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

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 [2–4]). 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 [5–14].

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 [17–20]. In recent years, a few authors [17, 21–26] 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 [27–31].

In [27], the authors proposed a class of bidirectional Cohen-Grossberg neural networks with distributed delays as follows:d𝑥_𝑖(𝑡)=d−𝑡𝑎_𝑖d𝑦_𝑗𝑥_𝑖(𝑡)𝑏_𝑖𝑡,𝑥_𝑖(𝑡)−^𝑚_𝑗=1𝑝_𝑖𝑗(𝑡)^∞0𝐾_𝑗𝑖(𝑢)×𝑓_𝑗𝑡,𝜆_𝑗𝑦_𝑗(𝑡−𝑢)d𝑢−𝐼_𝑖(𝑡),𝑖=1,2,…,𝑛,(𝑡)=d−𝑡𝑐_𝑗𝑦_𝑗(𝑡)𝑑_𝑗𝑡,𝑦_𝑗(𝑡)−^𝑛_𝑖=1𝑞_𝑗𝑖(𝑡)^∞0𝐿_𝑖𝑗(𝑢)×𝑔_𝑖𝑡,𝜇_𝑖𝑥_𝑖(𝑡−𝑢)d𝑢−𝐽_𝑗(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: d𝑥_𝑖(𝑡)d𝑡=−𝑎^𝑖𝑥_𝑖(𝑡)𝑏_𝑖𝑥_𝑖(𝑡)−^𝑚_𝑗=1d𝑦_𝑗𝑝_𝑖𝑗(𝑡)𝑓_𝑗𝜆_𝑗𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−𝐼_𝑖(𝑡),𝑖=1,2,…,𝑛,(𝑡)d𝑡=−𝑐^𝑗𝑦_𝑗(𝑡)𝑑_𝑗𝑦_𝑗(𝑡)−^𝑛_𝑖=1𝑞_𝑗𝑖(𝑡)𝑔_𝑖𝜇_𝑖𝑥_𝑖𝑡−𝜎_𝑗𝑖(𝑡)−𝐽_𝑗(𝑡),𝑗=1,2,…,𝑚,(1.2) where 𝑛,𝑚≥2 are the number of neurons in the networks with initial value conditions: 𝑥_𝑖(𝜃)=𝜙_𝑖(𝜃),𝜃∈−𝑟₁,0,𝑦_𝑗(𝜃)=𝜙_𝑗(𝜃),𝜃∈−𝑟₂,0,(1.3) where 𝑟₁=max_{1≤𝑖≤𝑛,1≤𝑗≤𝑚,0≤𝑡≤𝜔}{𝜎_𝑗𝑖(𝑡)}, 𝑟₂=max_{1≤𝑖≤𝑛,1≤𝑗≤𝑚,0≤𝑡≤𝜔}{𝜏_𝑖𝑗(𝑡)}, 𝑎_𝑖(𝑥_𝑖(𝑡)), 𝑏_𝑖(𝑥_𝑖(𝑡)), 𝑐_𝑗(𝑦_𝑗(𝑡)), 𝑑_𝑗(𝑦_𝑗(𝑡)) are continuous functions, 𝑓_𝑗(𝜆_𝑗𝑦_𝑗(𝑡−𝜏_𝑗𝑖(𝑡))),𝑔_𝑖(𝜇_𝑖𝑥_𝑖(𝑡−𝛿_𝑖𝑗(𝑡))) are continuous functions, 𝜆_𝑗,𝜇_𝑖 are parameters, 𝐼_𝑖(𝑡) and 𝐽_𝑗(𝑡) are continuous functions, 𝑥_𝑖 and 𝑦_𝑗 denote the state variables of the 𝑖th neurons from the neural field 𝐹_𝑈 and the 𝑗th neurons from the neural field 𝐹_𝑉 at time 𝑡, respectively, 𝑎_𝑖(𝑥_𝑖(𝑡))>0,𝑐_𝑗(𝑦_𝑗(𝑡))>0 represent amplification functions of the 𝑖th neurons from the neural field 𝐹_𝑈 and the 𝑗th neurons from the neural field 𝐹_𝑉, respectively, 𝑏_𝑖(𝑥_𝑖(𝑡)),𝑑_𝑗(𝑦_𝑗(𝑡)) are appropriately behaved functions of the 𝑖th neurons from the neural field 𝐹_𝑈 and the 𝑗th neurons from the neural field 𝐹_𝑉, respectively, 𝑓_𝑗,𝑔_𝑖 are the activation functions of the 𝑗th neurons from the neural field 𝐹_𝑉 and the 𝑖th neurons from the neural field 𝐹_𝑈, respectively, 𝐼_𝑖,𝐽_𝑗 are the exogenous inputs of the 𝑖th neurons from the neural field 𝐹_𝑈 and the jth neurons from the neural field 𝐹_𝑉, respectively, 𝑝_𝑖𝑗 and 𝑞_𝑗𝑖 are the connection weights, which denote the strengths of connectivity between the neuron 𝑗 from the neural field 𝐹_𝑉 and the neuron 𝑖 from the neural field 𝐹_𝑈, and 𝜏_𝑖𝑗(𝑡),𝜎_𝑖𝑗(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: d𝑥_𝑖(𝑡)d𝑡=−𝑎^𝑖𝑥_𝑖(𝑡)𝑏_𝑖𝑥_𝑖(𝑡)−^𝑚_𝑗=1−𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)_𝑚_𝑗=1d𝑦_𝑗𝑏_𝑖𝑗(𝑡)𝑓_𝑗𝑦_𝑗𝑡−𝜎_𝑖𝑗(𝑡)−𝐼_𝑖(𝑡),𝑖=1,2,…,𝑛,(𝑡)d𝑡=−𝑐^𝑗𝑦_𝑗(𝑡)𝑑_𝑗𝑦_𝑗(𝑡)−^𝑛_𝑖=1−𝑐_𝑗𝑖(𝑡)g_𝑖𝑥_𝑖𝑡−𝑝_𝑗𝑖(𝑡)_𝑛_𝑖=1𝑑_𝑗𝑖(𝑡)𝑔_𝑖𝑥_𝑖𝑡−𝑞_𝑗𝑖(𝑡)−𝐽_𝑗(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: Dom𝐿⊂𝑋→𝑍 a linear mapping, and 𝑁∶𝑋→𝑍 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 𝑃∶𝑋→Ker𝐿 and 𝑄∶𝑍→𝑍/Im𝐿 such that Im𝑃=Ker𝐿 and Im𝐿=Ker𝑄=Im(𝐼−𝑄). It follows that 𝐿/_{Dom𝐿∩Ker𝑃}∶(𝐼−𝑃)𝑋→Im𝐿 is invertible. We denote the inverse of the map 𝐿/_{Dom𝐿∩Ker𝑃} by 𝐾_𝑝. 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 𝐾_𝑝(𝐼−𝑄)𝑁∶Ω→𝑋 is compact. Since Im 𝑄 is isomorphic to Ker 𝐿, there exists an isomorphism 𝐽∶Im𝑄→Ker𝐿.

