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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 805846, 21 pages
http://dx.doi.org/10.1155/2012/805846
Research Article

Existence and Global Exponential Stability of Periodic Solution to Cohen-Grossberg BAM Neural Networks with Time-Varying Delays

College of Mathematics and Econometrics, Hunan University, Changsha 410082, China

Received 16 December 2011; Revised 4 February 2012; Accepted 6 February 2012

Academic Editor: Agacik Zafer

Copyright © 2012 Kaiyu Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate first the existence of periodic solution in general Cohen-Grossberg BAM neural networks with multiple time-varying delays by means of using degree theory. Then using the existence result of periodic solution and constructing a Lyapunov functional, we discuss global exponential stability of periodic solution for the above neural networks. Our result on global exponential stability of periodic solution is different from the existing results. In our result, the hypothesis for monotonicity ineqiality conditions in the works of Xia (2010) Chen and Cao (2007) on the behaved functions is removed and the assumption for boundedness in the works of Zhang et al. (2011) and Li et al. (2009) is also removed. We just require that the behaved functions satisfy sign conditions and activation functions are globally Lipschitz continuous.

1. Introduction

In 1983, Cohen and Grossberg [1] constructed a kind of simplified neural networks that are now called Cohen-Grossberg neural networks (CGNNs); they have received increasing interesting due to their promising potential applications in many fields such as pattern recognition, parallel computing, associative memory, and combinatorial optimization. Such applications heavily depend on the dynamical behaviors. Thus, the qualitative analysis of the dynamical behaviors is a necessary step for the practical design and application of neural networks (or neural system [24]). The stability of Cohen-Grossberg neural network with or without delays has been widely studied by many researchers, and various interesting results have been reported [514].

On the other hand, since the pioneering work of Kosko [15, 16], a series of neural networks related to bidirectional associative memory models have been proposed. These models generalized the single-layer autoassociative Hebbian correlator to a class of two-layer pattern-matched heteroassociative circuits. Bidirectional associative memory neural networks have also been used in many fields such as pattern recognition and automatic control and image and signal processing. During the last years, many authors have discussed the existence and global stability of BAM neural networks [1720]. In recent years, a few authors [17, 2126] discussed global stability of Cohen-Grossberg BAM neural networks.

As is well known, the studies on neural dynamical system not only involve a discussion of stability properties but also involve other dynamic behavior, such as periodic oscillatory behavior, chaos, and bifurcation. In many applications, periodic oscillatory behavior is of great interest; it has been found in applications in learning theory. Hence, it is of prime importance to study periodic oscillatory solutions of neural networks.

This motivates us to consider periodic solutions of Cohen-Grossberg BAM neural networks. Recently, a few authors discussed the existence and stability of periodic solution to Cohen-Grossberg BAM neural networks with delays [2731].

In [27], the authors proposed a class of bidirectional Cohen-Grossberg neural networks with distributed delays as follows:d𝑥𝑖(𝑡)d𝑡=𝑎𝑖𝑥𝑖𝑏(𝑡)𝑖𝑡,𝑥𝑖(𝑡)𝑚𝑗=1𝑝𝑖𝑗(𝑡)0𝐾𝑗𝑖(𝑢)×𝑓𝑗𝑡,𝜆𝑗𝑦𝑗(𝑡𝑢)d𝑢𝐼𝑖(𝑡),𝑖=1,2,,𝑛,d𝑦𝑗(𝑡)d𝑡=𝑐𝑗𝑦𝑗𝑑(𝑡)𝑗𝑡,𝑦𝑗(𝑡)𝑛𝑖=1𝑞𝑗𝑖(𝑡)0𝐿𝑖𝑗(𝑢)×𝑔𝑖𝑡,𝜇𝑖𝑥𝑖(𝑡𝑢)d𝑢𝐽𝑗(𝑡),𝑗=1,2,,𝑚.(1.1) By using the Lyapunov functional method and some analytical techniques, some sufficient conditions were obtained for global exponential stability of periodic solutions to these networks.

In [28], the authors discussed the following Cohen-Grossberg-type BAM neural networks with time-varying delays: d𝑥𝑖(𝑡)d𝑡=𝑎𝑖𝑥𝑖𝑏(𝑡)𝑖𝑥𝑖(𝑡)𝑚𝑗=1𝑝𝑖𝑗(𝑡)𝑓𝑗𝜆𝑗𝑦𝑗𝑡𝜏𝑖𝑗(𝑡)𝐼𝑖(𝑡),𝑖=1,2,,𝑛,d𝑦𝑗(𝑡)d𝑡=𝑐𝑗𝑦𝑗𝑑(𝑡)𝑗𝑦𝑗(𝑡)𝑛𝑖=1𝑞𝑗𝑖(𝑡)𝑔𝑖𝜇𝑖𝑥𝑖𝑡𝜎𝑗𝑖(𝑡)𝐽𝑗(𝑡),𝑗=1,2,,𝑚,(1.2) where 𝑛,𝑚2 are the number of neurons in the networks with initial value conditions: 𝑥𝑖(𝜃)=𝜙𝑖(𝜃),𝜃𝑟1,0,𝑦𝑗(𝜃)=𝜙𝑗(𝜃),𝜃𝑟2,0,(1.3) where 𝑟1=max1𝑖𝑛,1𝑗𝑚,0𝑡𝜔{𝜎𝑗𝑖(𝑡)}, 𝑟2=max1𝑖𝑛,1𝑗𝑚,0𝑡𝜔{𝜏𝑖𝑗(𝑡)}, 𝑎𝑖(𝑥𝑖(𝑡)), 𝑏𝑖(𝑥𝑖(𝑡)), 𝑐𝑗(𝑦𝑗(𝑡)), 𝑑𝑗(𝑦𝑗(𝑡)) are continuous functions, 𝑓𝑗(𝜆𝑗𝑦𝑗(𝑡𝜏𝑗𝑖(𝑡))),𝑔𝑖(𝜇𝑖𝑥𝑖(𝑡𝛿𝑖𝑗(𝑡))) are continuous functions, 𝜆𝑗,𝜇𝑖 are parameters, 𝐼𝑖(𝑡) and 𝐽𝑗(𝑡) are continuous functions, 𝑥𝑖 and 𝑦𝑗 denote the state variables of the 𝑖th neurons from the neural field 𝐹𝑈 and the 𝑗th neurons from the neural field 𝐹𝑉 at time 𝑡, respectively, 𝑎𝑖(𝑥𝑖(𝑡))>0,𝑐𝑗(𝑦𝑗(𝑡))>0 represent amplification functions of the 𝑖th neurons from the neural field 𝐹𝑈 and the 𝑗th neurons from the neural field 𝐹𝑉, respectively, 𝑏𝑖(𝑥𝑖(𝑡)),𝑑𝑗(𝑦𝑗(𝑡)) are appropriately behaved functions of the 𝑖th neurons from the neural field 𝐹𝑈 and the 𝑗th neurons from the neural field 𝐹𝑉, respectively, 𝑓𝑗,𝑔𝑖 are the activation functions of the 𝑗th neurons from the neural field 𝐹𝑉 and the 𝑖th neurons from the neural field 𝐹𝑈, respectively, 𝐼𝑖,𝐽𝑗 are the exogenous inputs of the 𝑖th neurons from the neural field 𝐹𝑈 and the jth neurons from the neural field 𝐹𝑉, respectively, 𝑝𝑖𝑗 and 𝑞𝑗𝑖 are the connection weights, which denote the strengths of connectivity between the neuron 𝑗 from the neural field 𝐹𝑉 and the neuron 𝑖 from the neural field 𝐹𝑈, and 𝜏𝑖𝑗(𝑡),𝜎𝑖𝑗(𝑡) correspond to the transmission time delays.

