\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 61, 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 (login: ftp)} \thanks{\copyright 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/61\hfil Periodic solutions for planar systems] {Periodic solutions for planar systems with time-varying delays} \author[C. Huang, L. Huang, Y. Meng\hfil EJDE-2005/61\hfilneg] {Chuangxia Huang, Lihong Huang, Yimin Meng} % in alphabetical order \address{Chuangxia Huang\hfill\break College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China} \email{huangchuangxia@sina.com} \address{Lihong Huang\hfill\break College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China} \email{lhhuang@hnu.cn} \address{Yimin Meng \hfill\break College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China} \email{mym1986cs@sina.com} \date{} \thanks{Submitted September 22, 2004. Published June 10, 2005.} \thanks{Supported by grant 10371034 from the National Natural Science Foundation of China} \subjclass[2000]{34K13, 92B20} \keywords{Differential system; neural network; periodic solution} \begin{abstract} This paper concerns delay differential systems that can be regarded as a model of two-neuron artificial neural network with delayed feedback. Some interesting results are obtained for the existence of a periodic solution for the system. Our approach is based on the continuation theorem of coincidence degree, and a-priori estimates of the periodic solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \section{Introduction} Neural networks are complex and large-scale nonlinear dynamics, while the dynamics of the delayed neural network are even richer and more complicated \cite{w1}. To obtain a deep and clear understanding of the dynamics of neural networks, there has been an increasing interest in the investigations of delayed neural network models with two neurons, see \cite{b1,f1,g1,g3,o1,r1,t1,z1}. T\'{a}boas \cite{t1} considered the system of delay differential equations \begin{equation} \begin{gathered} \dot x_1(t) = -x_1(t) +\alpha f_{1}(x_1(t-\tau), x_2(t-\tau)),\\ \dot x_2(t) = -x_2(t) +\alpha f_{2}(x_1(t-\tau), x_2(t-\tau)), \end{gathered} \label{e1.1} \end{equation} which arises as a model for a network of two saturating amplifiers (or neurons) with delayed outputs, where $ \alpha >0 $ is a constant, $ f_{1} , f_{2}$, are bounded $C^{3}$ functions on $\mathbb{R}^{2}$ satisfying $$ \frac {\partial f_{1} }{\partial x_2 }( 0,0 ) \neq0 \quad\mbox{and} \quad \frac {\partial f_{2} }{\partial x_1 }( 0,0 )\neq0, $$ and the negative feedback conditions : $ x_2f_{1} (x_1, x_2) >0$, $x_2\neq0$; $x_1f_{2} (x_1, x_2) <0$, $x_1\neq0$. T\'{a}boas showed that there is an $\alpha_{0}>0$ such that for $\alpha>\alpha_{0}$, there exists a non-constant periodic solution with period greater than $4$. Further study on the global existence of periodic solutions to system \eqref{e1.1} can be found in \cite{b1} and \cite{g3}. All together