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)𝐿𝑥≠𝜆𝑁(𝑥),forall𝜆∈(0,1),𝑥∈𝜕Ω,(b)𝑄𝑁(𝑥)≠0,forall𝑥∈Ker𝐿∩𝜕Ω,(c)deg(𝐽𝑄𝑁𝑥,Ω∩Ker𝐿,0)≠0.
Then, 𝐿𝑥=𝑁𝑥 has at least one solution in Dom𝐿∩Ω.

For the sake of convenience, we introduce some notations.

|⋅| denotes the norm in 𝑅, 𝑓=max_{0≤𝑡≤𝜔}|𝑓(𝑡)|,𝑓=min_{0≤𝑡≤𝜔}|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)𝑝_𝑖𝑗(𝑡),𝑞_𝑗𝑖(𝑡),𝐼_𝑖(𝑡),𝐽_𝑗(𝑡) are continuously periodic functions on 𝑡∈[0,+∞) with common period 𝜔>0,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚;(ii)𝑎_𝑖(⋅) and 𝑐_𝑗(⋅) are continuously bounded, that is, there exist positive constants 𝑙_𝑖,𝑙^∗𝑖,𝑘_𝑗,𝑘^∗𝑗(𝑖=1,…,𝑛,𝑗=1,…,𝑚) such that𝑘𝑙_𝑖𝑗≤𝑎_𝑖≤𝑙^∗𝑖,≤𝑐_𝑗≤𝑘^∗𝑗;(2.1)(iii)𝑏_𝑖(𝑥_𝑖(𝑡)) and 𝑑_𝑗(𝑦_𝑗(𝑡)) are continuous and there exist positive constants 𝑀_𝑖,𝑁_𝑗(𝑖=1,…,𝑛,𝑗=1,…,𝑚) such that for all 𝑥,𝑦≠𝑥∈𝑅,ssiiggnn((𝑥𝑥−−𝑦𝑦))𝑑𝑏_𝑖𝑗(𝑥)−𝑏_𝑖(𝑦)≥𝑀_𝑖||𝑥−𝑦||,(𝑥)−𝑑_𝑗(𝑦)≥𝑁_𝑗||𝑥−𝑦||;(2.2)(iv) there exist positive constants A_𝑗,𝐵_𝑖(𝑖=1,…,𝑛,𝑗=1,2,…,𝑚) such that for all 𝑥,𝑦∈𝑅,||𝑓||𝑔_𝑗(𝑥)−𝑓_𝑗(𝑦)||≤𝐴_𝑗||𝑥−𝑦_𝑖||,(𝑥)−𝑔_𝑖(𝑦)||≤𝐵_𝑖||𝑥−𝑦||;(2.3)(v) there exist two positive constants 𝑟_𝑖>1,𝑖=1,2 with 𝜏^𝑖𝑗<min{1,1−𝑟¹⁻¹}<1 and 𝜎^𝑗𝑖<min{1,1−𝑟²⁻¹}<1 such that for 𝑖=1,…,𝑛;𝑗=1,…,𝑚,𝑙_𝑖𝑀_𝑖>^𝑚_𝑗=1𝑘_𝑗𝑙^∗𝑖𝑝_𝑖𝑗𝐴_𝑗𝜆_𝑗√𝑟₁,𝑁_𝑗>^𝑛_𝑖=1𝑘^∗𝑗𝑞_𝑗𝑖𝐵_𝑖𝜇_𝑖√𝑟₂.(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 𝑋=𝑢=𝑥₁,𝑥₂𝑍,…=,𝑥𝑧_𝑛,∈𝑦₁𝐶,𝑦₂𝑅,…,𝑅,𝑦_{𝑚𝑚+𝑛}^𝑇∈𝐶𝑅,𝑅^𝑚+𝑛∶𝑢(𝑡+𝜔)=𝑢(𝑡),∶𝑧(𝑡+𝜔)=𝑧(𝑡).(2.5) Define ‖𝑢‖=max_{𝑛𝑡∈[0,𝜔]}_𝑖=1||𝑥_𝑖(𝑡)||+max_{𝑚𝑡∈[0,𝜔]}_𝑗=1||𝑦_𝑗(𝑡)||,𝑢∈𝑋or𝑍.(2.6) Equipped with the above norm ‖⋅‖,𝑋 and 𝑍 are Banach spaces.