By using the analysis method and inequality technique, some sufficient conditions were obtained to ensure the existence, uniqueness, global attractivity, and exponential stability of the periodic solution to this neural networks.

In [29, 30], the authors discussed, respectively, two Cohen-Grossberg BAM neural networks on time scales. When time scale 𝑇 becomes 𝑅, the existence and global exponential stability of periodic solution are obtained in [29, 30] under the assumptions that activation functions satisfy global Lipschitz conditions and boundedness conditions and behaved functions satisfy some inequality conditions.

In [31], the authors discussed the following Cohen-Grossberg BAM neural networks of neutral type with delays: d𝑥𝑖(𝑡)d𝑡=𝑎𝑖𝑥𝑖𝑏(𝑡)𝑖𝑥𝑖(𝑡)𝑚𝑗=1𝑎𝑖𝑗(𝑡)𝑓𝑗𝑦𝑗𝑡𝜏𝑖𝑗(𝑡)𝑚𝑗=1𝑏𝑖𝑗(𝑡)𝑓𝑗𝑦𝑗𝑡𝜎𝑖𝑗(𝑡)𝐼𝑖(𝑡),𝑖=1,2,,𝑛,d𝑦𝑗(𝑡)d𝑡=𝑐𝑗𝑦𝑗𝑑(𝑡)𝑗𝑦𝑗(𝑡)𝑛𝑖=1𝑐𝑗𝑖(𝑡)g𝑖𝑥𝑖𝑡𝑝𝑗𝑖(𝑡)𝑛𝑖=1𝑑𝑗𝑖(𝑡)𝑔𝑖𝑥𝑖𝑡𝑞𝑗𝑖(𝑡)𝐽𝑗(𝑡),𝑗=1,2,,𝑚.(1.4) Under the assumptions that activation functions satisfy global Lipschitz conditions and behaved functions satisfy some inequality conditions, global exponential stability of periodic solution is obtained for system (1.4).

In this paper, our purpose is to obtain a new sufficient condition for the existence and global exponential stability of periodic solution of system (1.2). The paper is organized as follows. In Section 2, we discuss the existence of periodic solution of system (1.2) by using coincidence degree theory and inequality technique. In Section 3, we study the global exponential stability of periodic solution of system (1.2) by using the existence result of periodic solution and constituting Lyapunov functional. Our result on global exponential stability of periodic solution is different from the existing results. In our result, the hypotheses for monotonicity inequalities in [27, 28] on behaved functions are replaced with sign conditions and the assumption for boundedness in [29, 30] on activation functions is removed.

2. Existence of Periodic Solution

In this section, we first establish the existence of at least a periodic solution by applying the coincidence degree theory. To establish the existence of at least a periodic solution by applying the coincidence degree theory, we recall some basic tools in the frame work of Mawhin’s coincidence degree [32] that will be used to investigate the existence of periodic solutions.

Let 𝑋, 𝑍 be Banach spaces, 𝐿: Dom𝐿𝑋𝑍 a linear mapping, and 𝑁𝑋𝑍 a continuous mapping. The mapping 𝐿 will be called a Fredholm mapping of index zero if dimKer𝐿=codimIm𝐿< and Im𝐿 is closed in 𝑍. If 𝐿 is a Fredholm mapping of index zero, then there exist continuous projectors 𝑃𝑋Ker𝐿 and 𝑄𝑍𝑍/Im𝐿 such that Im𝑃=Ker𝐿 and Im𝐿=Ker𝑄=Im(𝐼𝑄). It follows that 𝐿/Dom𝐿Ker𝑃(𝐼𝑃)𝑋Im𝐿 is invertible. We denote the inverse of the map 𝐿/Dom𝐿Ker𝑃 by 𝐾𝑝. If Ω is an open bounded subset of 𝑋, the mapping 𝑁 will be called 𝐿-compact on Ω if (𝑄𝑁)(Ω) is bounded and 𝐾𝑝(𝐼𝑄)𝑁Ω𝑋 is compact. Since Im 𝑄 is isomorphic to Ker 𝐿, there exists an isomorphism 𝐽Im𝑄Ker𝐿.

In the proof of our existence theorem, we will use the continuation theorem of Gaines and Mawhin [32].

Lemma 2.1 (continuation theorem). Let 𝐿 be a Fredholm mapping of index zero, and let 𝑁 be 𝐿-compact on Ω. Suppose(a)𝐿𝑥𝜆𝑁(𝑥),forall𝜆(0,1),𝑥𝜕Ω,(b)𝑄𝑁(𝑥)0,forall𝑥Ker𝐿𝜕Ω,(c)deg(𝐽𝑄𝑁𝑥,ΩKer𝐿,0)0.
Then, 𝐿𝑥=𝑁𝑥 has at least one solution in Dom𝐿Ω.

For the sake of convenience, we introduce some notations.

|| denotes the norm in 𝑅, 𝑓=max0𝑡𝜔|𝑓(𝑡)|,𝑓=min0𝑡𝜔|𝑓(𝑡)|, where 𝑓(𝑡) is a continuously periodic function with common period 𝜔. Our main result on the existence of at least a periodic solution for system (1.2) is stated in the following theorem.

Theorem 2.2. One assume that the following conditions holds:(i)𝑝𝑖𝑗(𝑡),𝑞𝑗𝑖(𝑡),𝐼𝑖(𝑡),𝐽𝑗(𝑡) are continuously periodic functions on 𝑡[0,+) with common period 𝜔>0,𝑖=1,2,,𝑛,𝑗=1,2,,𝑚;(ii)𝑎𝑖() and 𝑐𝑗() are continuously bounded, that is, there exist positive constants 𝑙𝑖,𝑙𝑖,𝑘𝑗,𝑘𝑗(𝑖=1,,𝑛,𝑗=1,,𝑚) such that𝑙𝑖𝑎𝑖𝑙𝑖,𝑘𝑗𝑐𝑗𝑘𝑗;(2.1)(iii)𝑏𝑖(𝑥𝑖(𝑡)) and 𝑑𝑗(𝑦𝑗(𝑡)) are continuous and there exist positive constants 𝑀𝑖,𝑁𝑗(𝑖=1,,𝑛,𝑗=1,,𝑚) such that for all 𝑥,𝑦𝑥𝑅,𝑏sign(𝑥𝑦)𝑖(𝑥)𝑏𝑖(𝑦)𝑀𝑖||||,𝑑𝑥𝑦sign(𝑥𝑦)𝑗(𝑥)𝑑𝑗(𝑦)𝑁𝑗||||;𝑥𝑦(2.2)(iv) there exist positive constants A𝑗,𝐵𝑖(𝑖=1,,𝑛,𝑗=1,2,,𝑚) such that for all 𝑥,𝑦𝑅,||𝑓𝑗(𝑥)𝑓𝑗||(𝑦)𝐴𝑗||||,||𝑔𝑥𝑦𝑖(𝑥)𝑔𝑖||(𝑦)𝐵𝑖||||;𝑥𝑦(2.3)(v) there exist two positive constants 𝑟𝑖>1,𝑖=1,2 with 𝜏𝑖𝑗<min{1,1𝑟11}<1 and 𝜎𝑗𝑖<min{1,1𝑟21}<1 such that for 𝑖=1,,𝑛;𝑗=1,,𝑚,𝑙𝑖𝑀𝑖>𝑚𝑗=1𝑙𝑖𝑝𝑖𝑗𝐴𝑗𝜆𝑗𝑟1,𝑘𝑗𝑁𝑗>𝑛𝑖=1𝑘𝑗𝑞𝑗𝑖𝐵𝑖𝜇𝑖𝑟2.(2.4)
Then, system (1.2) has at least one 𝜔-periodic solution.

