\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 41, pp. 1--7.\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-2009/41\hfil Exponential convergence] {Exponential convergence of solutions of SICNNs with mixed delays} \author[H.-S. Ding, G.-R. Ye\hfil EJDE-2009/41\hfilneg] {Hui-Sheng Ding, Guo-Rong Ye} % in alphabetical order \address{College of Mathematics and Information Science, Jiangxi Normal University\\ Nanchang, Jiangxi 330022, China} \email[Ding]{dinghs@mail.ustc.edu.cn} \email[Ye]{yeguorong2006@sina.com} \thanks{Submitted November 17, 2008. Published March 19, 2009.} \thanks{Supported by the NSF of China (10826066), the NSF of Jiangxi province of China \hfill\break\indent(2008GQS0057), the Youth Foundation of Jiangxi Provincial Education Department \hfill\break\indent(GJJ09456), and the Youth Foundation of Jiangxi Normal University.} \subjclass[2000]{34K25, 34K20} \keywords{Exponential convergence behavior; delay; \hfill\break\indent shunting inhibitory cellular neural networks} \begin{abstract} In this paper, we discuss shunting inhibitory cellular neural networks (SICNNs) with mixed delays and time-varying coefficients. We establish conditions for all solutions of SICNNs to converge exponentially to zero. Our theorem improve some known results and allow for more general activation functions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \section{Introduction} In this article, we study the following shunting inhibitory cellular neural networks with mixed delays and time-varying coefficients: \begin{equation} \label{equation} \begin{aligned} x'_{ij}(t) &=-a_{ij}(t)x_{ij}(t)-\sum_{C_{kl}\in N_r(i,j)}C_{ij}^{kl}(t)f[x_{kl}(t-\tau_{ij}(t))]x_{ij}(t) \\ &\quad-\sum_{C_{kl}\in N_q(i,j)}B_{ij}^{kl}(t)\int^{\infty}_0 k_{ij}(u)g[x_{kl}(t-u)]du\cdot x_{ij}(t)+L_{ij}(t), \end{aligned} \end{equation} where $i=1,2,\dots,m$, $j=1,2,\dots,n$; $C_{ij}$ denotes the cell at the $(i,j)$ position of the lattice; $x_{ij}$ is the activity of the cell $C_{ij}$; the $r$-neighborhood $N_r(i,j)$ of $C_{ij}$ is defined as $$ N_r(i,j)=\{C_{kl}:\max(|k-i|,|l-j|)\leq r, 1\leq k \leq m, 1\leq l \leq n\} $$ and $N_q(i,j)$ is similarly defined; $L_{ij}(t)$ is the external input to $C_{ij}$; $a_{ij}> 0$ represents the passive decay rate of the cell activity; $C_{ij}^{kl}\geq 0$ and $B_{ij}^{kl}\geq 0$ are the connection or coupling strength of postsynaptic activity of the cell $C_{kl}$ transmitted to the cell $C_{ij}$; the activation functions $f,g$ are continuous functions representing the output or firing rate of the cell $C_{kl}$; and $\tau_{ij}(t)\geq 0$ are the transmission delays. Recall that in 1990s, Bouzerdoum and Pinter \cite{Bouzerdoum91,Bouzerdoum92,Bouzerdoum} introduced and analyzed the networks commonly called shunting inhibitory cellular neural networks (SICNNs). Now, SICNNs have been extensively applied in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing (see, e.g., \cite{Bouzerdoum1,Bouzerdoum2} and references therein). It is well known that analysis of dynamic behaviors is very important for design of neural networks. Therefore, there has been of great interest for many authors to study all kinds of dynamic behaviors for SICNNs and its variants (see, e.g., \cite{ding,liu2009,liyaqiong,liyaqiong-huang,liyaqiong2008,liu,liu071,liu072}). Especially, there are many interesting and important works about exponential convergence behavior of solutions to SICNNs. For example, in \cite{liyaqiong2008}, the authors studied the following SICNNs with delays \begin{equation}\label{1} x'_{ij}(t)=-a_{ij}(t)x_{ij}(t)-\sum_{C_{kl}\in N_r(i,j)}C_{ij}^{kl}(t)f[x_{kl}(t-\tau(t))]x_{ij}(t)+L_{ij}(t), \end{equation} where $i=1,2,\dots,m, j=1,2,\dots,n,$ and established a theorem which ensure that all the solutions of \eqref{1} converge exponentially to zero. Also, in \cite{liyaqiong}, the authors considered the same problem for the the following SICNNs with distributed delays $$ x'_{ij}(t)=-a_{ij}(t)x_{ij}(t)-\sum_{C_{kl}\in N_r(i,j)}C_{ij}^{kl}(t)\int^{\infty}_0 k_{ij}(u)f[x_{kl}(t-u)]du\cdot x_{ij}(t)+L_{ij}(t), $$ where $i=1,2,\dots,m, j=1,2,\dots,n$. In addition, the authors in \cite{liyaqiong-huang} studied the convergence behavior of solutions for the SICNNs \eqref{equation}. In \cite{liyaqiong,liyaqiong-huang,liyaqiong2008}, the activity functions $f$ and $g$ are assumed to be bounded. Recently, in \cite{liu2009}, the assumption is weakened into \begin{itemize} \item[(H0)] There exist constants $m\geq 1$, $n\geq 1$, $L_f$ and $L_g$ such that for all $u\in\mathbb{R}$, $$ |f(u)|\leq L_f |u|^m,\quad |g(u)|\leq L_g |u|^n. $$ \end{itemize} In this paper, we allow for more general activity functions $f$ and $g$; i.e., we only assume that \begin{itemize} \item[(H1)] $f$ and $g$ are continuous functions on $\mathbb{R}$. \end{itemize} In addition, we do not need the restrictive condition used in \cite{liu2009} (see remark \ref{remark}). Throughout this paper, for $i=1,2,\dots,m$, $j=1,2,\dots,n,$ $k_{ij}:[0,+\infty)\to\mathbb{R}$ are continuous integrable functions, $a_{ij},C_{ij}^{kl},B_{ij}^{kl},\tau_{ij}$ are continuous functions, and $L_{ij}$ are continuous bounded functions. Moreover, for real functions $u(t)$ and $v(t)$, we write $u(t)=O(v(t))$ if there exists a constant $M \geq 0$ such that for some $N > 0$, $$ |u(t)|\leq M|v(t)|, \quad \forall t \geq N. $$ Since $f$ and $g$ are continuous functions, we define the following functions on $[0,+\infty)$: $$ F(x)=\max_{|t|\leq x}|f(t)|,\quad G(x)=\max_{|t|\leq x}|g(t)|. $$ \section{Main results} In the proof of our results, we will use the following assumptions: \begin{itemize} \item[(H2)] There exist constants $\eta > 0$ and $\lambda > 0$ such that $$ [\lambda -a_{ij}(t)]+\sum_{C_{kl}\in N_r(i,j)} C_{ij}^{kl}(t) F(\beta)+ \sum_{C_{kl}\in N_q(i,j)} B_{ij}^{kl}(t) G(\beta) \int_0^\infty |k_{ij}(u)|du < -\eta, $$ for all $t>0$, $i\in\{1,2,\dots,m\}$ and $j\in\{1,2,\dots,n\}$, where \[ \beta =\frac{ \max_{(i,j)}\{\sup _{t\geq 0}|L_{ij}(t)|\}}{\eta}. \] \item[(H3)] $L_{ij}(t) = O (e^{-\lambda t})$, $i=1,2,\dots,m$, $j = 1,2,\dots,n$. \end{itemize} \begin{lemma}\label{lem2.1} Assume that {\rm (H1)} and {\rm (H2)} hold. Then, for every solution $$ \{x_{ij}(t)\}=(x_{11}(t),\dots,x_{1n}(t),\dots,x_{i1}(t),\dots,x_{in}(t),\dots,x_{m1}(t),\dots,x_{mn}(t)), $$ of \eqref{equation} with initial condition $ \sup_{-\infty0:|x_{i_0j_0}(t)|> \beta\}\neq\emptyset. \end{equation} For each $k\in\{1,2,\dots,m\}$ and $l\in\{1,2,\dots,n\}$, let $$ T_{kl}=\begin{cases} \inf\{t>0:|x_{kl}(t)|> \beta\} & \{t>0:|x_{kl}(t)|> \beta\}\neq\emptyset,\\ +\infty & \{t>0:|x_{kl}(t)|> \beta\}=\emptyset. \end{cases} $$ Then $T_{kl}>0$ and \begin{equation}\label{03} |x_{kl}(t)|\leq \beta,\quad \forall t\leq T_{kl},\; k=1,2,\dots,m,\; l=1,2,\dots,n. \end{equation} We denote $T_0=T_{ij}=\min_{(k,l)}T_{kl}$, where $i\in\{1,2,\dots,m\}$ and $j\in\{1,2,\dots,n\}$. In view of \eqref{00}, we have $00:|x_{ij}(t)|> \beta\}$, we obtain \begin{equation}\label{05} |x_{ij}(T_0)|=\beta,\quad D^+(|x_{ij}(s)|)|_{s=T_0}\geq0. \end{equation} Combing (H2), \eqref{04} and \eqref{05}, we have \begin{align*} &D^+(|x_{ij}(s)|)|_{s=T_0}\\ & = \mathop{\rm sgn} (x_{ij}(T_0))\Big\{-a_{ij}(T_0)x_{ij}(T_0) -\sum_{C_{kl}\in N_r(i,j)}C_{ij}^{kl}(T_0)f[x_{kl}(T_0-\tau_{ij}(T_0))]x_{ij}(T_0) \\ &\quad -\sum_{C_{kl}\in N_q(i,j)}B_{ij}^{kl}(T_0)\int^{\infty}_0 k_{ij}(u)g[x_{kl}(T_0-u)]du\cdot x_{ij}(T_0)+L_{ij}(T_0)\Big\}\\ &\leq -a_{ij}(T_0)\cdot|x_{ij}(T_0)|+\sum_{C_{kl}\in N_r(i,j)}C_{ij}^{kl}(T_0)F(\beta)\cdot|x_{ij}(T_0)|\\ &\quad +\sum_{C_{kl}\in N_q(i,j)} B_{ij}^{kl}(T_0) G(\beta)\int_0^\infty |k_{ij}(u)|du\cdot|x_{ij}(T_0)|+|L_{ij}(T_0)|\\ &\leq \Big\{-a_{ij}(T_0) +\sum_{C_{kl}\in N_r(i,j)}C_{ij}^{kl}(T_0)F(\beta) \\ &\quad +\sum_{C_{kl}\in