there is only one delay appearing in both equations. Ruan and Wei \cite{r1} investigated the existence of non-constant periodic solutions of the following planar system with two delays \begin{equation} \begin{gathered} \dot x_1(t)=-a_{0}x_1(t) +a_{1}f_{1}(x_1(t-\tau_{1}), x_2(t-\tau_{2})),\\ \dot x_2(t)=-b_{0}x_2(t)+b_{1}f_{2}(x_1(t-\tau_{1}), x_2(t-\tau_{2})), \end{gathered}\label{e1.2} \end{equation} where $a_{0}>0$, $b_{0}>0$, $a_{1}$ and $b_{1}$ are constants, the function $f_{1}$ and $f_{2}$ satisfy $f_{j} \in C^{3} (\mathbb{R}^{2}), f_{j} (0,0)=0$, $\frac{\partial f_{j}}{\partial x_j}( 0,0 )=0$, $j=1,2$; $x_2f_{1}(x_1, x_2)\neq 0$ for $x_2 \neq0$; $x_1f_{2} (x_1, x_2)\neq0$ for $x_1 \neq0$; $\frac {\partial f_{1}}{\partial x_2}(0,0)\neq0$, $\frac {\partial f_{2}}{\partial x_1}( 0,0 )\neq 0$. Recently, Zhang and Wang \cite{z1} investigated the system \begin{equation} \begin{gathered} \dot x_1(t)=-a_{1}x_1(t)+b_{1}f_{1}(x_1(t-\tau_{1}), x_2(t-\tau_{2})),\\ \dot x_2(t)=-a_{2}x_2(t)+b_{2}f_{2}(x_1(t-\tau_{3}), x_2(t-\tau_{4})), \end{gathered} \label{e1.3} \end{equation} where $a_{1}, a_{2}, b_{1}, b_{2}, \tau_{1}, \tau_{2}, \tau_{3}, \tau_{4}$ are constants. By means of the continuation theorem of the coincidence degree, they get some results about the periodic solutions to system \eqref{e1.3}. However, delays considered in all above systems are constant. It is well known that the delays in artificial neural networks are usually time-varying, and sometimes vary violently with time due to the finite switching speed of amplifiers and faults in the electrical circuit. They slow down the transmission rate and tend to introduce some degree of instability in circuits. Therefore, fast response must be required in practical artificial neural-network designs. The technique to achieve fast response troubles many circuit designers. So, it is more important to investigate the dynamic behave of neural networks with time-varying delays. Keeping this in mind, in this paper, we consider the following planar system where coefficients and delays are all periodically varying in time: \begin{equation} \begin{gathered} \dot x_1(t)=-a_{1}(t)x_1(t)+b_{1}(t)f_{1}(x_1(t-\tau_{1} (t)),x_2(t-\tau_{2}(t))),\\ \dot x_2(t)=-a_{2}(t)x_2(t)+b_{2}(t)f_{2}(x_1(t-\tau_{3} (t)),x_2(t-\tau_{4}(t))), \end{gathered}\label{e1.4} \end{equation} where $a_{i}\in C( \mathbb{R}, (0 , \infty) )$, $b_{ i }\in C( \mathbb{R}, \mathbb{R})$, $i=1, 2$, are periodic with a common period $ \omega (> 0 )$, $f_{ i }\in C(\mathbb{R}^2, \mathbb{R})$, $i=1, 2$; $\tau _ { i }\in C( \mathbb{R}, [0,\infty))$, $i=1, 2, 3, 4$, being $\omega$-periodic. For a continuous function g: $[0, \omega]\to \mathbb{R}$, we introduce the following notation: \begin{gather*} \overline{g}=\frac{1}{\omega}\int^\omega_0 g(t){\rm d}t, \\ a(t)= \min\{a_1(t),a_2(t) \}, \quad b(t)= \max\{|b_1(t)|,|b_2(t)| \}. \end{gather*} Obviously, system \eqref{e1.4} is more general than system \eqref{e1.3}. To