Let for 𝑢∈𝑋𝑁𝑢=⎛⎜⎜⎝𝐻^𝑖𝐾(𝑡)_𝑗(𝑡)⎞⎟⎟⎠=⎛⎜⎜⎜⎜⎜⎜⎝−𝑎^𝑖−𝑐_𝑗𝑥_𝑖(𝑡)𝑏_𝑖𝑥_𝑖(𝑡)−^𝑚_𝑗=1𝑝_𝑖𝑗(𝑡)𝑓_𝑗𝜆_𝑗𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−𝐼_𝑖(𝑡)𝐿𝑢=𝑢_𝑦_𝑗(𝑡)𝑑_𝑗𝑦_𝑗(𝑡)−^𝑛_𝑖=1𝑞_𝑗𝑖(𝑡)𝑔_𝑖𝜇_𝑖𝑥_𝑖𝑡−𝜎_𝑗𝑖(𝑡)−𝐽_𝑗(𝑡)⎞⎟⎟⎟⎟⎟⎟⎠,=d𝑢(𝑡)d𝑡,𝑃𝑢=𝜔1^𝜔0𝑢(𝑡)d𝑡,𝑢∈𝑋,𝑄𝑧=1𝜔^𝜔0𝑧(𝑡)d𝑡,𝑧∈𝑍.(2.7) Then, it follows that Ker𝐿=𝑅^(𝑚+𝑛), Im𝐿={𝑧∈𝑍∶∫^𝜔0𝑧(𝑡)𝑑𝑡=0} is closed in 𝑍, dimKer𝐿=𝑚+𝑛=codimIm𝐿, and 𝑃,𝑄 are continuous projectors such that Im𝑃=Ker𝐿,Ker𝑄=Im𝐿=Im(𝐼−𝑄).(2.8) Hence, 𝐿 is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to 𝐿) 𝐾_𝑝∶Im𝐿→Ker𝑃∩Dom𝐿 is given by 𝐾_𝑝(𝑧)=^𝑡0𝑧(𝑠)d𝑠−1𝜔^𝜔0^𝑠0𝑧(𝑡)d𝑡d𝑠.(2.9) Then, 𝑄𝑁𝑢=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝1𝜔^𝜔01𝐻₁(𝑠)d𝑠𝜔^𝜔01𝐻⋮₂(𝑠)d𝑠𝜔^𝜔01𝐻_𝑛(𝑠)d𝑠𝜔^𝜔01𝐾₁(𝑠)d𝑠𝜔^𝜔01𝐾⋮₂(𝑠)d𝑠𝜔^𝜔0𝐾𝐾_𝑚(𝑠)d𝑠⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,(2.10)_𝑝(𝐼−𝑄)𝑁𝑢=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑓^𝑡0𝐻₁(𝑠)d𝑠−1𝜔^𝜔0^𝑡0𝐻₁(𝑠)d𝑠d𝑡+12−𝜔𝑡^𝜔0𝑓𝐻₁(𝑠)d𝑠_𝑡0𝐻₂(𝑠)d𝑠−1𝜔^𝜔0^𝑡0𝐻₂(𝑠)d𝑠d𝑡+12−𝜔𝑡^𝜔0𝑓⋮𝐻₂(𝑠)d𝑠_𝑡0𝐻_𝑛(𝑠)d𝑠−1𝜔^𝜔0^𝑡0𝐻_𝑛(𝑠)d𝑠d𝑡+12−𝜔𝑡^𝜔0𝑓𝐻_𝑛(𝑠)d𝑠_𝑡0𝐾₁(𝑠)d𝑠−1𝜔^𝜔0^𝑡0𝐾₁(𝑠)d𝑠d𝑡+12−𝜔𝑡^𝜔0𝑓⋮𝐾₁(𝑠)d𝑠_𝑡0𝐾_𝑚(𝑠)d𝑠−1𝜔^𝜔0^𝑡0𝐾_𝑚(𝑠)d𝑠d𝑡+12−𝜔𝑡^𝜔0𝐾_𝑚(𝑠)d𝑠⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(1) Obviously, 𝑄𝑁 and 𝐾_𝑃(𝐼−𝑄)𝑁 are continuous. It is not difficult to show that 𝐾_𝑝(𝐼−𝑄)𝑁(Ω) is compact for any open bounded set Ω⊂𝑋 by using the Arzela-Ascoli theorem. Moreover, 𝑄𝑁(Ω) is clearly bounded. Thus, 𝑁 is 𝐿-compact on Ω with any open bounded set Ω⊂𝑋.
Condition (iii) in Theorem 2.2 implies that for all 𝑥∈𝑅ssiiggnn𝑥𝑥𝑑𝑏_𝑖𝑗(𝑥)≥𝑀_𝑖|𝑥|+sign𝑥𝑏_𝑖(0),(𝑥)≥𝑁_𝑗|𝑥|+sign𝑥𝑑_𝑗(0).(2.11) Condition (iv) in Theorem 2.2 implies that for all𝑥∈𝑅||||𝑓𝑔_𝑗(𝑥)||≤𝐴_𝑗|𝑥|+||𝑓_𝑗(0)_𝑖||,(𝑥)||≤𝐵_𝑖|𝑥|+||𝑔_𝑗(0)||.(2.12) Corresponding to the operator equation 𝐿𝑥=𝜆𝑁𝑥,𝜆∈(0,1), we have for 𝑖=1,2,…,𝑛,𝑗=1,…,𝑚d𝑥_𝑖(𝑡)d𝑡=𝜆𝐻^𝑖d𝑦_𝑗(𝑡),(𝑡)d𝑡=𝜆𝐾^𝑗(𝑡).