $I_i(t)$ are almost periodic functions. According to \cite[pp. 80-112]{f1} and \cite[pp. 90-100]{h1}, we know that the auxiliary system \eqref{e2.1} has exactly one almost periodic solution \begin{align*} x^\varphi (t) &=(x_1^\varphi (t),\ x_2^\varphi (t),\dots ,x_n^\varphi (t))^T \\ &= \Big(\int_{-\infty }^te^{-c_1(t-s)} \Big[\sum_{j=1}^na_{1j}(s)g_j\big(\varphi_j(s-\tau _{1j}(s))\big)\\ &\quad +\sum_{j=1}^n\sum_{l=1}^nb_{1jl}(s)g_j(\varphi_j(s-\sigma _{1jl}(s))) g_l(\varphi _l(s-\nu _{1jl}(s)))+I_1(s)\Big]ds, \\ &\quad\dots ,\int_{-\infty}^te^{-c_n(t-s)} \Big[\sum_{j=1}^na_{nj}(s)g_j(\varphi _j(s-\tau _{nj}(s))) \\ &\quad+\sum_{j=1}^n\sum_{l=1}^nb_{njl}(s)g_j(\varphi _j(s-\sigma _{njl}(s)))g_l(\varphi _l(s-\nu _{njl}(s)))+I_n(s)\Big]ds\Big)^T. \end{align*} % \eqref{e2.2} Now, we define a mapping $T:B\to B$ by setting \[ T(\varphi )(t)=x^\varphi (t),\quad \forall \varphi \in B. \] Since $B^{*}=\{\varphi \in B,\Vert \varphi -\varphi _0\Vert _B\le \frac{\delta L}{1-\delta }\}$, it is easy to see that $B^{*}$ is a closed convex subset of $B$. According to the definition of the norm of Banach space $B$, we get \begin{align*} \Vert \varphi _0\Vert _B &=\sup_{t\in \mathbb{R}}\max_{1\leq i\leq n} \big\{\int_{-\infty}^tI_i(s)\exp (-c_i(t-s))ds\big\}\\ &\le \sup_{t\in \mathbb{R}}\max_{1\leq i\leq n} \{\frac{\overline{I_i}}{c_i}\}\\ &=\max_{1\leq i\leq n}\{\frac{\overline{I_i}}{c_i}\}=L. \end{align*}% \eqref{e2.3} Therefore, for for all $\varphi \in B^{*}$, we have \begin{equation} \| \varphi \| _B\leq \| \varphi -\varphi _0\| _B+\| \varphi _0\| _B\leq \frac{\delta L}{1-\delta }+L=\frac L{1-\delta }\leq 1.\label{e2.4} \end{equation} In view of (H1), we have \begin{equation} |g_j(u)|\leq L_j^g|u|\quad \mbox{for all } u\in \mathbb{R},\; j=1,2,\dots ,n.\label{e2.5} \end{equation} Now, we prove that the mapping $T$ is a self-mapping from $B^{*}$ to $B^{*}$. In fact, for all $\varphi \in B^{*}$, from \eqref{e2.4} and \eqref{e2.5}, we obtain \begin{align*} &\Vert T\varphi -\varphi _0\Vert _B \\ &=\sup_{t\in \mathbb{R}}\max_{1\leq i\leq n} \big\{|\int_{-\infty }^te^{-c_i(t-s)}\Big[\sum_{j=1}^na_{ij}(s)g_j(\varphi _j(s-\tau _{ij}(s)))\\ &\quad +\sum_{j=1}^n\sum_{l=1}^nb_{ijl}(s) g_j(\varphi _j(s-\sigma _{ijl}(s)))g_l(\varphi _l(s-\nu _{ijl}(s)))\Big]ds|\big\} \\ &\le \sup_{t\in \mathbb{R}}\max_{1\leq i\leq n}\{\int_{-\infty }^te^{-c_i(t-s)}\Big[\sum_{j=1}^n\overline{a_{ij}}L_j^g\Vert \varphi \Vert _B+\sum_{j=1}^n\sum_{l=1}^n\overline{}b_{ijl}L_j^gL_l^g\Vert \varphi \Vert _B^2\Big]ds\} \\ &\le \sup_{t\in \mathbb{R}}\max_{1\leq i\leq n}\{\int_{-\infty }^te^{-c_i(t-s)}\Big[\sum_{j=1}^n\overline{a_{ij}}L_j^g\frac L{1-\delta }+\sum_{j=1}^n\sum_{l=1}^n\overline{}b_{ijl}L_j^gL_l^g(\frac L{1-\delta })^2\Big]ds\} \\ &\le \sup_{t\in \mathbb{R}}\max_{1\leq i\leq n}\{\int_{-\infty }^te^{-c_i(t-s)}\Big[\sum_{j=1}^n\overline{a_{ij}}L_j^g +\sum_{j=1}^n\sum_{l=1}^n\overline{b_{ijl}}L_j^gL_l^g\Big]ds\frac L{1-\delta }\} \\ &\le \max_{1\leq i\leq n}\{c_i^{-1}\Big[\sum_{j=1}^n\overline{a_{ij}} L_j^g+\sum_{j=1}^n\sum_{l=1}^n\overline{b_{ijl}}L_j^gL_l^g\Big]\}\frac L{1-\delta } \\ &=\frac{\delta L}{1-\delta }, \end{align*} where $\delta =\max_{1\leq i\leq n}\{c_i^{-1}[\sum_{j=1}^n\overline{a_{ij}}L_j^g +\sum_{j=1}^n\sum_{l=1}^n\overline{b_{ijl}}L_j^gL_l^g]\}$. This implies that $T(\varphi )(t)\in B^{*}$. Next, we prove that the mapping $T$ is a contraction mapping on $B^{*}$. In view of \eqref{e2.4} and (H1), for all $\phi ,\psi \in B^{*}$, we have \begin{align*} & |T(\phi (t))-T(\psi (t))| \\ &=\Big(|(T(\phi (t))-T(\psi (t)))_1|, \dots , |(T(\phi (t))-T(\psi (t)))_n|\Big)^T \\ &=\Big(|\int_{-\infty }^te^{-c_1(t-s)}\Big[\sum_{j=1}^na_{1j}(s)(g_j(\phi _j(s-\tau _{1j}(s)))-g_j(\psi _j(s-\tau _{1j}(s))))\\ &\quad +\sum_{j=1}^n\sum_{l=1}^nb_{1jl}(s) \big(g_j(\phi _j(s-\sigma _{1jl}(s)))g_l(\phi _l(s-\nu_{1jl}(s)))\\ &\quad -g_j(\psi _j(s-\sigma _{1jl}(s)))g_l(\psi _l(s-\nu _{1jl}(s)))\big)\Big]ds|, \dots, \\ &\quad |\int_{-\infty}^te^{-c_n(t-s)}\Big[\sum_{j=1}^na_{nj}(s)(g_j(\phi _j(s-\tau _{nj}(s)))-g_j(\psi _j(s-\tau _{nj}(s))))\\ &\quad +\sum_{j=1}^n\sum_{l=1}^nb_{njl}(s) \big(g_j(\phi _j(s-\sigma _{njl}(s)))g_l(\phi _l(s-\nu _{njl}(s)))\\ &\quad -g_j(\psi _j(s-\sigma _{njl}(s)))g_l(\psi _l(s-\nu _{njl}(s)))\big)\Big]ds|\Big)^T \\ &\leq \Big(\int_{-\infty }^te^{-c_1(t-s)}\Big[\sum_{j=1}^n\overline{a_{1j}} L_j^g\sup_{t\in \mathbb{R}}|\phi _j(t)-\psi _j(t)|\\ &\quad +\sum_{j=1}^n\sum_{l=1}^n\overline{ b_{1jl}}(|g_j(\phi _j(s-\sigma _{1jl}(s))) g_l(\phi _l(s-\nu _{1jl}(s)))\\ &\quad -g_j(\psi _j(s-\sigma_{1jl}(s)))g_l(\phi _l(s-\nu _{1jl}(s)))| \\ &\quad +|g_j(\psi _j(s-\sigma _{1jl}(s)))g_l(\phi _l(s-\nu _{1jl}(s))) \\ &\quad -g_j(\psi _j(s-\sigma _{1jl}(s)))g_l(\psi _l(s-\nu _{1jl}(s)))|)\Big]ds, \\ &\quad \dots , \int_{-\infty }^te^{-c_n(t-s)}\Big[\sum_{j=1}^n\overline{a_{nj}} L_j^g\sup_{t\in \mathbb{R}}|\phi _j(t)-\psi _j(t)|\\ &\quad +\sum_{j=1}^n\sum_{l=1}^n\overline{b_{njl}} (|g_j(\phi _j(s-\sigma _{njl}(s))) g_l(\phi _l(s-\nu _{njl}(s))) \\ &\quad -g_j(\psi _j(s-\sigma_{njl}(s)))g_l(\phi _l(s-\nu _{njl}(s)))|\\ &\quad +|g_j(\psi _j(s-\sigma _{njl}(s)))g_l(\phi _l(s-\nu _{njl}(s))) \\ &\quad -g_j(\psi_j(s-\sigma _{njl}(s)))g_l(\psi _l(s-\nu _{njl}(s)))|)\Big] ds\Big)^T \\ &\leq \Big(\int_{-\infty }^te^{-c_1(t-s)}[\sum_{j=1}^n\overline{a_{1j}} L_j^g\sup_{t\in \mathbb{R}}|\phi _j(t)-\psi _j(t)|\\ &\quad +\sum_{j=1}^n\sum_{l=1}^n\overline{b_{1jl}}L_j^gL_l^g (\sup_{t\in \mathbb{R}}|\phi _l(t)|+\sup_{t\in \mathbb{R}}|\psi _j(t)|) \Vert \phi -\psi \Vert _B]ds, \dots ,\\ &\quad \int_{-\infty}^te^{-c_n(t-s)}[\sum_{j=1}^n\overline{a_{nj}}L_j^g \sup_{t\in \mathbb{R}}|\phi_j(t)-\psi _j(t)| \\ &\quad+\sum_{j=1}^n\sum_{l=1}^n\overline{b_{njl}}L_j^gL_l^g (\sup_{t\in \mathbb{R}}|\phi_l(t)|+\sup_{t\in \mathbb{R}}|\psi _j(t)|) \Vert \phi -\psi \Vert _B]ds\Big)^T \\ &\leq \Big(c_1^{-1}(\sum_{j=1}^n\overline{a_{1j}}L_j^g+\frac{2L}{1-\delta } \sum_{j=1}^n\sum_{l=1}^n\overline{b_{1jl}}L_j^gL_l^g)\Vert \phi -\psi \Vert _B, \\ &\quad \dots , c_n^{-1}(\sum_{j=1}^n\overline{a_{nj}}L_j^g +\frac{2L}{1-\delta }\sum_{j=1}^n\sum_{l=1}^n\overline{b_{njl}} L_j^gL_l^g)\Vert \phi -\psi \Vert _B\Big)^T, \end{align*} which implies \begin{align*} \| T(\phi )-T(\psi )\| _B &\leq \max_{1\leq i\leq n}\{c_i^{-1}(\sum_{j=1}^n\overline{a_{ij}}L_j^g+\frac{2L}{1-\delta } \sum_{j=1}^n\sum_{l=1}^n\overline{b_{ijl}}L_j^gL_l^g)\}\| \phi -\psi \| _B \\ &=q\| \phi -\psi \| _B. \end{align*} Note that $q=\max_{1\leq i\leq n}\{c_i^{-1} (\sum_{j=1}^n\overline{a_{ij}}L_j^g+\frac{2L}{1-\delta } \sum_{j=1}^n\sum_{l=1}^n\overline{b_{ijl}}L_j^gL_l^g)\}<1$; it is clear that the mapping $T$ is a contraction. Therefore the mapping $T$ possesses a unique fixed point $Z^{*}\in B^{*}$, $TZ^{*}=Z^{*}$. By \eqref{e2.1}, $Z^{*}$ satisfies \eqref{e1.1}. So $Z^{*}$ is an almost periodic solution of \eqref{e1.1} in $B^{*}$. The proof is complete. \end{proof} \section{Stability of the