our best knowledge, the existence of $\omega$-periodic solution of the system \eqref{e1.4} has not been studied in pervious works. We shall employ the powerful method of coincidence degree to establish the existence of a periodic solution to \eqref{e1.4}. These conditions in our results are very simple and easy to be verified. \section{Existence of Periodic Solution} In this section, we use the coincidence degree theory to obtain the existence of an $\omega$-periodic solution to \eqref{e1.4}. For the sake of convenience, we briefly summarize the theory as below. Let $X$ and $Z$ be normed spaces, $L$: $\mathop{\rm Dom}L\subset X\to Z$ be a linear mapping and $N:X\to Z$ be a continuous mapping. The mapping $L$ will be called a Fredholm mapping of index zero if $\mathop{\rm dimKer}L=\mathop{\rm codimIm}L < \infty$ and $\mathop{\rm Im}L$ is closed in $Z$. If $L$ is a Fredholm mapping of index zero, there exist continuous projectors $P: X\to X$ and $Q: Z\to Z$ such that $\mathop{\rm Im}P=\ker L$ and $\mathop{\rm Im}L=\ker Q=\mathop{\rm Im}(I-Q)$. It follows that $L\big| \mathop{\rm Dom}L\cap \ker P: (I-P)X\to \mathop{\rm Im}L$ is invertible. We denote the inverse of this map by $K_p$. If $\Omega$ is a bounded open subset of $X$, the mapping $N$ is called $L$-compact on $\overline{\Omega}$ if $QN(\overline{\Omega})$ is bounded and $K_p(I-Q)N: \overline{\Omega}\to X$ is compact. Because $\mathop{\rm Im}Q$ is isomorphic to $\ker L$, there exists an isomorphism $J: \mathop{\rm Im}Q\to \ker L$. Let $\Omega\subset \mathbb{R}^n$ be open and bounded, $f\in C^1(\Omega, \mathbb{R}^n)\cap C(\overline{\Omega}, \mathbb{R}^n)$ and $y\in \mathbb{R}^n\backslash f(\partial\Omega \cup S_f)$, i.e., $y$ is a regular value of $f$. Here, $S_f=\{x\in \Omega: J_f(x)=0\}$, the critical set of $f$, and $J_f$ is the Jacobian of $f$ at $x$. Then the degree $\deg\{f, \Omega, y\}$ is defined by $$ \deg\{f, \Omega,y\}=\sum_{x\in f^{-1}(y)}\mathop{\rm sgn} J_f (x) $$ with the agreement that the above sum is zero if $f^{-1}(y)=\emptyset$. For more details about degree theory, we refer to the book by Deimiling \cite{d1}. Now, with the above notation, we are ready to state the continuation theorem. \begin{lemma}[{Continuation Theorem \cite[P.40]{g2}}] \label{lem2.1} Let $L$ be a Fredholm mapping of index zero and let $N$ be $L$-compact on $\overline{\Omega}$. Suppose \begin{itemize} \item[(a)] For each $\lambda\in (0,1)$, every solution $x$ of $Lx=\lambda Nx$ is such that $x\notin \partial\Omega$ \item[(b)] $QNx\neq 0 $ for each $x\in \partial\Omega\cap \ker L$ and $$ \deg\{JQN, \Omega\cap \ker L,0\}\neq0. $$ Then the equation $Lx=Nx$ has at least one solution lying in $\mathop{\rm Dom}L\cap\overline{\Omega}$. \end{itemize} The following is the main result of this section. \end{lemma} \begin{theorem} \label{thm2.2} Suppose that $| f_i(x_1, x_2)|\leq\alpha_i| x_1|+\beta_i| x_2|+M_i $ and $\frac{D_1}{D}>0$, $\frac{D_2}{D}>0$, where $\alpha_i\geq0$, $\beta_i\geq 