(2.13) Assume that 𝑢∈𝑋 is a solution of system (2.13) for some 𝜆∈(0,1). Multiplying the first equation of system (2.13) by 𝑥_𝑖(𝑡) and integrating over [0,𝜔], we have ^𝜔0×𝑥_𝑖(𝑡)𝑎sign𝑥_𝑖(𝑡)sign𝑥_𝑖(𝑡)_𝑖𝑥_𝑖(𝑡)𝑏_𝑖𝑥_𝑖(𝑡)−^𝑚_𝑗=1𝑝_𝑖𝑗(𝑡)𝑓_𝑗𝜆_𝑗𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−𝐼_𝑖(𝑡)d𝑡=0.(2.14) Multiplying the second equation of system (2.13) by 𝑦_𝑗(𝑡) and integrating over [0,𝜔], we have ^𝜔0×𝑦_𝑗(𝑡)𝑐sign𝑦_𝑗(𝑡)sign𝑦_𝑗(𝑡)_𝑗𝑦_𝑗(𝑡)𝑑_𝑗𝑦_𝑗(𝑡)−^𝑛_𝑖=1𝑞_𝑗𝑖(𝑡)𝑔_𝑖𝜇_𝑖𝑥_𝑖𝑡−𝜎_𝑗𝑖(𝑡)−𝐽_𝑗(𝑡)d𝑡=0.(2.15) From (2.14) and (2.15), we obtain 𝑙_𝑖𝑀_𝑖^𝜔0≤𝑙_∗𝑖||𝑥_𝑖(𝑡)||²d𝑡^𝜔0||𝑥_𝑖(𝑡)||−𝑎_𝑖sign𝑥_𝑖(𝑡)𝑏_𝑖(0)+^𝑚_𝑗=1𝑝_𝑖𝑗𝐴_𝑗𝜆_𝑗||𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)||+||𝑓_𝑗(0)||+𝐼_𝑖𝑘d𝑡,(2.16)_𝑖𝑁_𝑗^𝜔0≤𝑘_∗𝑗||𝑦_𝑗(𝑡)||²d𝑡^𝜔0||𝑦_𝑗(𝑡)||−𝑐_𝑗sign𝑦_𝑗(𝑡)𝑑_𝑗(0)+^𝑛_𝑖=1𝑞_𝑗𝑖𝐵_𝑖𝜇_𝑖||𝑥_𝑖𝑡−𝜎_𝑗𝑖(𝑡)||+||𝑔_𝑖(0)||+𝐽_𝑗d𝑡.(2.17) Hence, 𝑙_𝑖𝑀_𝑖^𝜔0≤𝑙_∗𝑖||𝑥_𝑖(𝑡)||²d𝑡^𝜔0||𝑥_𝑖(𝑡)||²d𝑡^1/2×^𝑚_𝑗=1𝑝_𝑖𝑗𝐴^𝑗𝜆_𝑗^𝜔0||𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)||²d𝑡^1/2+√𝜔||𝑓_𝑗(0)||+𝑙^∗𝑖||𝑏_𝑖(0)||+√𝜔𝐼_𝑖𝑘,(2.18)_𝑗𝑁_𝑗^𝜔0≤𝑘_∗𝑗||𝑦_𝑗(𝑡)||²d𝑡^𝜔0||𝑦_𝑗(𝑡)||²d𝑡^1/2×^𝑛_𝑖=1𝑞_𝑗𝑖𝐵^𝑖𝜇_𝑖^𝜔0||𝑥_𝑖𝑡−𝜎_𝑗𝑖(𝑡)||²d𝑡^1/2+√𝜔||𝑔_𝑖(0)||+𝑘^∗𝑗||𝑑_𝑗(0)||+√𝜔𝐽_𝑗(2.19) Denoting 𝑠=𝑡−𝜏_𝑖𝑗(𝑡)=𝑔(𝑡),𝜎=𝑡−𝜎_𝑗𝑖(𝑡)=ℎ(𝑡), then ^𝜔0||𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)||²d𝑡^1/2=^𝜔0||𝑦_𝑗(𝑠)||²1−𝜏^𝑖𝑗𝑔⁻¹(𝑠)d𝑠^1/2,(2.20)_𝜔0||𝑥_𝑖𝑡−𝜎_𝑗𝑖(𝑡)||²d𝑡^1/2=^𝜔0||𝑥_𝑖(𝜎)||²1−𝜎^𝑗𝑖ℎ⁻¹(𝜎)d𝜎^1/2.(2.21) Substituting (2.20) into (2.18) and substituting (2.21) into (2.19) give for 𝑖=1,…,𝑛,𝑗=1,…,𝑚𝑙_𝑖𝑀_𝑖^𝜔0≤𝑙_∗𝑖||𝑥_𝑖(𝑡)||²d𝑡^𝜔0||𝑥_𝑖(𝑡)||²d𝑡^1/2×^𝑚_𝑗=1𝑝_𝑖𝑗𝐴_𝑗𝜆_𝑗√𝑟₁^𝜔0||𝑦_𝑗(𝑡)||²d𝑡^1/2+√𝜔^𝑚_𝑗=1𝑝_𝑖𝑗||𝑓_𝑗(0)||+||𝑏_𝑖(0)||+𝐼_𝑖𝑘,(2.22)_𝑗𝑁_𝑗^𝜔0≤𝑘_∗𝑗||𝑦_𝑗(𝑡)||²d𝑡^𝜔0||𝑦_𝑗(𝑡)||²^1/2×^𝑛_𝑖=1𝑞_𝑗𝑖𝐵_𝑖𝜇_𝑖√𝑟₂^𝜔0||𝑥_𝑖(𝑡)||²d𝑡^1/2+√𝜔^𝑛_𝑖=1𝑞_𝑗𝑖||𝑔_𝑖(0)||+||𝑑_𝑗(0)||+𝐽_𝑗.