Proof. In order to apply Lemma 2.1 to system (1.2), let 𝑥𝑋=𝑢=1,𝑥2,,𝑥𝑛,𝑦1,𝑦2,,𝑦𝑚𝑇𝐶𝑅,𝑅𝑚+𝑛,𝑢(𝑡+𝜔)=𝑢(𝑡)𝑍=𝑧𝐶𝑅,𝑅𝑚+𝑛.𝑧(𝑡+𝜔)=𝑧(𝑡)(2.5) Define 𝑢=max𝑛𝑡[0,𝜔]𝑖=1||𝑥𝑖||(𝑡)+max𝑚𝑡[0,𝜔]𝑗=1||𝑦𝑗||(𝑡),𝑢𝑋or𝑍.(2.6) Equipped with the above norm ,𝑋 and 𝑍 are Banach spaces.
Let for 𝑢𝑋𝐻𝑁𝑢=𝑖𝐾(𝑡)𝑗=(𝑡)𝑎𝑖𝑥𝑖𝑏(𝑡)𝑖𝑥𝑖(𝑡)𝑚𝑗=1𝑝𝑖𝑗(𝑡)𝑓𝑗𝜆𝑗𝑦𝑗𝑡𝜏𝑖𝑗(𝑡)𝐼𝑖(𝑡)𝑐𝑗𝑦𝑗𝑑(𝑡)𝑗𝑦𝑗(𝑡)𝑛𝑖=1𝑞𝑗𝑖(𝑡)𝑔𝑖𝜇𝑖𝑥𝑖𝑡𝜎𝑗𝑖(𝑡)𝐽𝑗,(𝑡)𝐿𝑢=𝑢=d𝑢(𝑡)1d𝑡,𝑃𝑢=𝜔𝜔01𝑢(𝑡)d𝑡,𝑢𝑋,𝑄𝑧=𝜔𝜔0𝑧(𝑡)d𝑡,𝑧𝑍.(2.7) Then, it follows that Ker𝐿=𝑅(𝑚+𝑛), Im𝐿={𝑧𝑍𝜔0𝑧(𝑡)𝑑𝑡=0} is closed in 𝑍, dimKer𝐿=𝑚+𝑛=codimIm𝐿, and 𝑃,𝑄 are continuous projectors such that Im𝑃=Ker𝐿,Ker𝑄=Im𝐿=Im(𝐼𝑄).(2.8) Hence, 𝐿 is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to 𝐿) 𝐾𝑝Im𝐿Ker𝑃Dom𝐿 is given by 𝐾𝑝(𝑧)=𝑡01𝑧(𝑠)d𝑠𝜔𝜔0𝑠0𝑧(𝑡)d𝑡d𝑠.(2.9) Then, 1𝑄𝑁𝑢=𝜔𝜔0𝐻11(𝑠)d𝑠𝜔𝜔0𝐻21(𝑠)d𝑠𝜔𝜔0𝐻𝑛1(𝑠)d𝑠𝜔𝜔0𝐾11(𝑠)d𝑠𝜔𝜔0𝐾21(𝑠)d𝑠𝜔𝜔0𝐾𝑚𝐾(𝑠)d𝑠,(2.10)𝑝𝑓(𝐼𝑄)𝑁𝑢=𝑡0𝐻11(𝑠)d𝑠𝜔𝜔0𝑡0𝐻11(𝑠)d𝑠d𝑡+2𝑡𝜔𝜔0𝐻1𝑓(𝑠)d𝑠𝑡0𝐻21(𝑠)d𝑠𝜔𝜔0𝑡0𝐻21(𝑠)d𝑠d𝑡+2𝑡𝜔𝜔0𝐻2𝑓(𝑠)d𝑠𝑡0𝐻𝑛1(𝑠)d𝑠𝜔𝜔0𝑡0𝐻𝑛1(𝑠)d𝑠d𝑡+2𝑡𝜔𝜔0𝐻𝑛𝑓(𝑠)d𝑠𝑡0𝐾11(𝑠)d𝑠𝜔𝜔0𝑡0𝐾11(𝑠)d𝑠d𝑡+2𝑡𝜔𝜔0𝐾1𝑓(𝑠)d𝑠𝑡0𝐾𝑚1(𝑠)d𝑠𝜔𝜔0𝑡0𝐾𝑚1(𝑠)d𝑠d𝑡+2𝑡𝜔𝜔0𝐾𝑚.(𝑠)d𝑠(1) Obviously, 𝑄𝑁 and 𝐾𝑃(𝐼𝑄)𝑁 are continuous. It is not difficult to show that 𝐾𝑝(𝐼𝑄)𝑁(Ω) is compact for any open bounded set Ω𝑋 by using the Arzela-Ascoli theorem. Moreover, 𝑄𝑁(Ω) is clearly bounded. Thus, 𝑁 is 𝐿-compact on Ω with any open bounded set Ω𝑋.
Condition (iii) in Theorem 2.2 implies that for all 𝑥𝑅sign𝑥𝑏𝑖(𝑥)𝑀𝑖|𝑥|+sign𝑥𝑏𝑖(0),sign𝑥𝑑𝑗(𝑥)𝑁𝑗|𝑥|+sign𝑥𝑑𝑗(0).(2.11) Condition (iv) in Theorem 2.2 implies that for all𝑥𝑅||𝑓𝑗||(𝑥)𝐴𝑗||𝑓|𝑥|+𝑗||,||𝑔(0)𝑖||(𝑥)𝐵𝑖||𝑔|𝑥|+𝑗||.(0)(2.12) Corresponding to the operator equation 𝐿𝑥=𝜆𝑁𝑥,𝜆(0,1), we have for 𝑖=1,2,,𝑛,𝑗=1,,𝑚d𝑥𝑖(𝑡)d𝑡=𝜆𝐻𝑖(𝑡),d𝑦𝑗(𝑡)d𝑡=𝜆𝐾𝑗(𝑡).(2.13) Assume that 𝑢𝑋 is a solution of system (2.13) for some 𝜆(0,1). Multiplying the first equation of system (2.13) by 𝑥𝑖(𝑡) and integrating over [0,𝜔], we have 𝜔0𝑥𝑖(𝑡)sign𝑥𝑖(𝑡)sign𝑥𝑖×𝑎(𝑡)𝑖𝑥𝑖𝑏(𝑡)𝑖𝑥𝑖(𝑡)𝑚𝑗=1𝑝𝑖𝑗(𝑡)𝑓𝑗𝜆𝑗𝑦𝑗𝑡𝜏𝑖𝑗(𝑡)𝐼𝑖(𝑡)d𝑡=0.(2.14) Multiplying the second equation of system (2.13) by 𝑦𝑗(𝑡) and integrating over [0,𝜔], we have 𝜔0𝑦𝑗(𝑡)sign𝑦𝑗(𝑡)sign𝑦𝑗×𝑐(𝑡)𝑗𝑦𝑗𝑑(𝑡)𝑗𝑦𝑗(𝑡)𝑛𝑖=1𝑞𝑗𝑖(𝑡)𝑔𝑖𝜇𝑖𝑥𝑖𝑡𝜎𝑗𝑖(𝑡)𝐽𝑗(𝑡)d𝑡=0.(2.15) From (2.14) and (2.15), we obtain 𝑙𝑖𝑀𝑖𝜔0||𝑥𝑖||(𝑡)2d𝑡𝑙𝑖𝜔0||𝑥𝑖||(𝑡)𝑎𝑖sign𝑥𝑖(𝑡)𝑏𝑖(0)+𝑚𝑗=1𝑝𝑖𝑗𝐴𝑗𝜆𝑗||𝑦𝑗𝑡𝜏𝑖𝑗||+||𝑓(𝑡)𝑗||+(0)𝐼𝑖𝑘d𝑡,(2.16)𝑖𝑁𝑗𝜔0||𝑦𝑗(||𝑡)2d𝑡𝑘𝑗𝜔0||𝑦𝑗||(𝑡)𝑐𝑗sign𝑦𝑗(𝑡)𝑑𝑗(0)+𝑛𝑖=1𝑞𝑗𝑖𝐵𝑖𝜇𝑖||𝑥𝑖𝑡𝜎𝑗𝑖||+||𝑔(𝑡)𝑖||+(0)𝐽𝑗d𝑡.