almost periodic solution} In this section, we establish some results for the stability of the almost periodic solution of \eqref{e1.1}. \begin{theorem} \label{thm3.1} Let \[ \max_{1\leq i\leq n}\{c_i^{-1}[\sum_{j=1}^n\overline{a_{ij}}L_j^g +\sum_{j=1}^n\sum_{l=1}^n\overline{b_{ijl}}L_j^gL_l^g(1+2\frac L{1-\delta })]\} <1. \] Suppose that all the conditions of Theorem \ref{thm2.1} are satisfied. Then \eqref{e1.1} has exactly one almost periodic solution $Z^{*}(t)=(x_1^{*}(t),x_2^{*}(t),\dots ,x_n^{*}(t))^T\in B^{*}$. Moreover, $Z^{*}(t)$ is locally exponentially stable, the domain of the attraction of $Z^{*}(t)$ is the set \[ G_1(Z^{*})=\{\varphi |\varphi =(\varphi _1(t),\varphi _2(t),\dots ,\varphi _n(t))^T\in C([-\tau ,\ 0];\ R^n),\ \Vert \varphi -\varphi ^{*}\Vert _1<1\}, \] where $\varphi ^{*}=\{\varphi _j^{*}(t)\}$, $\varphi_j^{*}(t)=x_j^{*}(t)$, $j=1, 2, \dots , n$, $t\in [-\tau , 0]$, and $\Vert \varphi -\varphi ^{*}\Vert _1=\sup_{-\tau \leq s\leq 0}\max_{1\leq j\leq n}|\varphi _j(s)-\varphi _j^{*}(s)|$. Namely, there exist constants $\lambda >0$ and $M>1$ such that for every solution $Z(t)=\{x_j(t)\}$ to system \eqref{e1.1} with initial value $\varphi =\{\varphi_j(t)\}\in G_1(Z^{*})$, we have \[ |x_i(t)-x_i^{*}(t)|\leq M\Vert \varphi -\varphi ^{*}\Vert _1e^{-\lambda t},\quad \forall t>0,\; i=1, 2, \dots , n. \] \end{theorem} \begin{proof} From Theorem \ref{thm2.1}, system \eqref{e1.1} has exactly one almost periodic solution $Z^{*}(t)=\{x^{*}_{ j}(t) \} \in B^{*}$. Let $Z(t)=\{x_{j}(t) \} $ be an arbitrary solution of system \eqref{e1.1} with initial value $\varphi=\{\varphi_{ j}(t) \}\in G_{1}(Z^{*})$, let $y(t)=\{y_{ j}(t)\}=\{x_{ j}(t)-x^{*}_{ j}(t) \}=Z(t)-Z^{*}(t)$. Then \begin{equation} \begin{aligned} y_{i}'(t)&=-c_{i}y_{i}(t)+\sum^n_{j=1}a_{ij}(t)(g_{j}(x_{j} (t-\tau_{ij}(t)))-g_{j}(x^{*}_{j}(t-\tau_{ij}(t)))) \\ &\quad +\sum^n_{j=1}\sum^n_{l=1}b_{ijl}(t)(g_{j}(x_{j}(t-\sigma_{ijl}(t))) g_{l}(x_{l}(t-\nu_{ijl}(t))) \\ &\quad -g_{j}(x^{*}_{j}(t-\sigma_{ijl}(t)))g_{l}(x^{*}_{l}(t-\nu_{ijl}(t)))), \quad i=1, 2, \dots, n. \end{aligned} \label{e3.1} \end{equation} Since $\max_{1\leq i \leq n}\{ c^{-1}_{i} [\sum^n_{j=1} \overline{a_{ij}}L^{g}_{j} +\sum^n_{j=1}\sum^n_{l=1} \overline{b_{ijl}} L^{g}_{j}L^{g}_{l} ( 1 + 2 \frac{L}{1-\delta}) ]\} <1$, we can