(2.23) Denoting for the sake of convenience max_{1≤𝑖≤𝑛}^𝜔0||𝑥_𝑖(𝑡)||²d𝑡^1/2=^𝜔0max_{1≤𝑗≤𝑚}||𝑥_𝑖0(𝑡)||²d𝑡^1/2,∫^𝜔0||𝑦_𝑗(𝑡)||²d𝑡^1/2=∫^𝜔0||𝑦_𝑗0(𝑡)||²d𝑡^1/2,(2.24) where, 𝑖₀∈{1,2,…,𝑛},𝑗₀∈{1,2,…,𝑚}, and from (2.22) and (2.23), we obtain 𝑙_𝑖0𝑀_𝑖0^𝜔0||𝑥_𝑖0(𝑡)||²d𝑡^1/2≤𝑙^∗𝑖₀^𝑚_𝑗=1𝑝_𝑖0𝑗𝐴_𝑗𝜆_𝑗√𝑟₁^𝜔0||𝑦_𝑗0(𝑡)||²d𝑡^1/2+𝑙^∗𝑖₀√𝜔^𝑚_𝑗=1𝑝_𝑖0𝑗||𝑓_𝑗(0)||+||𝑏_𝑖0(0)||+𝐼_𝑖0𝑘,(2.25)_𝑗0𝑁_𝑗0^𝜔0||𝑦_𝑗0(𝑡)||²d𝑡^1/2≤𝑘^∗𝑗₀^𝑛_𝑖=1𝑞_𝑗0𝑖𝐵_𝑖𝜇_𝑖√𝑟₂^𝜔0||𝑥_𝑖0(𝑡)||²d𝑡^1/2+𝑘^∗𝑗₀√𝜔^𝑛_𝑖=1𝑞_𝑗0𝑖||𝑔_𝑖(0)||+||𝑑_𝑗0(0)||+𝐽_𝑗0.(2.26) Now we consider two possible cases for (2.26) and (2.25): ((iii))^𝜔0_𝜔0||𝑦_𝑗0(𝑡)||²d𝑡^1/2≤^𝜔0||𝑥_𝑖0(𝑡)||²d𝑡^1/2,||𝑦_𝑗0(𝑡)||²d𝑡^1/2>^𝜔0||𝑥_𝑖0(𝑡)||²d𝑡^1/2.(2.27) When (∫^𝜔0|𝑦_𝑗0(𝑡)|²d𝑡)^1/2≤(∫^𝜔0|𝑥_𝑖0(𝑡)|²d𝑡)^1/2, from (2.25), we have 𝑙^𝑖0𝑀_𝑖0−𝑙^∗𝑖₀^𝑚_𝑗=1𝑝_𝑖0𝑗𝐴_𝑗𝜆_𝑗√𝑟₁^𝜔0||𝑥_𝑖0(𝑡)||²d𝑡^1/2≤𝑙^∗𝑖₀√𝜔^𝑚_𝑗=1𝑝_𝑖0𝑗||𝑓_𝑗(0)||+||𝑏_𝑖0(0)||+𝐼_𝑖0.(2.28) Thus, ^𝜔0||𝑥_𝑖0(𝑡)||²d𝑡^1/2≤𝑙^∗𝑖0√𝜔∑^𝑚𝑗=1𝑝_𝑖0𝑗||𝑓_𝑗(0)||+||𝑏_𝑖0(0)||+𝐼_𝑖0𝑙_𝑖0𝑀_𝑖0−𝑙^∗𝑖₀∑^𝑚𝑗=1𝑝_𝑖0𝑗𝐴_𝑗𝜆_𝑗√𝑟₁≤max_{1≤𝑖≤𝑛}⎧⎪⎨⎪⎩𝑙^∗𝑖√𝜔∑^𝑚𝑗=1𝑝_𝑖𝑗||𝑓_𝑗(0)||+||𝑏_𝑖(0)||+𝐼_𝑖𝑙_𝑖𝑀_𝑖−𝑙^∗𝑖∑^𝑚𝑗=1𝑝_𝑖𝑗𝐴_𝑗𝜆_𝑗√𝑟₁⎫⎪⎬⎪⎭_def=𝑑₁.(2.29) Therefore, ^𝜔0||𝑦_𝑗0(𝑡)||²d𝑡^1/2≤^𝜔0||𝑥_𝑖0(𝑡)||²d𝑡^1/2≤𝑑₁.(2.30)
(ii) When (∫^𝜔0|𝑦_𝑗0(𝑡)|²d𝑡)^1/2>(∫^𝜔0|𝑥_𝑖0(𝑡)|²d𝑡)^1/2, from (2.26), we have 𝑘^𝑗0𝑁_𝑗0−𝑘^∗𝑗₀^𝑛_𝑖=1𝑞_𝑗0𝑖𝐵_𝑖𝜇_𝑖√𝑟₂^𝜔0||𝑦_𝑗0(𝑡)||²d𝑡^1/2≤𝑘^∗𝑗₀√𝜔^𝑛_𝑖=1𝑞_𝑗0𝑖||𝑔_𝑖(0)||+||𝑑_𝑗0(0)||+𝐽_𝑗0.(2.31) Thus, ^𝜔0||𝑦_𝑗0(𝑡)||²d𝑡^1/2≤𝑘^∗𝑗0√𝜔∑^𝑛𝑖=1𝑞_𝑗0𝑖||𝑔_𝑖(0)||+||𝑑_𝑗0(0)||+𝐽_𝑗0𝑘_𝑗0𝑁_𝑗0−𝑘^∗𝑗₀∑^𝑛𝑖=1𝑞_𝑗0𝑖𝐵_𝑖𝜇_𝑖√𝑟₂≤max_{1≤𝑗≤𝑚}⎧⎪⎨⎪⎩𝑘^∗𝑗√𝜔∑^𝑛𝑖=1𝑞_𝑗𝑖||𝑔_𝑖(0)||+||𝑑_𝑗(0)||+𝐽_𝑗𝑘_𝑗𝑁_𝑗−𝑘^∗𝑗∑^𝑛𝑖=1𝑞_𝑗𝑖𝐵_𝑖𝜇_𝑖√𝑟₂⎫⎪⎬⎪⎭_def=𝑑₂.(2.32) Therefore, ^𝜔0||𝑥_𝑖0(𝑡)||²d𝑡^1/2≤^𝜔0||𝑦_𝑗0(𝑡)||²d𝑡^1/2≤𝑑₂.(2.33) Hence, from (2.30) and (2.33), we have for 𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚,𝑡∈[0,𝜔]^𝜔0||𝑥_𝑖(𝑡)||²d𝑡^1/2<max𝑑₁,𝑑₂=^def𝑑,(2.34)_𝜔0||𝑦_𝑗(𝑡)||²d𝑡^1/2<max𝑑₁,𝑑₂=𝑑.