\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 50, pp. 1--16.\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-2007/50\hfil Impulsive fuzzy cellular neural networks] {Behavior of impulsive fuzzy cellular neural networks with distributed delays} \author[Kelin Li, Zuoan Li, Qiankun Song\hfil EJDE-2007/50\hfilneg] {Kelin Li, Zuoan Li, Qiankun Song} % in alphabetical order \address{Kelin Li \newline Department of Mathematics, Sichuan University of Science \& Engineering, Sichuan 643000, China} \email{lkl@suse.edu.cn} \address{Zuoan Li \newline Department of Mathematics, Sichuan University of Science \& Engineering, Sichuan 643000, China} \email{lizuoan@suse.edu.cn} \address{Qiankun Song \newline Department of Mathematics, Chongqing Jiaotong University, Chongqing, 400074, China} \email{qiankunsong@163.com} \thanks{Submitted October 2, 2006. Published April 5, 2007.} \subjclass[2000]{92B20, 34K20, 34K13} \keywords{Fuzzy cellular neural networks; impulses; distributed delays; \hfill\break\indent global exponential stability; periodic oscillatory solution} \begin{abstract} In this paper, we investigate a generalized model of fuzzy cellular neural networks with distributed delays and impulses. By employing the theory of topological degree, $M$-matrix and Lypunov functional, we find sufficient conditions for the existence, uniqueness and global exponential stability of both the equilibrium point and the periodic solution. Two examples are given to illustrate the results obtained here. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} Since cellular neural networks (CNN) was introduced by Chua and Yang in \cite{c3,c4}, many researchers have done extensive works on this subject due to their applications in classification of patterns, associative memories, image processing, quadratic optimization, and other areas, e.g., \cite{a2,c1,c2,r1,s1,z1}. However, in mathematical modelling of real world problems, we encounter inconveniences, namely, the complexity and the uncertainty or vagueness. In order to take vagueness into consideration, fuzzy theory is considered as a suitable setting. Based on traditional CNN, Yang and Yang proposed the fuzzy cellular neural networks (FCNN) \cite{y2,y3}, which integrates fuzzy logic into the structure of the traditional CNN and maintains local connectedness among cells. Unlike previous CNN structures, FCNN has fuzzy logic between its template input and/or output besides the sum of product operation. FCNN is very useful paradigm for image processing problems, which is a cornerstone in image processing and pattern recognition. In such applications, it is of prime importance to ensure that the designed FCNN be stable. In \cite{y2,y3}, the authors have obtained some conditions for the existence and the global stability of the equilibrium point of FCNN without delays. In \cite{l2}, Liu and Tang have considered FCNN with either constant delays or time-varying delays, several sufficient conditions have been obtained to ensure the existence and uniqueness of the equilibrium point and its global exponential stability. Yuan, Cao and Deng have given several novel criteria of exponential stability and periodic solutions for FCNN with time-varying delays \cite{y4}. Recently, Huang has considered the stability of FCNN with diffusion terms and time-varying delay \cite{h3}, at the same time, Huang has investigated the exponential stability of FCNN with distributed delay \cite{h2}. However, besides delay effect, impulsive effect likewise exists in a wide variety of evolutionary processes in which states are changed abruptly at certain moments of time, involving such fields as medicine and biology, economics, mechanics, electronics and telecommunications, etc. Many interesting results on impulsive effect have been gained, e.g., Refs. \cite{a1,b1,g1,g2,l1,l3,x1,y1}. As artificial electronic systems, neural networks such as CNN, bidirectional neural networks and recurrent neural networks often are subject to impulsive perturbations which can affect dynamical behaviors of the systems just as time delays. Therefore, it is necessary to consider both impulsive effect and delay effect on the stability of neural networks. Motivated by the above discussions, in this paper, on the basis of the structure of FCNN, we consider a class of impulsive fuzzy neural networks with distributed delays described by the following system of integro-differential equations: \begin{equation} \begin{gathered} \begin{aligned} \frac{dx_i(t)}{dt} &=- d_ix_i(t)+\sum_{j=1}^{n}a_{ij}f_j(x_{j}(t)) +\sum_{j=1}^{n}\tilde{a}_{ij}u_{j}(t)+I_{i}(t)\\ &\quad +\bigwedge_{j=1}^{n}b_{ij}\int_{-\infty}^tK_{ij}(t-s)f_{j}(x_{j}(s))ds +\bigvee_{j=1}^{n}\tilde{b}_{ij} \int_{-\infty}^tK_{ij}(t-s)f_{j}(x_{j}(s))ds \\ &\quad +\bigwedge_{j=1}^{n}T_{ij}u_{j}(t)+\bigvee_{j=1}^{n}H_{ij}u_{j}(t), \quad t\neq t_{k}, \end{aligned} \\ \Delta x_{i}(t_k)=x_i(t_k^+)-x_i(t_k^-)=\Delta_{k}(x_i(t_k)),\quad t=t_k, \end{gathered} \label{e1} \end{equation} for $i=1,2,\dots,n$. Where the fixed times $t_{k}$ satisfy $t_10$ such that $$ |f_i(x)-f_i(y)|\leq F_i|x-y|,\quad i=1,2,\dots,n $$ for any $x,y\in R$. \item[(H2)] The delay kernels $K_{ij}:[0,+\infty)\to[0,+\infty)$ are piecewise continuous functions and satisfies: \begin{itemize} \item[(i)] $\int_0^\infty K_{ij}(s)ds=1,\quad i,j=1,2,\dots,n$. \item[(ii)] $\int_0^\infty sK_{ij}(s)ds<\infty,\quad i,j=1,2,\dots,n$. \item[(iii)] There exists a positive number $\mu$ such that $$ \int_0^\infty se^{\mu s}K_{ij}(s)ds<\infty,\quad i,j=1,2,\dots,n. $$ \end{itemize} \end{itemize} Let $C=C((-\infty,0],\mathbb{R}^n)$ be the linear space of bounded and continuous functions which map $(-\infty,0]$ into $\mathbb{R}^n$. The initial conditions associated with model \eqref{e1} are of the form \begin{equation} \label{e2} x_i(t)=\varphi_i(t),\quad -\inftyB$ represents $a_{ij}>b_{ij}$ for all $i,j=1,2,\dots,n$. Let a vector norm $\|x\|_p$ ($p=1,\infty$) (simply denoted by $\|x\|$) for $x\in \mathbb{R}^n$ be defined as $$ \|x\|_1=\sum_{i=1}^n|x_i|,\quad \|x\|_{\infty}=\max_{1\leq i\leq n}|x_i|. $$ For $\varphi\in C$, $\|\varphi\|_{\infty}$ is defined as $$ \|\varphi\|_{\infty}=\sup_{-\infty0$ or $D^Td >0$. \end{itemize} \end{lemma} \begin{lemma} [\cite{y2}] \label{lem2} Suppose $y$ and $\bar{y}$ are two state of model \eqref{e1}, then we have \begin{gather*} \Big|\bigwedge_{j=1}^{n}\alpha_{ij}f_{j}(y_{j})-\bigwedge_{j=1}^{n}\alpha_{ij}f_{j}(\bar{y}_{j}) \Big| \leq \sum_{j=1}^{n}\Big|\alpha_{ij}\Big|\cdot \Big| f_{j}(y_{j})-f_{j}(\bar{y}_{j}) \Big|, \\ \Big| \bigvee_{j=1}^{n}\beta_{ij}f_{j}(y_{j})-\bigvee_{j=1}^{n}\beta_{ij}f_{j}(\bar{y}_{j}) \Big|\leq\sum_{j=1}^{n}\Big|\beta_{ij}\Big|\cdot\Big| f_{j}(y_{j})-f_{j}(\bar{y}_{j}) \Big|. \end{gather*} \end{lemma} \section{Global exponential stability of equilibrium point} In this section, we will give several sufficient conditions on the global exponential stability of equilibrium point for the impulsive FCNN with distributed delays. Consider the case of model \eqref{e1} as $I_i(t)=I_i$, $u_i(t)=u_i$, $i=1,2,\dots,n$, and let $\tilde{I}_i=\sum_{j=1}^{n}\tilde{a}_{ij} u_{j}+I_{i}+\bigwedge_{j=1}^{n}T_{ij}u_{j}+\bigvee_{j=1}^{n}H_{ij}u_{j}$, then model \eqref{e1} becomes \begin{equation} \label{e3} \begin{gathered} \begin{aligned} \frac{dx_i(t)}{dt}&=- d_ix_i(t)+\sum_{j=1}^{n}a_{ij}f_j(x_{j}(t)) +\bigwedge_{j=1}^{n}b_{ij}\int_{-\infty}^tK_{ij}(t-s)f_{j}(x_{j}(s))ds \\ &\quad+\bigvee_{j=1}^{n}\tilde{b}_{ij}\int_{-\infty}^tK_{ij}(t-s) f_{j}(x_{j}(s))ds +\tilde{I}_i, \quad t\neq t_{k}, \end{aligned} \\ \Delta x_{i}(t_k)=x_i(t_k^+)-x_i(t_k^-)=\Delta_{k}(x_i(t_k)),\quad t=t_k, \end{gathered} \end{equation} for $i=1,2,\dots,n$. \begin{theorem} \label{thm1} Under assumptions {\rm (H1), (H2)}, the first equation in system \eqref{e3} has a unique equilibrium point if $D-(|A|+|B|+|\tilde{B}|)F$ is a nonsingular $M$-matrix, where $D=\mathop{\rm diag} (d_1,d_2,\dots,d_n)$, $A=(a_{ij})_{n\times n}$, $B=(b_{ij})_{n\times n}$, $\tilde{B}=(\tilde{b}_{ij})_{n\times n}$, $F=\mathop{\rm diag} (F_1,F_2,\dots,F_n)$. \end{theorem} \begin{proof} Let $x^*=(x_1^*,x_2^*,\dots,x_n^*)^T$ denote an equilibrium point of the first equation in model \eqref{e3}. Then $x^*$ satisfies \begin{equation} d_ix_i^*-\sum_{j=1}^na_{ij}f_j(x_j^*)-\bigwedge_{j=1}^nb_{ij}f_j(x_j^*) -\bigvee_{j=1}^n\tilde{b}_{ij}f_j(x_j^*)-\tilde{I}_i=0,\quad i=1,2,\dots,n. \label{e4} \end{equation} Let \[ h_i(x_i)=d_ix_i-\sum_{j=1}^na_{ij}f_j(x_j)-\bigwedge_{j=1}^nb_{ij}f_j(x_j) -\bigvee_{j=1}^n\tilde{b}_{ij}f_j(x_j)-\tilde{I}_i=0,\quad i=1,2,\dots,n. \] Obviously, the solutions of the above system are the equilibrium point of model \eqref{e3}. Let us define homotopic mapping \[ H(x,\lambda)=\lambda h(x)+(1-\lambda)x, \] where $\lambda\in[0,1]$, and \begin{gather*} h(x)=(h_1(x_1),h_2(x_2),\dots,h_n(x_n))^T, \\ H(x,\lambda)=(H_1(x_1,\lambda),H_2(x_2,\lambda),\dots,H_n(x_n,\lambda))^T, \end{gather*} then for $i\in\{1,2,\dots,n\}$, from (H1) and Lemma \ref{lem2}, we have \begin{align*} &|H_i(x,\lambda)|\\ &=\Big|\lambda\Big[d_ix_i-\sum_{j=1}^na_{ij}f_j(x_j)-\bigwedge_{j=1}^nb_{ij}f_j(x_j) -\bigvee_{j=1}^n\tilde{b}_{ij}f_j(x_j)-\tilde{I}_i\Big]+(1-\lambda)x_i\Big| \\ &\geq |\lambda d_ix_i+(1-\lambda)x_i|-\lambda\sum_{j=1}^n|a_{ij}||f_j(x_j)| -\lambda\sum_{j=1}^n|b_{ij}||f_j(x_j)|\\ &\quad -\lambda\sum_{j=1}^n|\tilde{b}_{ij}||f_j(x_j)|-\lambda|\tilde{I}_i| \\ &\geq [1+\lambda(d_i-1)]|x_i|-\lambda\sum_{j=1}^n|a_{ij}|F_j|x_j| -\lambda\sum_{j=1}^n|b_{ij}|F_j|x_j| -\lambda\sum_{j=1}^n|\tilde{b}_{ij} |F_j|x_j| \\ &\quad-\lambda\Big[|\tilde{I}_i|+\sum_{j=1}^n|a_{ij}||f_j(0)|+\sum_{j=1}^n|b_{ij}|f_j(0)|+\sum_{j=1}^n|\tilde{b}_{ij}|f_j(0)|\Big]. \\ &=[1+\lambda(d_i-1)]|x_i|-\lambda\sum_{j=1}^nF_j|x_j|\Big(|a_{ij}|+|b_{ij}|+|\tilde{b}_{ij}|\Big) \\ &\quad -\lambda\Big[|\tilde{I}_i|+\sum_{j=1}^n|f_j(0)|\Big(|a_{ij}| +|b_{ij}|+|\tilde{b}_{ij}|\Big)\Big]. \end{align*} Since $D-(|A|+|B|+|\tilde{B}|)F$ is a nonsingular $M$-matrix, there exist constants $l_i>0$ such that \[ l_id_i-F_i\sum_{j=1}^nl_j\Big(|a_{ji}|+|b_{ji}|+|\tilde{b}_{ji}|\Big)>0,\quad i=1,2,\dots,n, \] then, we have \begin{align*} &\sum_{i=1}^nl_i|H_i(x,\lambda)|\\ &\geq \sum_{i=1}^nl_i(1-\lambda)|x_i| +\lambda\sum_{i=1}^n\Big[d_il_i|x_i|-l_i\sum_{j=1}^nF_j|x_j|\Big(|a_{ij}|+|b_{ij}|+|\tilde{b}_{ij}|\Big)\Big] \\ &\quad -\lambda\sum_{i=1}^nl_i\Big[|\tilde{I}_i|+\sum_{j=1}^n|f_j(0)|\Big(|a_{ij}|+|b_{ij}|+|\tilde{b}_{ij}|\Big)\Big] \\ &\geq \lambda\sum_{i=1}^n\Big[d_il_i|x_i|-l_i\sum_{j=1}^nF_j|x_j|\Big(|a_{ij}|+|b_{ij}|+|\tilde{b}_{ij}|\Big)\Big] \\ &\quad -\lambda\sum_{i=1}^nl_i\Big[|\tilde{I}_i|+\sum_{j=1}^n|f_j(0)|\Big(|a_{ij}|+|b_{ij}|+|\tilde{b}_{ij}|\Big)\Big] \\ &=\lambda\sum_{i=1}^n\Big[l_id_i-F_i\sum_{j=1}^nl_j\Big(|a_{ji}|+|b_{ji}|+|\tilde{b}_{ji}|\Big)\Big]|x_i| \\ &\quad -\lambda\sum_{i=1}^nl_i\Big[|\tilde{I}_i|+\sum_{j=1}^n|f_j(0)|\Big(|a_{ij}|+|b_{ij}|+|\tilde{b}_{ij}|\Big)\Big] \\ &\geq\lambda l_0\|x\|_1-\lambda nI_0. \end{align*} Define \begin{gather*} l_0=\min_{1\leq i\leq n}\Big\{l_id_i-F_i\sum_{j=1}^nl_j\Big(|a_{ji}| +|b_{ji}|+|\tilde{b}_{ji}|\Big)\Big\}, \\ I_0=\max_{1\leq i\leq n}\Big\{l_i\Big(|\tilde{I}_i| +\sum_{j=1}^n|f_j(0)|\Big(|a_{ij}|+|b_{ij}|+|\tilde{b}_{ij}|\Big)\Big\}, \end{gather*} and let \[ %8 \Gamma=\big\{x : \|x\|_1 \leq \frac{n(I_0+1)}{l_0}\big\}. \] Then it follows that $\|x\|_1=n(I_0+1)/l_0$ for any $x\in\partial\Gamma$, and \[ \sum_{j=1}^nl_i|H_i(x,\lambda)|\geq\lambda l_0\frac{n(I_0+1)}{l_0}-\lambda nI_0>0,\quad \forall\lambda\in(0,1], \] that is $F(x,\lambda)\neq0$, for any $x\in\partial\Gamma$, $\lambda\in(0,1]$. Also, as $\lambda=0$, $H(x,\lambda)=i_d(x)=x\neq 0$, for any $x\in\partial\Gamma$, here, $i_d$ is identity mapping. Hence, we have $H(x,\lambda)\neq0$, for any $x\in\partial\Gamma$, $\lambda\in[0,1]$. From (H1), it is easy to prove $\deg(i_d,\Gamma,0)=1$ thus we have from homotopy invariance theorem \cite{c5} that $$ \deg(h,\Gamma,0)=deg(i_d,\Gamma,0)=1. $$ By the topological degree theory, we can conclude that \eqref{e3} has at least one solution in $\Gamma$. That is, model \eqref{e3} has at least an equilibrium point. Suppose $y^*=(y_1^*,y_2^*,\dots,y_n^*)^T$ is also an equilibrium point of model \eqref{e3}, then we have \begin{gather*} d_ix_i^*-\sum_{j=1}^na_{ij}f_j(x_j^*)-\bigwedge_{j=1}^nb_{ij}f_j(x_j^*) -\bigvee_{j=1}^n\tilde{b}_{ij}f_j(x_j^*)-\tilde{I}_i=0, \\ d_iy_i^*-\sum_{j=1}^na_{ij}f_j(y_j^*)-\bigwedge_{j=1}^nb_{ij}f_j(y_j^*) -\bigvee_{j=1}^n\tilde{b}_{ij}f_j(y_j^*)-\tilde{I}_i=0, \end{gather*} this implies \begin{align*} d_i(x_i^*-y_i^*) &=\sum_{j=1}^na_{ij}(f_j(x_j^*)-f_j(y_j^*))+\bigwedge_{j=1}^nb_{ij}f_j(x_j^*) -\bigwedge_{j=1}^nb_{ij}f_j(y_j^*) \\ &\quad +\bigvee_{j=1}^n\tilde{b}_{ij}f_j(x_j^*) -\bigvee_{j=1}^n\tilde{b}_{ij}f_j(y_j^*) \\ &\leq \sum_{j=1}^n|a_{ij}||f_j(x_j^*)-f_j(y_j^*)| +|\bigwedge_{j=1}^nb_{ij}f_j(x_j^*) -\bigwedge_{j=1}^nb_{ij}f_j(y_j^*)| \\ &\quad +|\bigvee_{j=1}^n\tilde{b}_{ij}f_j(x_j^*) -\bigvee_{j=1}^n\tilde{b}_{ij}f_j(y_j^*)| \end{align*} for $i=1,2,\dots,n$. By using (H1) and Lemma \ref{lem2}, we have \[ %9 d_i|x_i^*-y_i^*|\leq\sum_{j=1}^nF_j|x_j^*-y_j^*|\Big(|a_{ij}|+|b_{ij}|+|\tilde{b}_{ij}|\Big),\quad i=1,2,\dots,n, \] which can be rewritten as \[ %10 (D-(|A|+|B|+|\tilde{B}|)F)(|x_1^*-y_1^*|,|x_2^*-y_2^*|,\dots,|x_n^*-y_n^*|)^T\leq0. \] Since $D-(|A|+|B|+|\tilde{B}|)F$ is a nonsingular $M$-matrix, $(D-(|A|+|B|+|\tilde{B}|)F)^{-1}$ is a nonnegative matrix. Thus multiplying both sides of the above inequality by $(D-(|A|+|B|+|\tilde{B}|)F)^{-1}$ does not change the inequality direction, it follows that $$ (|x_1^*-y_1^*|,|x_2^*-y_2^*|,\dots,|x_n^*-y_n^*|)^T\leq0. $$ This implies that $x^*=y^*$. Therefore, the system \eqref{e3} has one unique equilibrium point. \end{proof} \begin{theorem} \label{thm2} Assume that {\rm (H1), (H2)} hold, and $D-(|A|+|B|+|\tilde{B}|)F$ is a nonsingular $M$-matrix, furthermore, suppose that the impulsive operator $\Delta_k(x_i(t_k))$ satisfies \begin{equation} \label{e11} \Delta_k(x_i(t_k))=-\delta_{ik}(x_i(t_k)-x_i^*),\quad 0<\delta_{ik}<2,\quad i=1,2,\dots,n, \; k=1,2,\dots. \end{equation} Then the equilibrium point $x^*=(x_1^*,x_2^*,\dots,x_n^*)^T$ of system \eqref{e3} is globally exponentially stable. \end{theorem} \begin{proof} From \eqref{e11}, we have $\Delta_k(x_i^*)=0$, and by Theorem \ref{thm1}, $x^*=(x_1^*,x_2^*,\dots,x_n^*)^T$ is only equilibrium point of the system \eqref{e3}. Let $x(t)=(x_1(t),x_2(t),\dots,x_n(t))^T$ be an arbitrary solution of the system \eqref{e3}. From assumption (H1) and Lemma \ref{lem2}, we obtain that \begin{equation} \label{e12} \begin{aligned} &\frac{d^+|x_i(t)-x_i^*|}{dt}\\ &=\mathop{\rm sign}(x_i(t)-x_i^*)\frac{d(x_i(t)-x_i^*)}{dt} \\ &\leq -d_i|x_i(t)-x_i^*|+\sum_{j=1}^n|a_{ij}||f_j(x_j(t))-f_j(x_j^*)| \\ &\quad +|\bigwedge_{j=1}^nb_{ij}\int_{-\infty}^tK_{ij}(t-s)f_j(x_j(s))ds -\bigwedge_{j=1}^nb_{ij}\int_{-\infty}^tK_{ij}(t-s)f_j(x_j^*)ds| \\ &\quad +|\bigvee_{j=1}^n\tilde{b}_{ij}\int_{-\infty}^tK_{ij}(t-s)f_j(x_j(s))ds -\bigvee_{j=1}^n\tilde{b}_{ij}\int_{-\infty}^tK_{ij}(t-s)f_j(x_j^*)ds| \\ &\leq -d_i|x_i(t)-x_i^*|+\sum_{j=1}^n|a_{ij}||f_j(x_j(t))-f_j(x_j^*)| \\ &\quad +|\sum_{j=1}^n|b_{ij}|\int_{-\infty}^tK_{ij}(t-s)|f_j(x_j(s))-f_j(x_j^*)|ds \\ &\quad +|\sum_{j=1}^n|\tilde{b}_{ij}|\int_{-\infty}^tK_{ij}(t-s)|f_j(x_j(s))-f_j(x_j^*)|ds \\ &\leq -d_i|x_i(t)-x_i^*|+\sum_{j=1}^n|a_{ij}|F_j|x_j(t)-x_j^*| \\ &\quad +\sum_{j=1}^n\Big(|b_{ij}|+|\tilde{b}_{ij}|\Big)\int_{-\infty}^tK_{ij}(t-s)F_j|x_j(s)-x_j^*|ds \\ &=-d_i|x_i(t)-x_i^*|+\sum_{j=1}^nF_j|a_{ij}||x_j(t)-x_j^*| \\ &\quad +\sum_{j=1}^nF_j\Big(|b_{ij}|+|\tilde{b}_{ij}|\Big) \int_{0}^{+\infty}K_{ij}(s)|x_j(t-s)-x_j^*|ds \end{aligned} \end{equation} for $t>0$, $i=1,2,\dots,n$, $t\neq t_k$, $k=1,2,\dots$. Also, \[ x_i(t_k^+)-x_i^*=-\delta_{ik}(x_i(t_k)-x_i^*)+x_i(t_k)-x_i^* =(1-\delta_{ik})(x_i(t_k)-x_i^*) \] for $i=1,2,\dots,n, \ k=1,2,\dots$. Hence \begin{equation} \label{e13} |x_i(t_k^+)-x_i^*|\leq|1-\delta_{ik}||x_i(t_k)-x_i^*|\leq|x_i(t_k)-x_i^*| \end{equation} for $i=1,2,\dots,n$, $k=1,2,\dots$. Since $D-(|A|+|B|+|\tilde{B}|)F$ is a nonsingular $M$-matrix, there exist constants $l_i>0$ such that \begin{equation} \label{e14} l_id_i-F_i\sum_{j=1}^nl_j\Big(|a_{ji}|+|b_{ji}|+|\tilde{b}_{ji}|\Big)>0,\quad