easily get \begin{equation} -c_{i}+ \sum^n_{j=1} \overline{a_{ij}}L^{g}_{j} +\sum^n_{j=1}\sum^n_{l=1} \overline{b_{ijl}} L^{g}_{j}L^{g}_{l} ( 1 + 2 \frac{L}{1-\delta})<0 , \quad i=1, 2, \dots,n, \label{e3.2} \end{equation} which implies that we can choose a positive constant $\lambda $ such that \begin{equation} ( \lambda - c_{i } )+\sum^n_{j=1} \overline{a_{ij}}L^{g}_{j} e^{\lambda \tau} +\sum^n_{j=1}\sum^n_{l=1} \overline{b_{ijl}} L^{g}_{j}L^{g}_{l} ( e^{2\lambda \tau} + 2e^{\lambda \tau}\frac{L}{1-\delta})<0, \label{e3.3} \end{equation} for $i=1, 2, \dots, n$. We consider the Lyapunov functional \begin{equation} V_i(t)=|y_i(t)|e^{\lambda t},\quad i=1,2,\dots ,n.\label{e3.4} \end{equation} Calculating the upper right derivative of $V_i(t)$ along the solution $y(t)=\{y_j(t)\}$ of system \eqref{e3.1} with the initial value $\bar{\varphi}=\varphi -\varphi ^{*}$, we have from \eqref{e2.4}, \eqref{e2.5}, \eqref{e3.1} and (H1) that \begin{align} &D^{+}(V_i(t)) \nonumber\\ &\leq -c_i|y_i(t)|e^{\lambda t}+\Big[\sum_{j=1}^n|a_{ij}(t)||g_j(x_j(t-\tau _{ij}(t)))-g_j(x_j^{*}(t-\tau _{ij}(t)))| \nonumber \\ &\quad+\sum_{j=1}^n\sum_{l=1}^n|b_{ijl}(t)||g_j(x_j(t-\sigma _{ijl}(t)))g_l(x_l(t-\nu _{ijl}(t)))\nonumber \\ &\quad -g_j(x_j^{*}(t-\sigma _{ijl}(t))) g_l(x_l^{*}(t-\nu _{ijl}(t)))|\Big] e^{\lambda t}+\lambda |y_i(t)|e^{\lambda t} \nonumber \\ &\leq (\lambda -c_i)|y_i(t)|e^{\lambda t} +\sum_{j=1}^n|a_{ij}(t)|L_j^g|y_j(t-\tau _{ij}(t))| \nonumber \\ &\quad +\Big[\sum_{j=1}^n\sum_{l=1}^n|b_{ijl}(t)|(|g_j(x_j(t-\sigma _{ijl}(t))) g_l(x_l(t-\nu _{ijl}(t))) \nonumber\\ &\quad -g_j(x_j^{*}(t-\sigma_{ijl}(t)))g_l(x_l(t-\nu _{ijl}(t)))| +|g_j(x_j^{*}(t-\sigma _{ijl}(t))) g_l(x_l(t-\nu _{ijl}(t))) \nonumber\\ &\quad -g_j(x_j^{*}(t-\sigma_{ijl}(t)))g_l(x_l^{*}(t-\nu _{ijl}(t)))|)\Big] e^{\lambda t} \nonumber \\ &\leq (\lambda -c_i)|y_i(t)|e^{\lambda t} +[\sum_{j=1}^n\overline{a_{ij}}L_j^g|y_j(t-\tau _{ij}(t))| \nonumber\\ &\quad +\sum_{j=1}^n\sum_{l=1}^n\overline{b_{ijl}} L_j^gL_l^g(|y_j(t-\sigma _{ijl}(t))| |y_l(t-\nu _{ijl}(t)) \nonumber \\ &\quad +x_l^{*}(t-\nu _{ijl}(t))|+|x_j^{*}(t-\sigma _{ijl}(t))||y_l(t-\nu _{ijl}(t))|)]e^{\lambda t} \nonumber \\ &\leq (\lambda -c_i)|y_i(t)|e^{\lambda t}+\Big[\sum_{j=1}^n\overline{a_{ij}} L_j^g|y_j(t-\tau _{ij}(t))| \nonumber\\ &\quad +\sum_{j=1}^n\sum_{l=1}^n\overline{b_{ijl}} L_j^gL_l^g(|y_j(t-\sigma _{ijl}(t))| |y_l(t-\nu _{ijl}(t))| \nonumber \\ &\quad +|y_j(t-\sigma _{ijl}(t))|\frac L{1-\delta} +\frac L{1-\delta }|y_l(t-\nu _{ijl}(t))|)\Big]e^{\lambda t}, \label{e3.5} \end{align} where $i=1, 2,\dots ,n$. Set \[ \|\varphi-\varphi^{*}\|_{1} =\sup_{-\tau\leq s\leq0}\max_{1\leq j\leq n }|\varphi_{ j}(s)-\varphi_{ j}^{*}(s)|>0. \] Since $\|\varphi-\varphi^{*}\|_{1}<1$, we can choose a positive constant $M>1 $ such that \begin{equation} M\|\varphi-\varphi^{*}\|_{1} <1, \quad (M\|\varphi-\varphi^{*}\|_{1})^{2}0,\; i=1,2,\dots ,n. \label{e3.7} \end{equation} Contrarily, there must exist an $i\in \{1,2,\dots ,n\}$ and $t_i>0$ such that $$ V_i(t_i)=M\Vert \varphi -\varphi ^{*}\Vert _1\quad \mbox{and}\quad V_j(t)0,\; i=1,2,\dots , n. %{e3.11} $$ 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 following HHNNs with delays: \begin{equation} \begin{aligned} x_1'(t) & = -x_1(t)+\frac 1{16}(\sin t)g_1(x_1(t-\sin ^2t))+\frac 1{16}(\cos 3t)g_2(x_2(t-7\sin ^2t)) \\ &\quad +\frac 18(\cos t)g_1(x_1(t-5\sin ^2t))g_2(x_2(t-2\sin ^2t))+\frac 34\sin (\sqrt{2}t), \\ x_2'(t) & = -x_2(t)+\frac 1{16}(\sin 2t)g_1(x_1(t-\cos ^2t))+\frac 1{16}(\cos 8t)g_2(x_2(t-5\sin ^2t)) \\ & \quad +\frac 18(\cos 4t)g_1(x_1(t-\sin ^2t))g_2(x_2(t-4\sin ^2t))+\frac 34\cos (\sqrt{2}t), \end{aligned} \label{e4.1} \end{equation} where $g_1(x)=g_2(x)=|x|$. Observe that $c_1=c_2=L_1^g=L_2^g=1$, $\overline{a_{ij}}=\frac 1{16}$, $i,j=1,2,\overline{b_{112}}=\overline{b_{212}}=\frac 18$, $\overline{b_{ijl}}=0$, $i,j,l=1,2$, $ijl\neq 112$, $ijl\neq 212$. Then \begin{gather*} L=\frac 34,\quad \delta =\max_{1\leq i\leq 2}\{c_i^{-1}[\sum_{j=1}^2\overline{a_{ij}}L_j^g +\sum_{j=1}^2\sum_{l=1}^2\overline{b_{ijl}}L_j^gL_l^g]\}=\frac 14<1, \\ q=\max_{1\leq i\leq 2}\{c_i^{-1}(\sum_{j=1}^2\overline{ a_{ij}}L_j^g+\frac{2L}{1-\delta }\sum_{j=1}^2\sum_{l=1}^2 \overline{b_{ijl}}L_j^gL_l^g)\}=\frac 38<1, \\ \max_{1\leq i\leq 2}\{c_i^{-1}[\sum_{j=1}^2\overline{a_{ij}} L_j^g+\sum_{j=1}^2\sum_{l=1}^2\overline{b_{ijl}} L_j^gL_l^g(1+2\frac L{1-\delta })]\}=\frac 12<1. \end{gather*} Therefore, By Theorem \ref{thm3.1}, system \eqref{e4.1} has a unique almost periodic solution $Z^{*}(t)$ in the region $\| \varphi -\varphi _0\| _B\leq 0.25$. Moreover, $Z^{*}(t)$ is