(2.35) Multiplying the first equation of system (2.13) by 𝑥^𝑖(𝑡) and integrating over [0,𝜔], from (2.20) and (2.35) and the fact that ^𝜔0𝑎_𝑖𝑥_𝑖(𝑡)𝑏_𝑖𝑥_𝑖(𝑡)𝑥^𝑖(𝑡)d𝑡=0,(2.36) it follows that ^𝜔0||𝑥^𝑖(𝑡)||²d𝑡^1/2≤𝑙^∗𝑖𝑚_𝑗=1𝑝_𝑖𝑗𝐴_𝑗𝜆_𝑗^𝜔0||𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)||d𝑡^1/2+𝑙^∗𝑖√𝜔^𝑚_𝑗=1𝑝_𝑖𝑗||𝑓_𝑗(0)||+𝐼_𝑖≤𝑙_∗𝑖𝑚_𝑗=1𝑝_𝑖𝑗𝐴_𝑗𝜆_𝑗√𝑟₁^𝜔0||𝑦_𝑗(𝑡)||²d𝑡^1/2+𝑙^∗𝑖√𝜔^𝑚_𝑗=1𝑝_𝑖𝑗||𝑓_𝑗(0)||+𝐼_𝑖<max_{1≤𝑖≤𝑛}𝑙^∗𝑖𝑚_𝑗=1𝑝_𝑖𝑗𝐴_𝑗𝜆_𝑗√𝑟₁𝑑+𝑙^∗𝑖√𝜔^𝑚_𝑗=1𝑝_𝑖𝑗||𝑓_𝑗(0)||+𝐼_𝑖^def=𝑐₁.(2.37) Similarly, multiplying the second equation of system (2.13) by 𝑦_𝑗(𝑡) and integrating over [0,𝜔], from (2.21) and (2.34) and the fact that ^𝜔0𝑐_𝑗𝑦_𝑗(𝑡)𝑑_𝑗𝑦_𝑗(𝑡)𝑦^𝑗(𝑡)d𝑡=0,(2.38) it follows that there exists a positive constant 𝑐₂ such that ^𝜔0||𝑦^𝑗(𝑡)||²d𝑡^1/2<𝑐₂.(2.39) From (2.34) and (2.35), it follows that there exist points 𝑡_𝑖 and 𝑡_𝑗 such that ||𝑥_𝑖𝑡_𝑖||<𝑑||𝑦√𝜔,(2.40)_𝑗𝑡_𝑗||<𝑑√𝜔.(2.41)
Since for all 𝑡∈[0,𝜔], ||𝑥_𝑖(𝑡)||≤≤||||𝑥𝑥_𝑖𝑡_𝑖||+^𝜔0||𝑥^𝑖(𝑡)||d𝑡_𝑖||𝑦𝑡_𝑖||+√𝜔^𝜔0||𝑥^𝑖(𝑡)||²^1/2,(2.42)_𝑗(𝑡)||≤≤||||𝑦𝑦_𝑗𝑡_𝑖||+^𝜔0||𝑦^𝑗(𝑡)||d𝑡_𝑗(𝑡)||+√𝜔^𝜔0||𝑦^𝑗(𝑡)||²^1/2,(2.43) then from (2.40)–(2.43), we have for 𝑡∈[0,𝜔],𝑖=1,…,𝑛,𝑗=1,…,𝑚||𝑥_𝑖(𝑡)||≤𝑑√𝜔+√||𝑦_𝑗𝜔𝑐₁,(𝑡)||≤𝑑√𝜔+√𝜔𝑐₂.(2.44)
Obviously, 𝑑/√𝜔,√𝜔𝑐₁, and √𝜔𝑐₂ are all independent of 𝜆. Now let Ω=𝑢‖=𝑢‖𝑥<₁,𝑛𝑥₂,…𝑑,𝑥_𝑛;𝑦₁,𝑦₂,…,𝑦_𝑚^𝑇∈𝑋∶√𝜔+𝑟¹+√𝜔𝑐₁+𝑚√𝑑𝜔+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∈𝜕Ω∩Ker𝐿=𝜕Ω∩𝑅^(𝑚+𝑛) we have 𝑄𝑁=𝑢1𝜔^𝜔0𝐻₁(𝑡)d𝑡,1𝜔^𝜔0𝐻₂(𝑡)d𝑡,…,1𝜔^𝜔0𝐻_𝑛(𝑡)d𝑡;1𝜔^𝜔0𝐾₁(𝑡)d𝑡,…,1𝜔^𝜔0𝐾_𝑚(𝑡)d𝑡^𝑇=(0,…,0)^𝑇.(2.46) Therefore, there exist points 𝜉_𝑖(𝑖=1,2,…,𝑛) and 𝜂_𝑗(𝑗=1,2,…,𝑚) such that 𝐾𝐻_𝑖𝜉_𝑖=0,_𝑗𝜂_𝑗=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]||𝑥_𝑖(𝑡)||<𝑑||𝑦_𝑗√𝜔,(𝑡)||<𝑑√𝜔.(2.48) Hence, ‖𝑢‖<𝑛𝑑√𝜔+𝑚√𝑑𝜔.(2.49) Thus, 𝑢∈Ω∩𝑅^(𝑚+𝑛). This contradicts the fact that 𝑢∈𝜕Ω∩𝑅^(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,Ω∩Ker𝐿,(0,0)^𝑇}≠(0,0,…,0)^𝑇. Now, we show that de=gd−eg𝐽𝑄𝑁𝑙₁𝑢,Ω∩Ker𝐿,(0,0,…,0)^𝑇𝑀₁𝑥₁,𝑙₂𝑀₂𝑥₂,…,𝑙_𝑛𝑀_𝑛𝑥_𝑛;𝑘₁𝑁₁𝑦₁,…,𝑘_𝑚𝑁_𝑚𝑦_𝑚^𝑇,Ω∩Ker𝐿,(0,…,0)^𝑇.