i=1,2,\dots,n. \end{equation} Now, for $i=1,2,\dots,n$, we define the functions $$ \tilde{h}_i(\alpha_i)=l_i(d_i-\alpha_i)-F_i\sum_{j=1}^nl_j\Big[|a_{ji}| +\Big(|b_{ji}|+|\tilde{b}_{ji}|\Big)\int_0^{+\infty}e^{\alpha_i s}K_{ji}(s)ds\Big], $$ where $\alpha_i\in[0,+\infty)$. Obviously, for $i\in\{1,2,\dots,n\}$, $\tilde{h}_i(\alpha_i)$ are continuous on $[0,+\infty)$, and from \eqref{e14}, we know that $\tilde{h}_i(0)>0$, for $i\in\{1,2,\dots,n\}$. Also, for $i\in\{1,2,\dots,n\}$, we have $\tilde{h}_i(\alpha_i)\to-\infty$ as $\alpha_i\to+\infty$. So there exists $\alpha_i^*$ such that $\tilde{h}_i(\alpha_i^*)=0$, $i\in\{1,2,\dots,n\}$. Let $\alpha=\min\{\alpha_1,\alpha_2,\dots,\alpha_n\}$, we get \begin{equation} \label{e15} \tilde{h}_i(\alpha)=l_i(d_i-\alpha)-F_i\sum_{j=1}^nl_j\Big[|a_{ji}| +\Big(|b_{ji}|+|\tilde{b}_{ji}|\Big)\int_0^{+\infty}e^{\alpha_i s}K_{ji}(s)ds\Big]\geq0 \end{equation} for $i=1,2,\dots,n$. Let $y_i(t)=e^{\alpha t}|x_i(t)-x_i^*|$, $i=1,2,\dots,n$. Then it follows from \eqref{e12} that \begin{equation} \label{e16} \begin{aligned} \frac{d^+y_i(t)}{dt} &=\alpha e^{\alpha t}|x_i(t)-x_i^*|+e^{\alpha t}\frac{d^+|x_i(t)-x_i^*|}{dt}\\ &\leq -(d_i-\alpha)y_i(t)+\sum_{j=1}^n|a_{ij}|F_jy_j(t) \\ &\quad +\sum_{j=1}^n\Big(|b_{ij}| +|\tilde{b}_{ij}|\Big)F_j\int_{0}^{+\infty}e^{\alpha s}K_{ij}(s)y_j(t-s)ds \end{aligned} \end{equation} for $t>0$, $i=1,2,\dots,n$, $t\neq t_k$, $k=1,2,\dots$. Also, from \eqref{e13}, we have \[ %17 y_i(t_k^+)=e^{\alpha t_k^+}|x_i(t_k^+)-x_i^*|\leq e^{\alpha t_k}|x_i(t_k)-x_i^*|=y_i(t_k)\] for $i=1,2,\dots,n$, $k=1,2,\dots$. Now, we construct the Lyapunov functional \begin{equation} \label{e18} V(t)=\sum_{i=1}^nl_i\Big[y_i(t)+\sum_{j=1}^nF_j\Big(|b_{ij}| +|\tilde{b}_{ij}|\Big) \int_{0}^{+\infty}e^{\alpha s}K_{ij}(s)\Big(\int_{t-s}^ty_j(r)dr\Big)ds\Big]. \end{equation} The derivative of $V(t)$ along with the trajectories of model \eqref{e3} is \begin{align*} &D^+V(t)\\ &=\sum_{i=1}^nl_i\Big[\frac{d^+y_i(t)}{dt} +\sum_{j=1}^nF_j\Big(|b_{ij}| +|\tilde{b}_{ij}|\Big)y_j(t)\int_{0}^{+\infty}e^{\alpha s}K_{ij}(s)ds \\ &\quad -\sum_{j=1}^nF_j\Big(|b_{ij}| +|\tilde{b}_{ij}|\Big)\int_{0}^{+\infty}e^{\alpha s}K_{ij}(s)y_j(t-s)ds\Big] \\ &\leq \sum_{i=1}^nl_i\Big[-(d_i-\alpha)y_i(t)+\sum_{j=1}^n|a_{ij}|F_jy_j(t) \\ &\quad +\sum_{j=1}^nF_j\Big(|b_{ij}| +|\tilde{b}_{ij}|\Big)\int_{0}^{+\infty}e^{\alpha s}K_{ij}(s)y_j(t-s)ds \\ &\quad +\sum_{j=1}^nF_j\Big(|b_{ij}| +|\tilde{b}_{ij}|\Big)y_j(t)\int_{0}^{+\infty}e^{\alpha s}K_{ij}(s)ds \\ &\quad -\sum_{j=1}^nF_j\Big(|b_{ij}| +|\tilde{b}_{ij}|\Big)\int_{0}^{+\infty}e^{\alpha s}K_{ij}(s)y_j(t-s)ds\Big] \\ &= -\sum_{i=1}^nl_i(d_i-\alpha)y_i(t)+\sum_{i=1}^n\sum_{j=1}^nl_iF_j|a_{ij}|y_j(t) \\ &\quad +\sum_{i=1}^n\sum_{j=1}^nl_iF_j\Big(|b_{ij}| +|\tilde{b}_{ij}|\Big)|y_j(t)\int_{0}^{+\infty}e^{\alpha s}K_{ij}(s)ds \\ &=\sum_{i=1}^n\Big\{-l_i(d_i-\alpha)+F_i\sum_{j=1}^nl_j\Big[|a_{ji}| +\Big(|b_{ji}|+|\tilde{b}_{ji}|\Big)\int_{0}^{+\infty}e^{\alpha s}K_{ji}(s)ds \Big]\Big\}y_i(t) \leq0 \end{align*} for $t>0$, $t\neq t_k$, $k=1,2,\dots$. Also, \begin{align*} V(t_k^+)&=\sum_{i=1}^nl_i\Big[y_i(t_k^+)+\sum_{j=1}^nF_j\Big(|b_{ij}| +|\tilde{b}_{ij}|\Big) \int_{0}^{+\infty}e^{\alpha