(2.17) Hence, 𝑙𝑖𝑀𝑖𝜔0||𝑥𝑖||(𝑡)2d𝑡𝑙𝑖𝜔0||𝑥𝑖||(𝑡)2d𝑡1/2×𝑚𝑗=1𝑝𝑖𝑗𝐴𝑗𝜆𝑗𝜔0||𝑦𝑗𝑡𝜏𝑖𝑗(||𝑡)2d𝑡1/2+𝜔||𝑓𝑗(||0)+𝑙𝑖||𝑏𝑖(||+0)𝜔𝐼𝑖,𝑘(2.18)𝑗𝑁𝑗𝜔0||𝑦𝑗||(𝑡)2d𝑡𝑘𝑗𝜔0||𝑦𝑗||(𝑡)2d𝑡1/2×𝑛𝑖=1𝑞𝑗𝑖𝐵𝑖𝜇𝑖𝜔0||𝑥𝑖𝑡𝜎𝑗𝑖||(𝑡)2d𝑡1/2+𝜔||𝑔𝑖||(0)+𝑘𝑗||𝑑𝑗||+(0)𝜔𝐽𝑗(2.19) Denoting 𝑠=𝑡𝜏𝑖𝑗(𝑡)=𝑔(𝑡),𝜎=𝑡𝜎𝑗𝑖(𝑡)=(𝑡), then 𝜔0||𝑦𝑗𝑡𝜏𝑖𝑗||(𝑡)2d𝑡1/2=𝜔0||𝑦𝑗||(𝑠)21𝜏𝑖𝑗𝑔1(𝑠)d𝑠1/2,(2.20)𝜔0||𝑥𝑖𝑡𝜎𝑗𝑖||(𝑡)2d𝑡1/2=𝜔0||𝑥𝑖||(𝜎)21𝜎𝑗𝑖1(𝜎)d𝜎1/2.(2.21) Substituting (2.20) into (2.18) and substituting (2.21) into (2.19) give for 𝑖=1,,𝑛,𝑗=1,,𝑚𝑙𝑖𝑀𝑖𝜔0||𝑥𝑖||(𝑡)2d𝑡𝑙𝑖𝜔0||𝑥𝑖||(𝑡)2d𝑡1/2×𝑚𝑗=1𝑝𝑖𝑗𝐴𝑗𝜆𝑗𝑟1𝜔0||𝑦𝑗(||𝑡)2d𝑡1/2+𝜔𝑚𝑗=1𝑝𝑖𝑗||𝑓𝑗(||+||𝑏0)𝑖(||+0)𝐼𝑖,𝑘(2.22)𝑗𝑁𝑗𝜔0||𝑦𝑗||(𝑡)2d𝑡𝑘𝑗𝜔0||𝑦𝑗||(𝑡)21/2×𝑛𝑖=1𝑞𝑗𝑖𝐵𝑖𝜇𝑖𝑟2𝜔0||𝑥𝑖||(𝑡)2d𝑡1/2+𝜔𝑛𝑖=1𝑞𝑗𝑖||𝑔𝑖||+||𝑑(0)𝑗||+(0)𝐽𝑗.(2.23) Denoting for the sake of convenience max1𝑖𝑛𝜔0||𝑥𝑖||(𝑡)2d𝑡1/2=𝜔0||𝑥𝑖0||(𝑡)2d𝑡1/2,max1𝑗𝑚𝜔0||𝑦𝑗||(𝑡)2d𝑡1/2=𝜔0||𝑦𝑗0||(𝑡)2d𝑡1/2,(2.24) where, 𝑖0{1,2,,𝑛},𝑗0{1,2,,𝑚}, and from (2.22) and (2.23), we obtain 𝑙𝑖0𝑀𝑖0𝜔0||𝑥𝑖0||(𝑡)2d𝑡1/2𝑙𝑖0𝑚𝑗=1𝑝𝑖0𝑗𝐴𝑗𝜆𝑗𝑟1𝜔0||𝑦𝑗0||(𝑡)2d𝑡1/2+𝑙𝑖0𝜔𝑚𝑗=1𝑝𝑖0𝑗||𝑓𝑗||+||𝑏(0)𝑖0||+(0)𝐼𝑖0,𝑘(2.25)𝑗0𝑁𝑗0𝜔0||𝑦𝑗0||(𝑡)2d𝑡1/2𝑘𝑗0𝑛𝑖=1𝑞𝑗0𝑖𝐵𝑖𝜇𝑖𝑟2𝜔0||𝑥𝑖0||(𝑡)2d𝑡1/2+𝑘𝑗0𝜔𝑛𝑖=1𝑞𝑗0𝑖||𝑔𝑖||+||𝑑(0)𝑗0||+(0)𝐽𝑗0.(2.26) Now we consider two possible cases for (2.26) and (2.25): (i)𝜔0||𝑦𝑗0||(𝑡)2d𝑡1/2𝜔0||𝑥𝑖0||(𝑡)2d𝑡1/2,(ii)𝜔0||𝑦𝑗0||(𝑡)2d𝑡1/2>𝜔0||𝑥𝑖0||(𝑡)2d𝑡1/2.(2.27) When (𝜔0|𝑦𝑗0(𝑡)|2d𝑡)1/2(𝜔0|𝑥𝑖0(𝑡)|2d𝑡)1/2, from (2.25), we have 𝑙𝑖0𝑀𝑖0𝑙𝑖0𝑚𝑗=1𝑝𝑖0𝑗𝐴𝑗𝜆𝑗𝑟1𝜔0||𝑥𝑖0||(𝑡)2d𝑡1/2𝑙𝑖0𝜔𝑚𝑗=1𝑝𝑖0𝑗||𝑓𝑗||+||𝑏(0)𝑖0||+(0)𝐼𝑖0.(2.28) Thus, 𝜔0||𝑥𝑖0||(𝑡)2d𝑡1/2𝑙𝑖0𝜔𝑚𝑗=1𝑝𝑖0𝑗||𝑓𝑗||+||𝑏(0)𝑖0||+(0)𝐼𝑖0𝑙𝑖0𝑀𝑖0𝑙𝑖0𝑚𝑗=1𝑝𝑖0𝑗𝐴𝑗𝜆𝑗𝑟1max1𝑖𝑛𝑙𝑖𝜔𝑚𝑗=1𝑝𝑖𝑗||𝑓𝑗||+||𝑏(0)𝑖||+(0)𝐼𝑖𝑙𝑖𝑀𝑖𝑙𝑖𝑚𝑗=1𝑝𝑖𝑗𝐴𝑗𝜆𝑗𝑟1def=𝑑1.(2.29) Therefore, 𝜔0||𝑦𝑗0||(𝑡)2d𝑡1/2𝜔0||𝑥𝑖0||(𝑡)2d𝑡1/2𝑑1.(2.30)
(ii) When (𝜔0|𝑦𝑗0(𝑡)|2d𝑡)1/2>(𝜔0|𝑥𝑖0(𝑡)|2d𝑡)1/2, from (2.26), we have 𝑘𝑗0𝑁𝑗0𝑘𝑗0𝑛𝑖=1𝑞𝑗0𝑖𝐵𝑖𝜇𝑖𝑟2𝜔0||𝑦𝑗0||(𝑡)2d𝑡1/2𝑘𝑗0𝜔𝑛𝑖=1𝑞𝑗0𝑖||𝑔𝑖||+||𝑑(0)𝑗0||+(0)𝐽𝑗0.(2.31) Thus, 𝜔0||𝑦𝑗0||(𝑡)2d𝑡1/2𝑘𝑗0𝜔𝑛𝑖=1𝑞𝑗0𝑖||𝑔𝑖||+||𝑑(0)𝑗0||+(0)𝐽𝑗0𝑘𝑗0𝑁𝑗0𝑘𝑗0𝑛𝑖=1𝑞𝑗0𝑖𝐵𝑖𝜇𝑖𝑟2max1𝑗𝑚𝑘𝑗𝜔𝑛𝑖=1𝑞𝑗𝑖||𝑔𝑖||+||𝑑(0)𝑗||+(0)𝐽𝑗𝑘𝑗𝑁𝑗𝑘𝑗𝑛𝑖=1𝑞𝑗𝑖𝐵𝑖𝜇𝑖𝑟2def=𝑑2.(2.32) Therefore, 𝜔0||𝑥𝑖0||(𝑡)2d𝑡1/2𝜔0||𝑦𝑗0||(𝑡)2d𝑡1/2𝑑2.(2.33) Hence, from (2.30) and (2.33), we have for 𝑖=1,2,,𝑛,𝑗=1,2,,𝑚,𝑡[0,𝜔]𝜔0||𝑥𝑖||(𝑡)2d𝑡1/2𝑑<max1,𝑑2def=𝑑,(2.34)𝜔0||𝑦𝑗||(𝑡)2d𝑡1/2𝑑<max1,𝑑2=𝑑.(2.35) Multiplying the first equation of system (2.13) by 𝑥𝑖(𝑡) and integrating over [0,𝜔], from (2.20) and (2.35) and the fact that 𝜔0𝑎𝑖𝑥𝑖𝑏(𝑡)𝑖𝑥𝑖𝑥(𝑡)𝑖(𝑡)d𝑡=0,(2.36) it follows that 𝜔0||𝑥𝑖||(𝑡)2d𝑡1/2𝑙𝑖𝑚𝑗=1𝑝𝑖𝑗𝐴𝑗𝜆𝑗𝜔0||𝑦𝑗𝑡𝜏𝑖𝑗||(𝑡)d𝑡1/2+𝑙𝑖𝜔𝑚𝑗=1𝑝𝑖𝑗||𝑓𝑗||+(0)𝐼𝑖𝑙𝑖𝑚𝑗=1𝑝𝑖𝑗𝐴𝑗𝜆𝑗𝑟1𝜔0||𝑦𝑗||(𝑡)2d𝑡1/2+𝑙𝑖𝜔𝑚𝑗=1𝑝𝑖𝑗||𝑓𝑗||+(0)𝐼𝑖<max1𝑖𝑛𝑙𝑖𝑚𝑗=1𝑝𝑖𝑗𝐴𝑗𝜆𝑗𝑟1𝑑+𝑙𝑖𝜔𝑚𝑗=1𝑝𝑖𝑗||𝑓𝑗||+(0)𝐼𝑖def=𝑐1.(2.37) Similarly, multiplying the second equation of system (2.13) by 𝑦𝑗(𝑡) and integrating over [0,𝜔], from (2.21) and (2.34) and the fact that 𝜔0𝑐𝑗𝑦𝑗𝑑(𝑡)𝑗𝑦𝑗𝑦(𝑡)𝑗(𝑡)d𝑡=0,(2.38) it follows that there exists a positive constant 𝑐2 such that 𝜔0||𝑦𝑗||(𝑡)2d𝑡1/2<𝑐2.(2.39) From (2.34) and (2.35), it follows that there exist points 𝑡𝑖 and 𝑡𝑗 such that ||𝑥𝑖𝑡𝑖||<𝑑𝜔||𝑦,(2.40)𝑗𝑡𝑗||<𝑑𝜔.(2.41)