(2.50) To this end, we define a mapping 𝜙∶Dom𝐿×[0,1]→𝑋 by 𝜙=𝑥₁−,𝑥𝜇₂,…,𝑥_𝑛;𝑦₁,𝑦₂,…,𝑦_𝑚,𝜇𝜔^𝜔0+(1−𝜇)𝐻𝑙₁₁(𝑡)d𝑡,^𝜔0𝐻₂(𝑡)d𝑡,…,^𝜔0𝐻_𝑛(𝑡)d𝑡,^𝜔0𝐾₁(𝑡)d𝑡,…,^𝜔0𝐾_𝑚(𝑡)d𝑡𝑀₁𝑥₁,𝑙₂𝑀₂𝑥₂,…,𝑙_𝑛𝑀_𝑛𝑥_𝑛;𝑘₁𝑁₁𝑦₁,…,𝑘_𝑚𝑁_𝑚𝑦_𝑚,(2.51) where 𝜇∈[0,1] is a parameter. We show that when 𝑢∈𝜕Ω∩Ker𝐿=𝜕Ω∩𝑅^(𝑚+𝑛), 𝜙(𝑥₁,𝑥₂,…,𝑥_𝑛;𝑦₁,…,𝑦_𝑚,𝜇)≠(0,0,…,0)^𝑇. If it is not true, then when 𝑢∈𝜕Ω∩Ker𝐿=𝜕Ω∩𝑅^(𝑚+𝑛), 𝜙(𝑥₁,𝑥₂,…,𝑥_𝑛;𝑦₁,…,𝑦_𝑚,𝜇)=(0,0,…,0)^𝑇. Thus, constant vector 𝑢 with 𝑢∈𝜕Ω satisfies for 𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚, 𝜇𝜔^𝜔0𝑎^𝑖𝑥_𝑖𝑏_𝑖𝑥_𝑖−𝑎_𝑖𝑥_𝑖^𝑚_𝑗=1𝜇𝑝_𝑖𝑗(𝑡)𝑓_𝑗𝜆_𝑗𝑦_𝑗−𝐼_𝑖(𝑡)d𝑡+(1−𝜇)𝑙_𝑖𝑀_𝑖𝑥_𝑖=0,𝜔^𝜔0𝑐^𝑗𝑦_𝑗𝑑_𝑗𝑦_𝑗−𝑐_𝑗𝑦_𝑗^𝑛_𝑖=1𝑞_𝑗𝑖(𝑡)𝑔_𝑖𝜇_𝑖𝑢_𝑖−𝐽_𝑗(𝑡)d𝑡+(1−𝜇)𝑘_𝑗𝑁_𝑗𝑦_𝑗=0.(2.52) That is, 𝜇𝜔^𝜔0sign𝑥_𝑖𝑎^𝑖+(1−𝜇)𝑙_𝑖𝑥_𝑖𝑏_𝑖𝑥_𝑖−𝑏_𝑖(0)+𝑎_𝑖𝑥_𝑖𝑏_𝑖(0)−𝑎_𝑖𝑥_𝑖^𝑚_𝑗=1𝑝_𝑖𝑗(𝑡)𝑓_𝑗𝜆_𝑗𝑦_𝑗−𝐼_𝑖(𝑡)d𝑡𝑀_𝑖||𝑥_𝑖||=0,(2.53)𝜇𝜔^𝜔0sign𝑦_𝑗𝑐^𝑗+(1−𝜇)𝑘_𝑗𝑦_𝑗𝑑_𝑗𝑦_𝑗−𝑑_𝑗(0)+𝑐_𝑗𝑦_𝑗𝑑_𝑗(0)−𝑐_𝑗𝑦_𝑗^𝑛_𝑖=1𝑞_𝑗𝑖(𝑡)𝑔_𝑖𝜇_𝑖𝑢_𝑖−𝐽_𝑗(𝑡)d𝑡𝑁_𝑗||𝑦_𝑗||=0.(2.54) Denote |𝑦_𝑗0|=max_{1≤𝑗≤𝑚}{|𝑦_𝑗|},|𝑥_𝑖0|=max_{1≤𝑖≤𝑛}{|𝑥_𝑖|}.

Claim 1. We claim that |𝑥_𝑖0|<(𝑑/√𝜔)+√𝜔𝑐₁+𝑟₁, otherwise, |𝑥_𝑖0|≥(𝑑/√𝜔)+√𝜔𝑐₁+𝑟₁. We consider two possible cases: (i) |𝑦_𝑗0|≤|𝑥_𝑖0| and (ii)|𝑦_𝑗0|>|𝑥_𝑖0|.(i)When |𝑦_𝑗0|≤|𝑥_𝑖0|, we have 𝜇𝜔^𝜔0sign𝑥_𝑖0𝑎^𝑖+(1−𝜇)𝑙_𝑖𝑥_𝑖0𝑏_𝑖𝑥_𝑖0−𝑏_𝑖(0)+𝑎_𝑖𝑥_𝑖0𝑏_𝑖(0)−^𝑚_𝑗=1𝑝_𝑖𝑗(𝑡)𝑓_𝑗𝜆_𝑗𝑦_𝑗−𝐼_𝑖(𝑡)d𝑡≥𝜇𝑙_𝑖𝑀_𝑖||𝑥_𝑖0||𝑀_𝑖||𝑥_𝑖0||−𝑙^∗𝑖||𝑏^𝑖(0)||+^𝑚_𝑗=1𝑝_𝑖𝑗𝜆_𝑗𝐴_𝑗||𝑦_𝑗||+||𝑓_𝑗(0)||+𝐼_𝑖+(1−𝜇)𝑙^𝑖≥𝑙_𝑖𝑀_𝑖||𝑥_𝑖0||𝑀_𝑖||𝑥_𝑖0||−𝑙^∗𝑖||𝑏^𝑖(0)||+^𝑚_𝑗=1𝑝_𝑖𝑗𝜆_𝑗𝐴_𝑗||𝑦_𝑗0||+||𝑓_𝑗(0)||+𝐼_𝑖≥𝑙_𝑖𝑀_𝑖−𝑙^∗𝑖𝑚_𝑗=1𝐴_𝑗𝜆_𝑗𝑝_𝑖𝑗||𝑥^𝑖0||−𝑙^∗𝑖||𝑏^𝑖(0)||+^𝑚_𝑗=1𝑝_𝑖𝑗𝜆_𝑗𝐴_𝑗||𝑦_𝑗0||+||𝑓_𝑗(0)||+𝐼_𝑖≥𝑙_𝑖𝑀_𝑖−𝑙^∗𝑖𝑚_𝑗=1𝐴_𝑗𝜆_𝑗𝑝_𝑖𝑗√𝑑¹𝜔+√𝜔𝑐₁+𝑟₁−𝑙^∗𝑖||𝑏^𝑖(0)||+^𝑚_𝑗=1𝑝_𝑖𝑗𝜆_𝑗𝐴_𝑗||𝑦_𝑗0||+||𝑓_𝑗(0)||+𝐼_𝑖>𝑙_𝑖𝑀_𝑖−𝑙^∗𝑖𝑚_𝑗=1𝐴_𝑗𝜆_𝑗𝑝_𝑖𝑗𝑟¹>0,(2.55)which contradicts (2.53).(ii)When |𝑦_𝑗0|>|𝑥_𝑖0|, we have 𝜇𝜔^𝜔0sign𝑦_𝑗0𝑐^𝑗+(1−𝜇)𝑘_𝑗𝑦_𝑗0𝑑_𝑗𝑦_𝑗0−𝑑_𝑗(0)+𝑐_𝑗𝑦_𝑗0𝑑_𝑗(0)−^𝑛_𝑖=1𝑞_𝑗𝑖(𝑡)𝑔_𝑖𝜇_𝑖𝑥_𝑖−𝐽_𝑗(𝑡)dt≥𝜇𝑘_𝑗𝑁_𝑗||𝑦_𝑗0||𝑁_𝑗||𝑦_𝑗0||−𝑘^∗𝑗||𝑑^𝑗(0)||+^𝑛_𝑖=1𝑞_𝑗𝑖𝜇_𝑖𝐵_𝑖||𝑥_𝑖||+||𝑔_𝑖(0)||+𝐽_𝑗+(1−𝜇)𝑘^𝑗≥𝑘_𝑗𝑁_𝑗||𝑦_𝑗0||𝑁_𝑗||𝑦_𝑗0||−𝑘^∗𝑗||𝑑^𝑗(0)||+^𝑛_𝑖=1𝑞_𝑗𝑖𝜇_𝑖𝐵_𝑖||𝑥_𝑖0||+||𝑔_𝑖(0)||+𝐽_𝑗≥𝑘_𝑗𝑁_𝑗−𝑘^∗𝑗𝑛_𝑖=1𝐵_𝑖𝜇_𝑖𝑞_𝑗𝑖||𝑦^𝑗0||−𝑘^∗𝑗||𝑑^𝑗(0)||+^𝑛_𝑖=1𝑞_𝑗𝑖𝜇_𝑖𝐵_𝑖||𝑥_𝑖0||+||𝑔_𝑖(0)||+𝐽_𝑗≥𝑘_𝑗𝑁_𝑗−𝑘^∗𝑗𝑛_𝑖=1𝐵_𝑖𝜇_𝑖𝑞_𝑗𝑖√𝑑²𝜔+√𝜔𝑐₁+𝑟₁−𝑘^∗𝑗||𝑑^𝑗(0)||+^𝑛_𝑖=1𝑞_𝑗𝑖𝜇_𝑖𝐵_𝑖||𝑥_𝑖0||+||𝑔_𝑖(0)||+𝐽_𝑗>𝑘_𝑗𝑁_𝑗−𝑘^∗𝑗𝑛_𝑖=1𝐵_𝑖𝜇_𝑖𝑞_𝑗𝑖𝑟¹>0,(2.56)which