N_q(i,j)} B_{ij}^{kl}(T_0) G(\beta) \int_0^\infty |k_{ij}(u)|du \Big\}\cdot \beta+|L_{ij}(T_0)|\\ & < -\eta\cdot \beta+|L_{ij}(T_0)|\\ &= -\max_{(i,j)}\{\sup _{t\geq 0}|L_{ij}(t)|\}+|L_{ij}(T_0)|\leq 0. \end{align*} This contradicts $D^+(|x_{ij}(s)|)|_{s=T_0}\geq0$. Thus, \eqref{longwei} holds. \end{proof} \begin{theorem}\label{theorem} Let {\rm (H1)--(H3)} hold. Then, for every solution $$ \{x_{ij}(t)\}=(x_{11}(t),\dots,x_{1n}(t),\dots,x_{i1}(t), \dots,x_{in}(t),\dots,x_{m1}(t),\dots,x_{mn}(t)) $$ of \eqref{equation} with initial condition $ \sup_{-\infty 0$ and $T> 0$ such that \begin{equation}\label{5} |L_{ij}(t)|< \frac{1}{2}\eta M e^{-\lambda t},\quad \forall t \geq T,\; ij=11,12,\dots,mn. \end{equation} Let $\{x_{ij}(t)\}=(x_{11}(t),\dots,x_{1n}(t),\dots,x_{i1}(t), \dots,x_{in}(t),\dots,x_{m1}(t),\dots,x_{mn}(t))$ be a solution of \eqref{equation} with initial condition $ \sup_{-\inftye^{\lambda t_0}|x_{ij}(t_0)|$. It follows form the continuity of $x_{ij}(t)$ that there exists $\delta_1> 0$ such that $$e^{\lambda t}|x_{ij}(t)| < V_{ij}(t_0),\quad \forall t \in (t_0,t_0+\delta_1).$$ Thus, we can conclude $$ V_{ij}(t)= V_{ij}(t_0),\quad \forall t \in (t_0,t_0+\delta_1). $$ \noindent\textbf{Case (ii)} $ V_{ij}(t_0)=e^{\lambda t_0}|x_{ij}(t_0)| 0$ such that $$ e^{\lambda t}|x_{ij}(t)| < M,\quad \forall t \in (t_0,t_0+\delta_2). $$ Therefore, $$ V_{ij}(t) 0$ such that $$ e^{\lambda t}|x_{ij}(t)|< e^{\lambda t_0}|x_{ij}(t_0)|= V_{ij}(t_0),\quad \forall t \in (t_0,t_0+\delta_3). $$ Then, we conclude that $$ V_{ij}(t)= V_{ij}(t_0), \quad \forall t \in (t_0,t_0+\delta_3). $$ In summary, for any given $ij\in\{11,12,\dots,mn\}$, for all $t_0\geq T$, there exists $\delta=\min\{\delta_1,\delta_2,\delta_3\} > 0$ such that $$ V_{ij}(t)\leq \max\{V_{ij}(t_0), M\},\quad \forall t \in (t_0,t_0+\delta). $$ Now, take $t_0=T$. Then there exists $\delta'>0$ such that $$ V_{ij}(t)\leq \max\{V_{ij}(T), M\},\quad \forall t \in (T,T+\delta'). $$ Since $V_{ij}$ is continuous, we have $$ V_{ij}(t)\leq \max\{V_{ij}(T), M\},\quad \forall t \in (T,T+\delta']. $$ Take $t_0=T+\delta'$. Then there exists $\delta''>0$ such that $$ V_{ij}(t)\leq \max\{V_{ij}(T+\delta'), M\}\leq \max\{V_{ij}(T), M\}, \quad \forall t \in (T+\delta',T+\delta'+\delta''). $$ Then $$ V_{ij}(t)\leq \max\{V_{ij}(T), M\},\quad \forall t \in (T,T+\delta'+\delta''). $$ Continuing the above step, at last, we get a maximal interval $(T,\alpha_{ij})$ such that $$ V_{ij}(t)\leq \max\{V_{ij}(T), M\},\quad \forall t \in (T,\alpha_{ij}). $$ Also, we have $\alpha_{ij}=+\infty $. In fact, if $\alpha_{ij}<+\infty $, then we have $$ V_{ij}(t)\leq \max\{V_{ij}(T), M\},\quad \forall t \in (T,\alpha_{ij}]. $$ Take $t_0=\alpha_{ij}$. Then there exists $\delta^*>0$ such that $$ V_{ij}(t)\leq \max\{V_{ij}(T), M\},\quad \forall t \in (T,\alpha_{ij}+\delta^*). $$ This is a contradiction. Therefore, $$ V_{ij}(t) \leq \max\{ V_{ij}(T),\ M \},\quad \forall t> T. $$ It follows that $$ e^{\lambda t}|x_{ij}(t)|\leq \max\{ V_{ij}(T),\ M \},\quad \forall t> T, $$ whichimplies $x_{ij}(t) =O(e^{-\lambda t})$. \end{proof} \begin{remark}\label{remark}\rm In \cite{liu2009}, it is assume that $\beta<1$. But in Theorem \ref{theorem}, we do not need this condition. In addition, it is not difficult to show that Theorem \cite[Theorem 2.1]{liu2009} is a corollary of Theorem \ref{theorem}. \end{remark} \section{Examples} In this section, we give an example to illustrate our results. \begin{example}\label{example} \rm Consider the SICNNs: \begin{equation}\label{6} x'_{ij}(t)=-a_{ij}(t)x_{ij}(t)-\sum_{C_{kl}\in N_1(i,j)}C_{ij}^{kl}(t)f[x_{kl}(t-\tau_{ij}(t))]x_{ij}(t)+L_{ij}(t), \end{equation} where $i=1,2,3$, $j=1,2,3$, $\tau_{ij}(t)=|\frac{1}{2}t \sin(i+j) t|$, $f(x) =e^x$, \begin{gather*} \begin{pmatrix} a_{11}(t) & a_{12}(t) & a_{13}(t)\\ a_{21}(t) & a_{22}(t) & a_{23}(t)\\ a_{31}(t)& a_{32}(t) & a_{33}(t) \end{pmatrix} = \begin{pmatrix} 5+\sin^2 t &5+|\sin t| &7+\sin t\\ 6+\sin t & 7+|\sin t| & 6+\sin t\\ 7+\sin t & 8+\sin t & 8+\sin t \end{pmatrix}, \\ \begin{pmatrix} c_{11}(t) & c_{12}(t) & c_{13}(t) \\ c_{21}(t) & c_{22}(t)& c_{23}(t) \\ c_{31}(t) &c_{32}(t) &c_{33}(t) \end{pmatrix} = \begin{pmatrix} 0.1 |\sin t| & 0.1\sin^2 t & 0.2|\sin t|\\ 0 & 0.2\sin^2 t & 0 \\ 0.1\sin^2 t &0.1|\sin t|& 0.2 \sin^2 t \end{pmatrix}, \\ \begin{pmatrix} L_{11}(t) & L_{12}(t) &L_{13}(t) \\ L_{21}(t) &L_{22}(t) & L_{23}(t)\\ L_{31}(t) & L_{32}(t) & L_{33}(t) \end{pmatrix} = \begin{pmatrix} \frac{1}{2}e^{-t} & e^{-2t} & 2e^{-2t}\\ e^{-2t} & e^{-t} &e^{-2t} \\ e^{-2t} & e^{-t} & e^{-2t} \end{pmatrix}. \end{gather*} Obviously, (H1) holds. By some calculations, it is easy to obtain that for all $t\in\mathbb{R}$, $$ a_{ij}(t)\geq 5.\quad \sum_{C_{kl}\in N_1(i,j)}C_{ij}^{kl}(t)\leq 1. $$ In addition, $$ \max_{(i,j)}\{\sup _{t\geq 0}|L_{ij}(t)|\}=2,\quad F(x)=e^x . $$ Let $\lambda=0.2$ and $\eta=2$. Then $\beta=1$ and \[ [\lambda -a_{ij}(t)]+\sum_{C_{kl}\in N_1(i,j)} C_{ij}^{kl}(t)F(\beta)\leq 0.2-5+e<-2=-\eta, \] for all $t>0$, $i=1,2,3$ and $j=1,2,3$. Therefore, (H2) holds. It is easy to verify that (H3) holds for $\lambda=0.2$. Now, by Theorem \ref{theorem}, all the solutions of \eqref{6} with initial condition $$ \sup_{-\infty