Since for all 𝑡[0,𝜔], ||𝑥𝑖||||𝑥(𝑡)𝑖𝑡𝑖||+𝜔0||𝑥𝑖||||𝑥(𝑡)d𝑡𝑖𝑡𝑖||+𝜔𝜔0||𝑥𝑖||(𝑡)21/2,||𝑦(2.42)𝑗(||||𝑦𝑡)𝑗𝑡𝑖||+𝜔0||𝑦𝑗(||||𝑦𝑡)d𝑡𝑗||+(𝑡)𝜔𝜔0||𝑦𝑗||(𝑡)21/2,(2.43) then from (2.40)–(2.43), we have for 𝑡[0,𝜔],𝑖=1,,𝑛,𝑗=1,,𝑚||𝑥𝑖||𝑑(𝑡)𝜔+𝜔𝑐1,||𝑦𝑗||𝑑(𝑡)𝜔+𝜔𝑐2.(2.44)
Obviously, 𝑑/𝜔,𝜔𝑐1, and 𝜔𝑐2 are all independent of 𝜆. Now let 𝑥Ω=𝑢=1,𝑥2,,𝑥𝑛;𝑦1,𝑦2,,𝑦𝑚𝑇𝑑𝑋𝑢<𝑛𝜔+𝑟1+𝜔𝑐1𝑑+𝑚𝜔+𝑟2+𝜔𝑐2,(2.45) where 𝑟1,𝑟2 are two chosen positive constants such that the bound of Ω is larger. Then, Ω is bounded open subset of 𝑋. Hence, Ω satisfies requirement (a) in Lemma 2.1. We prove that (b) in Lemma 2.1 holds. If it is not true, then when 𝑢𝜕ΩKer𝐿=𝜕Ω𝑅(𝑚+𝑛) we have =1𝑄𝑁𝑢𝜔𝜔0𝐻11(𝑡)d𝑡,𝜔𝜔0𝐻21(𝑡)d𝑡,,𝜔𝜔0𝐻𝑛1(𝑡)d𝑡;𝜔𝜔0𝐾11(𝑡)d𝑡,,𝜔𝜔0𝐾𝑚(𝑡)d𝑡𝑇=(0,,0)𝑇.(2.46) Therefore, there exist points 𝜉𝑖(𝑖=1,2,,𝑛) and 𝜂𝑗(𝑗=1,2,,𝑚) such that 𝐻𝑖𝜉𝑖𝐾=0,𝑗𝜂𝑗=0.(2.47) From this and following the arguments of (2.40) and (2.41), we have for forall 𝑖=1,2,,𝑛,𝑗=1,2,,𝑚,𝑡[0,𝜔]||𝑥𝑖||<𝑑(𝑡)𝜔,||𝑦𝑗||<𝑑(𝑡)𝜔.(2.48) Hence, 𝑑𝑢<𝑛𝜔𝑑+𝑚𝜔.(2.49) Thus, 𝑢Ω𝑅(𝑚+𝑛). This contradicts the fact that 𝑢𝜕Ω𝑅(𝑚+𝑛). Hence, this proves that (b) in Lemma 2.1 holds. Finally, we show that (c) in Lemma 2.1 holds. We only need to prove that deg{𝐽𝑄𝑁𝑢,ΩKer𝐿,(0,0)𝑇}(0,0,,0)𝑇. Now, we show that deg𝐽𝑄𝑁𝑢,ΩKer𝐿,(0,0,,0)𝑇𝑙=deg1𝑀1𝑥1,𝑙2𝑀2𝑥2,,𝑙𝑛𝑀𝑛𝑥𝑛;𝑘1𝑁1𝑦1,,𝑘𝑚𝑁𝑚𝑦𝑚𝑇,ΩKer𝐿,(0,,0)𝑇.(2.50) To this end, we define a mapping 𝜙Dom𝐿×[0,1]𝑋 by 𝜙𝑥1,𝑥2,,𝑥𝑛;𝑦1,𝑦2,,𝑦𝑚𝜇,𝜇=𝜔𝜔0𝐻1(𝑡)d𝑡,𝜔0𝐻2(𝑡)d𝑡,,𝜔0𝐻𝑛(𝑡)d𝑡,𝜔0𝐾1(𝑡)d𝑡,,𝜔0𝐾𝑚𝑙(𝑡)d𝑡+(1𝜇)1𝑀1𝑥1,𝑙2𝑀2𝑥2,,𝑙𝑛𝑀𝑛𝑥𝑛;𝑘1𝑁1𝑦1,,𝑘𝑚𝑁𝑚𝑦𝑚,(2.51) where 𝜇[0,1] is a parameter. We show that when 𝑢𝜕ΩKer𝐿=𝜕Ω𝑅(𝑚+𝑛), 𝜙(𝑥1,𝑥2,,𝑥𝑛;𝑦1,,𝑦𝑚,𝜇)(0,0,,0)𝑇. If it is not true, then when 𝑢𝜕ΩKer𝐿=𝜕Ω𝑅(𝑚+𝑛), 𝜙(𝑥1,𝑥2,,𝑥𝑛;𝑦1,,𝑦𝑚,𝜇)=(0,0,,0)𝑇. Thus, constant vector 𝑢 with 𝑢𝜕Ω satisfies for 𝑖=1,2,,𝑛,𝑗=1,2,,𝑚, 𝜇𝜔𝜔0𝑎𝑖𝑥𝑖𝑏𝑖𝑥𝑖𝑎𝑖𝑥𝑖𝑚𝑗=1𝑝𝑖𝑗(𝑡)𝑓𝑗𝜆𝑗𝑦𝑗𝐼𝑖(𝑡)d𝑡+(1𝜇)𝑙𝑖𝑀𝑖𝑥𝑖𝜇=0,𝜔𝜔0𝑐𝑗𝑦𝑗𝑑𝑗𝑦𝑗𝑐𝑗𝑦𝑗𝑛𝑖=1𝑞𝑗𝑖(𝑡)𝑔𝑖𝜇𝑖𝑢𝑖𝐽𝑗(𝑡)d𝑡+(1𝜇)𝑘𝑗𝑁𝑗𝑦𝑗=0.(2.52) That is, 𝜇𝜔𝜔0sign𝑥𝑖𝑎𝑖𝑥𝑖𝑏𝑖𝑥𝑖𝑏𝑖(0)+𝑎𝑖𝑥𝑖𝑏𝑖(0)𝑎𝑖𝑥𝑖𝑚𝑗=1𝑝𝑖𝑗(𝑡)𝑓𝑗𝜆𝑗𝑦𝑗𝐼𝑖(𝑡)d𝑡+(1𝜇)𝑙𝑖𝑀𝑖||𝑥𝑖||=0,(2.53)𝜇𝜔𝜔0sign𝑦𝑗𝑐𝑗𝑦𝑗𝑑𝑗𝑦𝑗𝑑𝑗(0)+𝑐𝑗𝑦𝑗𝑑𝑗(0)𝑐𝑗𝑦𝑗𝑛𝑖=1𝑞𝑗𝑖(𝑡)𝑔𝑖𝜇𝑖𝑢𝑖𝐽𝑗+(𝑡)d𝑡(1𝜇)𝑘𝑗𝑁𝑗||𝑦𝑗||=0.(2.54) Denote |𝑦𝑗0|=max1𝑗𝑚{|𝑦𝑗|},|𝑥𝑖0|=max1𝑖𝑛{|𝑥𝑖|}.