0$ and $M_i>0$ are constants for $i=1, 2$, $D=a^2-ab\alpha_1-ab\beta_2+b^2\alpha_1\beta_2-b^2\alpha_2\beta_1$, $D_1=abM_1+b^2M_2\beta_1-b^2M_1\beta_2$, $D_2=abM_2-b^2M_2\alpha_1+b^2M_1\alpha_2$, $a=\min_ { t\in [0 , \omega ]}a(t)$, $b=\max_ { t\in [0 , \omega ]}b(t)$. Then system \eqref{e1.4} has an $\omega$-periodic solution. \end{theorem} \begin{proof} Take $X=\{u(t)=(x_1(t), x_2(t) )^T\in C(\mathbb{R},\mathbb{R}^{2}): u(t)=u(t+\omega )\ \mbox{for} \ t\in \mathbb{R}\}$ and denote \begin{gather*} \| x_i\| =\max_{ t\in [0, \omega]} |x_i(t)|, \quad i=1, 2; \\ \| u\| _0=\max_{i=1, 2}\| x_i\|. \end{gather*} Equipped with the norm $\|.\| _0$, $X$ is a Banach space. For any $u(t)\in X$, because of the periodicity, it is easy to check that $$t\mapsto \begin{pmatrix} -a_1(t)x_1(t)+b_1(t)f_1(x_1(t-\tau_1(t)), x_2(t-\tau_2(t))), \\ -a_2(t)x_2(t)+b_2(t)f_2(x_1(t-\tau_3(t)), x_2(t-\tau_4(t))). \end{pmatrix}\in X. $$ Let $$ L: \mathop{\rm Dom}L=\{u\in X: u\in C^1(\mathbb{R},\mathbb{R}^2)\}\ni u \mapsto u'\in X, $$ $$ P:X\ni u\to \overline{u}\in X,\quad Q:X\ni x\mapsto \overline{x}\in X, $$ where for any $K=(k_1, k_2)^T\in \mathbb{R}^2$, we identify it as the constant function in X with the value vector K $=(k_1, k_2)^T$. Define $N:X\to X $ given by $$ (Nu)(t)= \begin{pmatrix} -a_1(t)x_1(t)+b_1(t)f_1(x_1(t-\tau_1(t)), x_2(t-\tau_2(t))),\\ -a_2(t)x_2(t)+b_2(t)f_2(x_1(t-\tau_3(t)), x_2(t-\tau_4(t))). \end{pmatrix}\in X. $$ Then system \eqref{e1.4} can be reduced to the operator equation $Lu=Nu$. Note that N is continuous, since $f_i$ are uniformly continuous on compact sets of $\mathbb{R}^2$. It is easy to see that \begin{gather*} \ker L=\mathbb{R}^2, \\ \mathop{\rm Im}L=\{x\in X: \overline{x}=0\}, \mbox{ which is closed in $X$},\\ \mathop{\rm dim\,ker}L = \mathop{\rm codim\,Im}L=2< \infty, \end{gather*} and $P$, $Q$ are continuous projectors such that $$ \mathop{\rm Im}P = \ker L,\quad \ker Q = \mathop{\rm Im}L = \mathop{\rm Im}(I-Q). $$ It follows that $L$ is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to $L$) $K_p: \mathop{\rm Im}L\to \ker P\cap \mathop{\rm Dom}L$ is given by $$ (K_p(u))(t)= \begin{pmatrix} \int ^t _0 x_1(s){\rm d}s - \frac{1}{\omega}\int^\omega _0\int^s _0 x_1(v){\rm d}v{\rm d}s\\ \int ^t _0 x_2(s){\rm d}s - \frac{1}{\omega}\int^\omega _0\int^s_0x_2(v){\rm d}v{\rm d}s \end{pmatrix}. $$ Thus, \[ (QNu)(t)= \begin{pmatrix} \frac{1}{\omega}\int^\omega _0\{ -a_1(s)x_1(s)+b_1(s)f_1(x_1(s-\tau_1(s)) x_2(s-\tau_2(s)))\}{\rm d}s\\ \frac{1}{\omega}\int^\omega _0\{-a_2(s)x_2(s)+b_2(s)f_2(x_1(s-\tau_3(s)) x_2(s-\tau_4(s)))\}{\rm d}s \end{pmatrix}, \] and \begin{align*} &(K_p(I-Q)Nu)(t)\\ &= \begin{pmatrix} \int ^t _0\{-a_1(s)x_1(s)+b_1(s)f_1(x_1(s-\tau_1(s)), x_2(s-\tau_2(s)))\}{\rm d}s\\ \int ^t _0\{-a_2(s)x_2(s)+b_2(s)f_2(x_1(s-\tau_3(s)), x_2(s-\tau_4(s)))\}{\rm d}s \end{pmatrix}\\ &\quad -\begin{pmatrix} \frac{1}{\omega} \int^\omega _0\int^s _0 \{-a_1(v)x_1(v)+b_1(v)f_1(x_1(v-\tau_1(v)), x_2(v-\tau_2(v)))\}{\rm