s}K_{ij}(s)\Big(\int_{t_k^+-s}^{t_k^+}y_j(r)dr\Big)ds\Big] \\ &\leq\sum_{i=1}^nl_i\Big[y_i(t_k)+\sum_{j=1}^nF_j\Big(|b_{ij}| +|\tilde{b}_{ij}|\Big) \int_{0}^{+\infty}e^{\alpha s}K_{ij}(s)\Big(\int_{t_k-s}^{t_k}y_j(r)dr\Big)ds\Big] \\ &=V(t_k), \quad k=1,2,\dots. \end{align*} So, we have $V(t)\leq V(0)$, for all $t>0$. From \eqref{e18}, we obtain \begin{equation} \label{e19} V(t)\geq \min_{1\leq i\leq n}\{l_i\}\sum_{i=1}^ny_i(t). \end{equation} Also, \begin{align*} %(20) & V(0)\\ &=\sum_{i=1}^nl_i\Big[y_i(0)+\sum_{j=1}^nF_j\Big(|b_{ij}| +|\tilde{b}_{ij}|\Big) \int_{0}^{+\infty}e^{\alpha s}K_{ij}(s)\Big(\int_{-s}^{0}y_j(r)dr\Big)ds\Big] \\ &\leq \max_{1\leq i\leq n}\{l_i\}\sum_{i=1}^n\Big[y_i(0)+\sum_{j=1}^nF_j\Big(|b_{ij}| +|\tilde{b}_{ij}|\Big) \int_{0}^{+\infty}e^{\alpha s}K_{ij}(s)\Big(\int_{-s}^{0}y_j(r)dr\Big)ds\Big] \\ &\leq \max_{1\leq i\leq n}\{l_i\}\sum_{i=1}^n\Big[y_i(0)+\sum_{j=1}^nF_j\Big(|b_{ij}| +|\tilde{b}_{ij}|\Big) \int_{0}^{+\infty}se^{\alpha s}K_{ij}(s)ds\Big(\sup_{-\infty0$. It follows from the definition of $y_i(t)$ and the above inequality that \[ \sum_{i=1}^n|x_i(t)-x_i^*|\leq Me^{-\alpha t}\sup_{-\infty0$, where $$ M=\frac{\max_{1\leq i\leq n}\{l_i\}}{\min_{1\leq i\leq n}\{l_i\}}\Big[1+\max_{1\leq i\leq n}\sum_{j=1}^nF_j\Big(|b_{ij}| +|\tilde{b}_{ij}|\Big)\int_{0}^{+\infty}se^{\alpha s}K_{ij}(s)ds\Big]. $$ The proof is completed. \end{proof} As a direct result of Theorems \ref{thm1}, \ref{thm2}, we have the following result. \begin{corollary} \label{coro1} Assume that {\rm (H1), (H2)} hold, then model \eqref{e3} has one unique equilibrium point, if any one of the following conditions is true: \begin{itemize} \item[(i)] $d_i>F_i\sum_{j=1}^n\Big(|a_{ji}|+|b_{ji}|+|\tilde{b}_{ji}|\Big)$, $i=1,2,\dots,n$. \item[(ii)] $d_i>\sum_{j=1}^nF_j\Big(|a_{ij}|+|b_{ij}|+|\tilde{b}_{ij}|\Big)$, $i=1,2,\dots,n$. \item[(iii)] There exists a positive vector $l=(l_1,l_2,\dots,l_n)^T>0$ such that $$ l_id_i>\sum_{j=1}^nl_jF_j\Big(|a_{ij}|+|b_{ij}|+|\tilde{b}_{ij}|\Big),\quad i=1,2,\dots,n. $$ \end{itemize} Furthermore, suppose that the impulsive operator $\Delta_{k}(x_i(t_k))$ satisfies \[ \Delta_{k}(x_i(t_k))=-\delta_{ik}(x_i(t_k)-x_i^*),\quad 0<\delta_{ik}<2,\quad i=1,2,\dots,n, \ k=1,2,\dots. \] Then the equilibrium point $x^*=(x_1^*,x_2^*,\dots,x_n^*)^T$ of the system \eqref{e3} is globally exponentially stable. \end{corollary} \begin{proof} In fact, any one of the conditions (i)-(iii) can assure, $D-(|A|+|B|+|\tilde{B}|)F$ is a nonsingular $M$-matrix. \end{proof} \section{Periodic oscillatory solution} In the section, we discuss the existence, uniqueness and global exponential stability of the periodic oscillatory solution of model \eqref{e1}. Let $I_i:R\to R$ and $u_i:R\to R$ be continuously periodic function with period $\omega$, i.e. $I_i(t+\omega)=I_i(t)$, $u_i(t+\omega)=u_i(t)$ for $i=1,2,\dots,n$. Furthermore, we assume that \begin{itemize} \item[(H3)] There exists a positive integer $q$ such that $$ t_{k+q}=t_k+\omega,\quad \delta_{i(k+q)}=\delta_{ik},\quad k=1,2,\dots, \ i=1,2,\dots,n, $$ where $\delta_{ik}$ satisfy $\Delta_k(x_i(t_k))=x_i(t_k^+)-x_i(t_k^-)=-\delta_{ik}x_i(t_k)$, $0<\delta_{ik}<2$. \end{itemize} \begin{theorem} \label{thm3} Under hypothesis {\rm (H1)--(H3)}, there exists exactly