Claim 1. We claim that |𝑥𝑖0|<(𝑑/𝜔)+𝜔𝑐1+𝑟1, otherwise, |𝑥𝑖0|(𝑑/𝜔)+𝜔𝑐1+𝑟1. We consider two possible cases: (i) |𝑦𝑗0||𝑥𝑖0| and (ii)|𝑦𝑗0|>|𝑥𝑖0|.(i)When |𝑦𝑗0||𝑥𝑖0|, we have 𝜇𝜔𝜔0sign𝑥𝑖0𝑎𝑖𝑥𝑖0𝑏𝑖𝑥𝑖0𝑏𝑖(0)+𝑎𝑖𝑥𝑖0𝑏𝑖(0)𝑚𝑗=1𝑝𝑖𝑗(𝑡)𝑓𝑗𝜆𝑗𝑦𝑗𝐼𝑖(𝑡)d𝑡+(1𝜇)𝑙𝑖𝑀𝑖||𝑥𝑖0||𝜇𝑙𝑖𝑀𝑖||𝑥𝑖0||𝑙𝑖||𝑏𝑖||+(0)𝑚𝑗=1𝑝𝑖𝑗𝜆𝑗𝐴𝑗||𝑦𝑗||+||𝑓𝑗||+(0)𝐼𝑖+(1𝜇)𝑙𝑖𝑀𝑖||𝑥𝑖0||𝑙𝑖𝑀𝑖||𝑥𝑖0||𝑙𝑖||𝑏𝑖||+(0)𝑚𝑗=1𝑝𝑖𝑗𝜆𝑗𝐴𝑗||𝑦𝑗0||+||𝑓𝑗||+(0)𝐼𝑖𝑙𝑖𝑀𝑖𝑙𝑖𝑚𝑗=1𝐴𝑗𝜆𝑗𝑝𝑖𝑗||𝑥𝑖0||𝑙𝑖||𝑏𝑖||+(0)𝑚𝑗=1𝑝𝑖𝑗𝜆𝑗𝐴𝑗||𝑦𝑗0||+||𝑓𝑗||+(0)𝐼𝑖𝑙𝑖𝑀𝑖𝑙𝑖𝑚𝑗=1𝐴𝑗𝜆𝑗𝑝𝑖𝑗𝑑1𝜔+𝜔𝑐1+𝑟1𝑙𝑖||𝑏𝑖||+(0)𝑚𝑗=1𝑝𝑖𝑗𝜆𝑗𝐴𝑗||𝑦𝑗0||+||𝑓𝑗||+(0)𝐼𝑖>𝑙𝑖𝑀𝑖𝑙𝑖𝑚𝑗=1𝐴𝑗𝜆𝑗𝑝𝑖𝑗𝑟1>0,(2.55)which contradicts (2.53).(ii)When |𝑦𝑗0|>|𝑥𝑖0|, we have 𝜇𝜔𝜔0sign𝑦𝑗0𝑐𝑗𝑦𝑗0𝑑𝑗𝑦𝑗0𝑑𝑗(0)+𝑐𝑗𝑦𝑗0𝑑𝑗(0)𝑛𝑖=1𝑞𝑗𝑖(𝑡)𝑔𝑖𝜇𝑖𝑥𝑖𝐽𝑗(𝑡)dt+(1𝜇)𝑘𝑗𝑁𝑗||𝑦𝑗0||𝜇𝑘𝑗𝑁𝑗||𝑦𝑗0||𝑘𝑗||𝑑𝑗||+(0)𝑛𝑖=1𝑞𝑗𝑖𝜇𝑖𝐵𝑖||𝑥𝑖||+||𝑔𝑖||+(0)𝐽𝑗+(1𝜇)𝑘𝑗𝑁𝑗||𝑦𝑗0||𝑘𝑗𝑁𝑗||𝑦𝑗0||𝑘𝑗||𝑑𝑗||+(0)𝑛𝑖=1𝑞𝑗𝑖𝜇𝑖𝐵𝑖||𝑥𝑖0||+||𝑔𝑖||+(0)𝐽𝑗𝑘𝑗𝑁𝑗𝑘𝑗𝑛𝑖=1𝐵𝑖𝜇𝑖𝑞𝑗𝑖||𝑦𝑗0||𝑘𝑗||𝑑𝑗||+(0)𝑛𝑖=1𝑞𝑗𝑖𝜇𝑖𝐵𝑖||𝑥𝑖0||+||𝑔𝑖||+(0)𝐽𝑗𝑘𝑗𝑁𝑗𝑘𝑗𝑛𝑖=1𝐵𝑖𝜇𝑖𝑞𝑗𝑖𝑑2𝜔+𝜔𝑐1+𝑟1𝑘𝑗||𝑑𝑗||+(0)𝑛𝑖=1𝑞𝑗𝑖𝜇𝑖𝐵𝑖||𝑥𝑖0||+||𝑔𝑖||+(0)𝐽𝑗>𝑘𝑗𝑁𝑗𝑘𝑗𝑛𝑖=1𝐵𝑖𝜇𝑖𝑞𝑗𝑖𝑟1>0,(2.56)which contradicts (2.54). From the discussion of (i) and (ii), Claim 1 holds.