d}v{\rm d}s\\ \frac{1}{\omega}\int^\omega _0\int^s _0 \{-a_2(v)x_2(v)+b_2(v)f_2(x_1(v-\tau_3(v)),x_2(v-\tau_4(v)))\}{\rm d}v{\rm d}s \end{pmatrix}\\ &\quad +\begin{pmatrix} (\frac{1}{2}-\frac{t}{\omega})\int^\omega _0\{ -a_1(s)x_1(s)+b_1(s)f_1(x_1(s-\tau_1(s)), x_2(s-\tau_2(s)))\}{\rm d}s\\ (\frac{1}{2}-\frac{t}{\omega})\int^\omega _0\{-a_2(s)x_2(s)+b_2(s)f_2(x_1(s-\tau_3(s)),x_2(s-\tau_4(s)))\}{\rm d}s \end{pmatrix}. \end{align*} Clearly, $QN$ and $K_p(I-Q)N$ are continuous. For any bounded open subset $\Omega\subset X$, $QN(\overline{\Omega})$ is obviously bounded. Moreover, applying the Arzela-Ascoli theorem, one can easily show that $\overline{K_p(I-Q)N(\overline{\Omega})}$ is compact. Note that $K_p(I-Q)N$ is a compact operator and $QN(\overline{\Omega})$ is bounded, therefore, N is $L$-compact on $\overline{\Omega}$ for any bounded open subset $\Omega\subset X$. Since $\mathop{\rm Im}Q=\ker L$, we take the isomorphism $J$ of $\mathop{\rm Im}Q$ onto $\ker L$ to be the identity mapping. Corresponding to equation $Lu=\lambda Nu, \lambda\in (0,1)$, we have \begin{equation} \begin{gathered} \dot x_1(t)=\lambda\{-a_{1}(t)x_1(t)+b_{1}(t)f_{1}(x_1(t-\tau_{1} (t)),x_2(t-\tau_{2}(t)))\},\\ \dot x_2(t)=\lambda\{-a_{2}(t)x_2(t)+b_{2}(t)f_{2}(x_1(t-\tau_{3} (t)),x_2(t-\tau_{4}(t)))\}. \end{gathered} \label{e2.1} \end{equation} Now we reach the position to search for an appropriate open bounded subset $\Omega$ for the application of the Lemma \ref{lem2.1}. Assume that $u=u(t)\in X$ is a solution of system \eqref{e2.1}. Then, the components $x_i(t) (i=1, 2 )$ of $u(t)$ are continuously differentiable. Thus, there exists $t_i\in [0, \omega]$ such that $| x_i(t_i)|=\max_{t\in [0, \omega]}| x_i(t)|$. Hence, $\dot x_i(t_i)=0$. This implies \begin{equation} a_{i}(t_i)x_i(t_i)=b_{1}(t_i)f_{i}(x_1(t_i-\tau_{1}(t_i)), x_2(t_i-\tau_{2}(t_i))),\label{e2.2} \end{equation} for $i=1, 2$. Since $$ | f_i(x_1, x_2)|\leq\alpha_i| x_1|+\beta_i| x_2|+M_i \quad \mbox{for} \quad i=1, 2, $$ we get \begin{equation} | x_i(t_i)| \leq\frac{\alpha_i b| x_1(t_i-\tau_{1}(t_i))|}{a}+\frac{\beta_i b| x_2(t_i-\tau_{2}(t_i))|}{a}+\frac{bM_i}{a}, \label{e2.3} \end{equation} for $i=1, 2$. From $k_1=\frac{D_1}{D}$, $k_2=\frac{D_2}{D}$, we find that \begin{equation} \begin{gathered} k_1=\frac{\alpha_1 b}{a}k_1+\frac{\beta_1 b}{a}k_2+\frac{bM_1}{a},\\ k_2=\frac{\alpha_2 b}{a}k_1+\frac{\beta_2 b}{a}k_2+\frac{bM_2}{a}. \end{gathered} \label{e2.4} \end{equation} Now, we choose a constant number $d>1$ and take $$ \Omega=\{(x_1, x_2)^T\in \mathbb{R}^2; | x_i|0$, $k_2=\frac{D_2}{D}>0$. We will show that $\Omega$ satisfies all the requirements given in Lemma \ref{lem2.1}. In fact, we will prove that if $x_i(t_i-\tau_{1}(t_i))\in \Omega$ then $x_i(t_i)\in \Omega $ for $i=1, 2$. Therefore, it means that $u=u(t)$ is uniformly bounded with respect to $\lambda$ when the initial value function belongs to $\Omega$. It follows from \eqref{e2.3} that \begin{align*} | x_i(t_i)| &\leq \frac{\alpha_ib| x_1(t_i-\tau_{1}(t_i))|}{a} +\frac{\beta_ib| x_2(t_i-\tau_{2}(t_i))|}{a}+\frac{bM_i}{a}\\ &< d\big(\frac{\alpha_i b}{a}k_1+\frac{\beta_ib}{a}k_2+\frac{bM_i}{a}\big). \end{align*} This, together with \eqref{e2.4}, implies $| x_i(t_i)| < dk_i$, for $i=1, 2$. Therefore, \begin{equation} \| x_i\| < dk_i\quad \mbox{for} \quad i=1, 2.