locally exponentially stable, the domain of the attraction of $Z^{*}(t)$ is the set $G_1(Z^{*})$. We remark that \eqref{e4.1} is a very simple form of HHNNs. Since $g_{1}(x)=g_{2}(x) = |x | $, one can observe that the condition (T0) is not satisfied. Therefore, all the results in \cite{c1,d1,j1,l1,w1,x1,y1} and the references cited therein can not be applicable to system \eqref{e4.1}. This implies that the results of this paper are essentially new. \begin{thebibliography}{00} \bibitem{c1} Jinde Cao, Jinling Liang and James Lam; \emph{Exponential stability of high-order bidirectional associative memory neural networks with time delays}, Physica D: Nonlinear Phenomena, 199 (3-4) (2004) 425-436. \bibitem{d1} A. Dembo, O. Farotimi, T. Kailath; \emph{High-order Absolutely Stable Neural Network,IEEE Trans on Circuits and System}, 8(1) (1991), 57-65 81, (1984) 3088-3092. \bibitem{f1} A. M. Fink; \emph{Almost periodic differential equations}, Lecture Notes in Mathematics, Vol. 377, Springer, Berlin, 1974, pp. 80-112. \bibitem{h1} C. Y. He; \emph{Almost periodic differential equation}, Higher Education Publishing House, Beijing, 1992 pp. 90-100. [In Chinese] \bibitem{j1} Haijun Jiang and Zhidong Teng; \emph{Boundedness and global stability for nonautonomous recurrent neural networks with distributed delays}, Chaos, Solitons \& Fractals, 30(1) (2006) 83-93. \bibitem{l1} Chuandong Li, Xiaofeng Liao, Rong Zhang; \emph{Delay-dependent exponential stability analysis of bi-directional associative memory neural networks with time delay: an LMI approach}, Chaos, Solitons \& Fractals, 24(4) (2005) 1119-1134. \bibitem{w1} P. G. Wang, H. R. Lian; \emph{Global exponential stability and periodic solutions of the high-order Hopfield type neural networks with time-varying coefficient}. Zeischrift fur Analysis und ihre Anwendungen, 24(2) (2005) 419-429. \bibitem{x1} Bingji Xu, Xinzhi Liu, Xiaoxin Liao; \emph{Global asymptotic stability of high-order Hopfield type neural networks with time delays}. Computers \& Mathematics with Applications, 45 ( 2003) 1729-1737 \bibitem{y1} Haifeng Yang, Tianguang Chu and Cishen Zhang; \emph{Exponential stability of neural networks with variable delays via LMI approach}, Chaos, Solitons \& Fractals, 30(1) (2006) 133-139. \end{thebibliography} \end{document}