Claim 2. We claim that |𝑦𝑗0|<(𝑑/𝜔)+𝜔𝑐2+𝑟2, otherwise, |𝑦𝑗0|(𝑑/𝜔)+𝜔𝑐2+𝑟2. We consider two possible cases: (i) |𝑥𝑖0||𝑦𝑗0| and (ii) |𝑥𝑖0|>|𝑦𝑗0|.
The proofs of (i) and (ii) are similar to those of (ii) and (1) in Claim 1, respectively, therefore Claim 2 holds.

Thus, |𝑥𝑖|<(𝑑1/𝜔)+𝑐1𝜔+𝑟1 and |𝑦𝑗|<(𝑑2/𝜔)+𝜔𝑐2+𝑟2. Thus, 𝑢Ω𝑅(𝑚+𝑛). This contradicts the fact that 𝑢𝜕Ω𝑅(𝑚+𝑛). According to the topological degree theory and by taking 𝐽=𝐼 since Ker𝐿=Im𝑄, we obtain deg𝐽𝑄𝑁𝑢,ΩKer𝐿,(0,0)𝑇𝜙𝑢=deg1,𝑢2,,𝑢𝑛;𝑣1,𝑣2,,𝑣𝑚,1,ΩKer𝐿,(0,0)𝑇𝜙𝑢=deg1,𝑢2,,𝑢𝑛;𝑣1,𝑣2,,𝑣𝑚,0,ΩKer𝐿,(0,0)𝑇𝑙=deg1𝑀1𝑥1,𝑙2𝑀2𝑥2,,𝑙𝑛𝑀𝑛𝑥𝑛;𝑘1𝑁1𝑦1,,𝑘𝑚𝑁𝑚𝑦𝑚𝑇,ΩKer𝐿,(0,,0)𝑇0.(2.57) So far, we have proved that Ω satisfies all the assumptions in Lemma 2.1. Therefore, system (1.2) has at least one 𝜔-periodic solution.