\label{e2.5} \end{equation} Clearly, $dk_i$, i=1,2, are independent of $\lambda$. It is easy to see that there are no $\lambda\in (0, 1)$ and $u\in \partial\Omega$ such that $Lu=\lambda Nu$. If $u=(x_1, x_2)^T \in \partial\Omega\cap \ker L=\partial\Omega\cap \mathbb{R}^2$, then $u$ is a constant vector in $\mathbb{R}^2$ with $| x_i| =dk_i$ for $i=1, 2$. Note that $QNu=JQNu$, we have \begin{equation} QNu= \begin{pmatrix} -\overline{a_{1}}x_1+\overline{b_{1}}f_1(x_1, x_2)\\ -\overline{a_{2}}x_2+\overline{b_{2}}f_2(x_1, x_2) \end{pmatrix} \label{e2.7} \end{equation} We claim that \begin{equation} |(QNu)_i|>0 \quad \mbox{for } i=1, 2. \label{e2.8} \end{equation} Contrarily, suppose that there exists some $i$ such that $|(QNu)_i|=0$, i.e., $\overline{a_{i}}x_1=\overline{b_{i}}f_i(x_1, x_2)$. So, we have \begin{equation} \begin{aligned} dk_i & = | x_i| \\ &\leq \frac{b}{a}| f_i(x_1, x_2)| \\ &\leq \frac{\alpha_ib}{a}dk_1+\frac{\beta_ib}{a}dk_2+\frac{bM_i}{a} \\ &< \frac{\alpha_ib}{a}dk_1+\frac{\beta_ib}{a}dk_2+d\frac{bM_i}{a} \\ & = dk_i, \end{aligned} \label{e2.9} \end{equation} this is a contradiction. Therefore, \eqref{e2.8} holds, and hence, \begin{equation} QNu\neq 0, \quad \mbox{for any } u\in \partial\Omega\cap \ker L=\partial\Omega\cap \mathbb{R}^2. \label{e2.10} \end{equation} Consider the homotopy $F: (\overline{\Omega}\cap \ker L)\times [0,1]\to \overline{\Omega}\cap \ker L$, defined by \begin{equation} F(u, \mu)=-\mu \mathop{\rm diag} (\overline{a_1}, \overline{a_2})u + (1-\mu)QNu, \label{e2.11} \end{equation} for all $u\in \overline{\Omega}\cap \ker L=\overline{\Omega}\cap \mathbb{R}^2$ and $\mu\in [0, 1]$. When $u\in \partial\Omega\cap \ker L=\partial\Omega\cap \mathbb{R}^2$ and $\mu\in [0, 1]$, $u=(x_1, x_2)^T$ is a constant vector in $\mathbb{R}^2$ with $| x_i| =dk_i$ for $i=1, 2$. Thus \begin{align*} \| F(u, \mu)\|_0 & = \max_{i=1, 2}|-\mu\overline{a_i}x_i + (1-\mu)[-\overline{a_i}x_i+\overline{b_i}f_i(x_1, x_2)]| \\ & = \max_{i=1, 2}|-\overline{a_i}x_i + (1-\mu)\overline{b_i}f_i(x_1, x_2)|. \end{align*} We claim that \begin{equation} \| F(u, \mu)\|_0>0\,. \label{e2.12} \end{equation} Contrarily, suppose that $\| F(u, \mu)\|_0=0$, then $$ \overline{a_i}x_i = (1-\mu)\overline{b_i}f_i(x_1, x_2) \quad \mbox{for } i=1, 2. $$ Thus \begin{align*} dk_i &= | x_i|\\ &= (1-\mu)\frac{\overline{b_i}}{\overline{a_i}}| f_i(x_1, x_2)|\\ &\leq \frac{b}{a}| f_i(x_1, x_2)|\\ &\leq \frac{\alpha_ib}{a}dk_1+ \frac{\beta_ib}{a}dk_2+\frac{bM_i}{a}\\ & < \frac{\alpha_ib}{a}dk_1+\frac{\beta_ib}{a}dk_2+d\frac{bM_i}{a}\\ & = dk_i. \end{align*} This is