one $\omega$-periodic solution of model \eqref{e1} and all other solutions of model \eqref{e1} converge exponentially to it as $t\to+\infty$, if $D-(|A|+|B|+|\tilde{B}|)F$ is a nonsingular $M$-matrix. \end{theorem} \begin{proof} Let $x(t,\phi)=(x_1(t,\phi),x_2(t,\phi),\dots,x_n(t,\phi))^T$ and let $$ x(t,\varphi)=(x_1(t,\varphi),x_2(t,\varphi),\dots,x_n(t,\varphi))^T $$ be an arbitrary pair of solutions of \eqref{e1}. Since $D-(|A|+|B|+|\tilde{B}|)F$ is a nonsingular $M$-matrix, \eqref{e14} and \eqref{e15} hold. Let $\tilde{y}(t)=e^{\alpha t}|x_i(t,\phi)-x_i(t,\varphi)|$, $i=1,2,\dots,n$, we easily obtain \begin{equation} \label{e21} \begin{aligned} \frac{d^+\tilde{y}_i(t)}{dt} &\leq-(d_i-\alpha)\tilde{y}_i(t)+\sum_{j=1}^n|a_{ij}|F_j\tilde{y}_j(t)\\ &\quad +\sum_{j=1}^nF_j\Big(|b_{ij}| +|\tilde{b}_{ij}|\Big)\int_{0}^{+\infty}e^{\alpha s}K_{ij}(s)\tilde{y}_j(t-s)ds, \end{aligned} \end{equation} and \begin{equation} \label{e22} \begin{aligned} \tilde{y}_i(t_k^+) &=e^{\alpha t_k^+}|x_i(t_k^+,\phi)-x_i(t_k^+,\varphi)| \\ &= e^{\alpha t_k}|x_i(t_k^-,\phi)-\delta_{ik}x_i(t_k,\phi)-x_i(t_k^-,\varphi)+\delta_{ik}x_i(t_k,\varphi)| \\ &= e^{\alpha t_k}|x_i(t_k,\phi)-\delta_{ik}x_i(t_k,\phi)-x_i(t_k,\varphi)+\delta_{ik}x_i(t_k,\varphi)| \\ &= e^{\alpha t_k}|1-\delta_{ik}||x_i(t_k,\phi)-x_i(t_k,\varphi)| \\ &\leq e^{\alpha t_k}|x_i(t_k,\phi)-x_i(t_k,\varphi)|=\tilde{y}_i(t_k). \end{aligned} \end{equation} Now, we construct the Lyapunov functional \[%23 V(t)=\sum_{i=1}^nl_i\Big[\tilde{y}_i(t)+\sum_{j=1}^nF_j\Big(|b_{ij}| +|\tilde{b}_{ij}|\Big) \int_{0}^{+\infty}e^{\alpha s}K_{ij}(s) \Big(\int_{t-s}^t\tilde{y}_j(r)dr\Big)ds\Big]. \] By a minor modification of the proof of Theorem \ref{thm2}, we can easily derive \[ \sum_{i=1}^n|x_i(t,\phi)-x_i(t,\varphi)|\leq Me^{-\alpha t}\sup_{-\inftyF_i\sum_{j=1}^n\Big(|a_{ji}|+|b_{ji}|+|\tilde{b}_{ji}|\Big)$, $i=1,2,\dots,n$. \item[(ii)] $d_i>\sum_{j=1}^nF_j\Big(|a_{ij}|+|b_{ij}|+|\tilde{b}_{ij}|\Big)$, $i=1,2,\dots,n$. \item[(iii)] There exists a positive vector $l=(l_1,l_2,\dots,l_n)^T>0$ such that $$ l_id_i>\sum_{j=1}^nl_jF_j\Big(|a_{ij}|+|b_{ij}|+|\tilde{b}_{ij}|\Big),\quad i=1,2,\dots,n. $$ \end{itemize} \end{corollary} \section{Examples} \begin{example} \label{exa1}\rm Consider the model \begin{equation} \label{e25} \begin{gathered} \begin{aligned} \frac{dx_1(t)}{dt} &=-d_1x_1(t)+\sum_{j=1}^2a_{1j}f_j(x_j(t))+\sum_{j=1}^2\tilde{a}_{1j}u_j+I_1\\ &\quad +\bigwedge_{j=1}^2b_{1j}\int_{-\infty}^{t}e^{-(t-s)}f_j(x_j(s))ds +\bigvee_{j=1}^2\tilde{b}_{1j}\int_{-\infty}^{t}e^{-(t-s)}f_j(x_j(s))ds\\ &\quad +\bigwedge_{j=1}^2T_{1j}u_j+\bigvee_{j=1}^2H_{1j}u_j, \quad t\geq0,\; t\neq t_k \end{aligned} \\ \Delta x_1(t_k)=-(1+\frac{1}{2}\sin(1+k))(x_1(t_k)-\frac{25}{32}),\quad k=1,2,\dots, \\ \begin{aligned} \frac{dx_2(t)}{dt} &=-d_2x_2(t)+\sum_{j=1}^2a_{2j}f_j(x_j(t)) +\sum_{j=1}^2\tilde{a}_{2j}u_j+I_2 \\ &\quad +\bigwedge_{j=1}^2b_{2j}\int_{-\infty}^{t}e^{-(t-s)}f_j(x_j(s))ds +\bigvee_{j=1}^2\tilde{b}_{2j}\int_{-\infty}^{t}e^{-(t-s)}f_j(x_j(s))ds \\ &\quad +\bigwedge_{j=1}^2T_{2j}u_j+\bigvee_{j=1}^2H_{2j}u_j, \quad t\geq0,\; t\neq t_k \end{aligned} \\ \Delta x_2(t_k)=-(1+\frac{2}{3}\cos(2k))(x_2(t_k)-\frac{1}{2}),\quad k=1,2,\dots, \end{gathered} \end{equation} where $0