3. Global Exponential Stability of Periodic Solution

In this section, by constructing a Lyapunov functional, we derive new sufficient conditions for global exponential stability of a periodic solution of system (1.2).

Theorem 3.1. In addition to all conditions in Theorem 2.2, one assumes further that the following conditions hold:(H1) there exists two positive constants 𝑟𝑖1(𝑖=1,2) with 𝑀𝑖>𝑚𝑗=1𝑞𝑗𝑖𝜇𝑖B𝑖𝑟2 and 𝑁𝑗>𝑛𝑖=1𝑝𝑖𝑗𝜆𝑗𝐴𝑗𝑟1 such that 𝜏𝑖𝑗<min{1,1𝑟11}<1 and 𝜎𝑗𝑖<min{1,1𝑟21}<1;(H2) there exist constants 𝜏𝑖𝑗 and 𝜎𝑗𝑖,𝑖=1,2,,𝑛,𝑗=1,2,,𝑚, such that0<𝜏𝑖𝑗(𝑡)<𝜏𝑖𝑗,0<𝜎𝑗𝑖(𝑡)<𝜎𝑗𝑖.(3.1) Then, the 𝜔 periodic solution of system (1.2) is globally exponentially stable.

Proof. By Theorem 2.2, system (1.2) has at least one 𝜔 periodic solution, say, 𝑢(𝑡)=(𝑥1(𝑡),𝑥2(𝑡),,𝑥𝑛(𝑡);𝑦1(𝑡),,𝑦𝑚(𝑡))𝑇. Suppose that 𝑢(𝑡)=(𝑥1(𝑡),𝑥2(𝑡),,𝑥𝑛(𝑡),𝑦1(𝑡),,𝑦𝑚(𝑡))𝑇 is an arbitrary 𝜔 periodic solution of system (1.2). From (H1), we can choose a suitable 𝜃 such that 𝑀𝑖>𝜃𝑙𝑖+𝑚𝑗=1𝑞𝑗𝑖𝜇𝑖𝐵𝑖𝑟2exp𝜃𝜏𝑖𝑗,𝑁𝑗>𝜃𝑘𝑗+𝑛𝑖=1𝑝𝑖𝑗𝜆𝑗𝐴𝑗𝑟𝑖exp𝜃𝜎𝑗𝑖.(3.2) We define a Lyapunov functional as follows for 𝑡>0,𝑖=1,2,,𝑛,𝑗=1,2,,𝑚𝑉(𝑡)=exp(𝜃𝑡)𝑛𝑖=1||||𝑥𝑖𝑥(𝑡)𝑖(𝑡)1𝑎𝑖||||+(𝑠)d𝑠𝑚𝑗=1||||𝑦𝑗𝑦(𝑡)𝑗(𝑡)1𝑐𝑗||||+(𝑠)d𝑠𝑛𝑚𝑖=1𝑗=1𝑝𝑖𝑗𝜆𝑗𝐴𝑗𝑡𝑡𝜏𝑖𝑗(𝑡)𝜃exp𝜎+𝜏𝑖𝑗𝑔1||𝑦(𝜎)𝑗(𝜎)𝑦𝑗||(𝜎)1𝜏𝑖𝑗𝑔1+(𝜎)d𝜎𝑛𝑚𝑖=1𝑗=1𝑞𝑗𝑖𝜇𝑖𝐵𝑖𝑡𝑡𝜎𝑗𝑖(𝑡)𝜃exp𝜎+𝜎𝑗𝑖1||𝑥(𝜎)𝑖(𝜎)𝑥𝑖||(𝜎)1𝜎𝑗𝑖1(𝜎)d𝜎,(3.3) where 𝑔(𝑡)=𝑡𝜏𝑖𝑗(𝑡),(𝑡)=𝑡𝜎𝑗𝑖(𝑡),𝑖=1,2,,𝑛,𝑗=1,,𝑚. Calculating the upper right derivative 𝐷+𝑉(𝑡) of 𝑉(𝑡) along the solutions of system (1.2), we obtain 𝐷+𝑉(𝑡)exp(𝜃𝑡)𝑛𝑖=1𝜃||||𝑥𝑖𝑥(𝑡)𝑖(𝑡)1𝑎𝑖||||(𝑠)d𝑠𝑀𝑖||𝑥𝑖(𝑡)𝑥𝑖||+(𝑡)𝑚𝑗=1𝑝𝑖𝑗𝜆𝑗𝐴𝑗||𝑦𝑗𝑡𝜏𝑖𝑗(𝑡)𝑦𝑗𝑡𝜏𝑖𝑗||(𝑡)+exp(𝜃𝑡)𝑚𝑗=1𝜃||||𝑦𝑗𝑦(𝑡)𝑗(𝑡)1𝑐𝑗||||(𝑠)d𝑠𝑁𝑗||𝑦𝑗(𝑡)𝑦𝑗||+(𝑡)𝑛𝑖=1𝑞𝑗𝑖𝜇𝑖𝐵𝑖||𝑥𝑖𝑡𝜎𝑗𝑖(𝑡)𝑥𝑖𝑡𝜏𝑗𝑖||(𝑡)+exp(𝜃𝑡)𝑛𝑚𝑖=1𝑗=1𝑝𝑖𝑗𝜆𝑗𝐴𝑗||𝑦𝑗(𝑡)𝑦𝑗||(𝑡)exp𝜃𝜏𝑖𝑗𝑠1(𝑡)1𝜏𝑖𝑗𝑠1(||𝑦𝑡)𝑗𝑡𝜏𝑖𝑗(𝑡)𝑦𝑗𝑡𝜏𝑖𝑗||(𝑡)+exp(𝜃𝑡)𝑛𝑚𝑖=1𝑗=1𝑞𝑗𝑖𝜇𝑖𝐵𝑖||𝑥𝑖(𝑡)𝑥𝑖||(𝑡)exp𝜃𝜎𝑗𝑖1(𝑡)1𝜎𝑗𝑖1||𝑥(𝑡)𝑖𝑡𝜎𝑗𝑖(𝑡)𝑥𝑖𝑡𝜎𝑗𝑖||.(𝑡)(3.4) Since there exist points 𝜉𝑖,𝜂𝑗 such that ||||𝑥𝑖𝑥(𝑡)𝑖(𝑡)1𝑎𝑖||||=1(𝑠)d𝑠𝑎𝑖𝜉𝑖||𝑥𝑖(𝑡)𝑥𝑖||,||||(𝑡)𝑦𝑗𝑦(𝑡)𝑗(𝑡)1𝑐j(||||=1𝑠)d𝑠𝑐𝑗𝜂𝑗||𝑦𝑗(𝑡)𝑦𝑗||,(𝑡)(3.5) from (3.4), we have 𝐷+𝑉(𝑡)exp(𝜃𝑡)𝑚𝑗=1𝑁𝑗𝜃𝑘𝑗𝑛𝑖=1𝑝𝑖𝑗𝜆𝑗𝐴𝑗𝑟1exp𝜃𝜎𝑗𝑖||𝑦𝑗(𝑡)𝑦𝑗||(𝑡)exp(𝜃𝑡)𝑛𝑖=1𝑀𝑖𝜃𝑙𝑖𝑚𝑗=1𝑞𝑗𝑖𝜇𝑖𝐵𝑖𝑟2exp𝜃𝜏𝑖𝑗||𝑥𝑖(𝑡)𝑥𝑖||.(𝑡)(3.6) In view of (3.2), it follows that 𝑉(𝑡)<𝑉(0). Therefore, exp(𝜃𝑡)𝑛𝑖=1||||𝑥𝑖𝑥(𝑡)𝑖(𝑡)1𝑎𝑖||||+(𝑠)d𝑠𝑚𝑗=1||||𝑦𝑗𝑦(𝑡)𝑗(𝑡)1𝑐𝑗||||(𝑠)d𝑠<𝑉(𝑡)<𝑉(0).(3.7) Equation (3.3) implies that 𝑉(0)<𝑛𝑖=11𝑙𝑖+𝑚𝑗=1𝑤𝑗𝑖𝜇𝑖𝐵𝑖𝑟20𝜎𝑗𝑖(0)𝜃exp𝜎+𝜎𝑗𝑖1(𝜎)d𝜎sup0𝜔||𝑥𝑖(𝑠)𝑥𝑖||+(𝑠)𝑚𝑗=11𝑘𝑗+𝑛𝑖=1𝑖𝑗𝜆𝑗𝐴𝑗𝑟10𝜏𝑖𝑗𝜃exp𝜎+𝜏𝑖𝑗𝑔1(𝜎)d𝜎sup0𝑠𝜔||𝑦𝑗(𝑠)𝑦𝑗||.(𝑠)(3.8) Substituting (3.8) into (3.7) gives 𝑛𝑖=1||𝑥𝑖(𝑡)𝑥𝑖||+(𝑡)𝑚𝑗=1||𝑦𝑗(𝑡)𝑦𝑗||<𝑀(𝑡)𝑁exp(𝜃𝑡)𝑛𝑖=1sup0𝑠𝜔||𝑥𝑖(𝑠)𝑥𝑖||+(𝑠)𝑚𝑗=1sup0𝑠𝜔||𝑦𝑗(𝑠)𝑦𝑗||,(𝑠)(3.9) where 𝑀=max1𝑖𝑛,1𝑗𝑚1𝑙𝑖+𝑛𝑖=1𝑖𝑗𝜆𝑗𝐴𝑗𝑟10𝜏𝑖𝑗(0)exp𝜃+𝜏𝑖𝑗𝑔11(𝜎)d𝜎,𝑘𝑗+𝑚𝑗=1𝑤𝑗𝑖𝜇𝑖𝐵𝑖𝑟20𝜎𝑗𝑖(0)exp𝜃+𝜎𝑗𝑖1,1(𝜎)d𝜎𝑁=min𝑙𝑖,1𝑘𝑗.(3.10) The proof of Theorem 3.1 is complete.

4. An Example

Example 4.1. Consider the following Cohen-Grossberg BAM neural networks with time-varying delays: d𝑥1(𝑡)d𝑡=2+sin𝑥1200𝑥1(𝑡)+100sin𝑥1|||𝑦(𝑡)(2+sin𝑡)1𝑡1+sin𝑡2|||,sin𝑡d𝑦1(𝑡)d𝑡=3+cos𝑦1200𝑦1(𝑡)+100sin𝑦1|||𝑥(𝑡)(2+cos𝑡)1𝑡1+sin𝑡3|||.cos𝑡(4.1)

In Theorem 3.1, 𝐴1=1,𝐵1=1,𝑙1=1,𝑙1=3,𝑘1=2,𝑘1=4,𝑀1𝑁=100,1=100,𝑝11=3,𝑞11=3,𝜆1=𝜇1𝜏=1,11=cos𝑡2,𝜎11=cos𝑡3.(4.2) Since 1cos𝑡21|co