impossible. Thus, \eqref{e2.12} holds. Therefore, $$ F(u, \mu)\neq0 \quad \mbox{for } (u, \mu)\in(\partial\Omega\cap \ker L)\times [0,1]. $$ From the property of invariance under a homotopy, it follows that \begin{align*} \deg\{JQN, \Omega\cap \ker L,0\} & = \deg\{F(\cdot\quad,0), \Omega\cap \ker L,0\} \\ & = \deg\{F(\cdot\quad,1), \Omega\cap \ker L,0\} \\ & = \mathop{\rm sgn} \Bigg| \begin{matrix} -\overline{a_1}&0 \\ 0&-\overline{a_2} \end{matrix}\Bigg| \\ & = \mathop{\rm sgn} \{\overline{a_1}\cdot\overline{a_2} \} \neq 0. \end{align*} We have shown that $\Omega$ satisfies all the assumptions of Lemma \ref{lem2.1}. Hence, $Lu=Nu$ has at least one $\omega$-periodic solution on $\mathop{\rm Dom}L\cap\overline{\Omega}$. This completes the proof. \end{proof} \begin{corollary}\label{coro2.3} Suppose there exist positive constants $M_i$ such that $| f_i(x_1, x_2)|\leq M_i$ for $i=1, 2$. Then system \eqref{e1.4} has at least an $\omega$-periodic solution. \end{corollary} \begin{proof} Since $| f_i(x_1, x_2)|\leq M_i$ ($i=1, 2,$) implies that $\alpha_i, \beta_i=0$, hence the conditions in Theorem \ref{thm2.2} are all satisfied. \end{proof} \subsection*{Acknowledgment} The authors wish to thank the anonymous referee for his/her valuable comments that led to the improvement of the original manuscript. \begin{thebibliography}{99} \bibitem{b1} M. Baptistini and P. T\'{a}boas; \emph{On the existence and global bifurcation of periodic solutions to planar differential delay equations}, J. Diff. Eqns., 127 (1996), 391-425. \bibitem{d1} K. Deimling; \emph{Nonlinear Functional Analysis}, Springer, NewYork, 1985. \bibitem{f1} T. Faria; \emph{On a planar system modelling a neuron network with memory}, J. Differential Equations, 168(2000), 129-149. \bibitem {g1} K. Gopalsamy and I. Leung; \emph{Delay induced periodicity in a neural netlet of excitation and inhibition}, Physica D 89(1996), 395-426. \bibitem{g2} R. E. Gaines and J. L. Mawhin; \emph{Coincidence Degree and Nonlinear Differenial Equations}, Springer-Verlag, Berlin, 1977. \bibitem{g3} S. M. S. Godoy and J. G. Dos. Reis; \emph{Stability and existence of periodic solutions of a functional differential equation}, J. Math. Analysis Appl., 198 (1996), 381-398. \bibitem{o1} L. Olien and J. B\'{e}lair; \emph{Bifurcations, stability, and monotonicity properties of a delayed neural network model}, Physica D 102(1997), 349-363. \bibitem{r1} S. Ruan and J. Wei; \emph{Periodic solutions of planar systems with two delays}, Proc. R. Soc. Edinb., 129A(1999), 1017-1032. \bibitem{t1} P. T\'{a}boas; \emph{Periodic solution of a planar delay equation}, Proc. R. Soc. Edinb., 116A (1990), 85-101. \bibitem{w1} J. Wu; \emph{Introduction to Neural Dynamics and Signal Transmission Delay}, Walter de Cruyter, Berlin, 2001. \bibitem{z1} Z. Q. Zhang and Z. C. Wang; \emph{Periodic solutions of planar systems with four delays}, Ann. of Diff. Eqs. 17: 4(2001), 400-408. \end{thebibliography} \end{document}