contradicts (2.54). From the discussion of (i) and (ii), Claim 1 holds.

Claim 2. We claim that |𝑦_𝑗0|<(𝑑/√𝜔)+√𝜔𝑐₂+𝑟₂, otherwise, |𝑦_𝑗0|≥(𝑑/√𝜔)+√𝜔𝑐₂+𝑟₂. We consider two possible cases: (i) |𝑥_𝑖0|≤|𝑦_𝑗0| and (ii) |𝑥_𝑖0|>|𝑦_𝑗0|.
The proofs of (i) and (ii) are similar to those of (ii) and (1) in Claim 1, respectively, therefore Claim 2 holds.

Thus, |𝑥_𝑖|<(𝑑₁/√𝜔)+𝑐₁√𝜔+𝑟₁ and |𝑦_𝑗|<(𝑑₂/√𝜔)+√𝜔𝑐₂+𝑟₂. Thus, 𝑢∈Ω∩𝑅^(𝑚+𝑛). This contradicts the fact that 𝑢∈𝜕Ω∩𝑅^(𝑚+𝑛). According to the topological degree theory and by taking 𝐽=𝐼 since Ker𝐿=Im𝑄, we obtain deg=−d𝐽e𝑄g𝑁𝜙𝑢,Ω𝑢₁∩Ker𝐿,(0,0)^𝑇=deg𝜙𝑢₁,𝑢₂,…,𝑢_𝑛;𝑣₁,𝑣₂,…,𝑣_𝑚,1,Ω∩Ker𝐿,(0,0)^𝑇=deg𝑙₁,𝑢₂,…,𝑢_𝑛;𝑣₁,𝑣₂,…,𝑣_𝑚,0,Ω∩Ker𝐿,(0,0)^𝑇≠0.𝑀₁𝑥₁,𝑙₂𝑀₂𝑥₂,…,𝑙_𝑛𝑀_𝑛𝑥_𝑛;𝑘₁𝑁₁𝑦₁,…,𝑘_𝑚𝑁_𝑚𝑦_𝑚^𝑇,Ω∩Ker𝐿,(0,…,0)^𝑇(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 𝑟_𝑖≥1(𝑖=1,2) with 𝑀_𝑖>∑^𝑚𝑗=1𝑞_𝑗𝑖𝜇_𝑖B_𝑖𝑟₂ and 𝑁_𝑗>∑^𝑛𝑖=1𝑝_𝑖𝑗𝜆_𝑗𝐴_𝑗𝑟₁ such that 𝜏^𝑖𝑗<min{1,1−𝑟¹⁻¹}<1 and 𝜎^𝑗𝑖<min{1,1−𝑟²⁻¹}<1;(H₂) there exist constants 𝜏_𝑖𝑗 and 𝜎_𝑗𝑖,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚, such that0<𝜏_𝑖𝑗(𝑡)<𝜏_𝑖𝑗,0<𝜎_𝑗𝑖(𝑡)<𝜎_𝑗𝑖.(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)=(𝑥^∗1(𝑡),𝑥^∗2(𝑡),…,𝑥^∗𝑛(𝑡);𝑦^∗1(𝑡),…,𝑦^∗𝑚(𝑡))^𝑇. Suppose that 𝑢(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡),𝑦₁(𝑡),…,𝑦_𝑚(𝑡))^𝑇 is an arbitrary 𝜔 periodic solution of system (1.2). From (H₁), we can choose a suitable 𝜃 such that 𝑀_𝑖>𝜃𝑙_𝑖+^𝑚_𝑗=1𝑁_𝑗𝑞_𝑗𝑖𝜇_𝑖𝐵_𝑖𝑟₂exp𝜃𝜏_𝑖𝑗,>𝜃𝑘_𝑗+^𝑛_𝑖=1𝑝_𝑖𝑗𝜆_𝑗𝐴_𝑗𝑟_𝑖exp𝜃𝜎_𝑗𝑖.(3.2) We define a Lyapunov functional as follows for 𝑡>0,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚∶𝑉(𝑡)=exp(𝜃𝑡)^𝑛_𝑖=1||||^𝑥_𝑖^𝑥(𝑡)_{∗𝑖(𝑡)}1𝑎_𝑖(𝑠)d𝑠||||+^𝑚_𝑗=1||||^𝑦_𝑗^𝑦(𝑡)_{∗𝑗(𝑡)}1+𝑐_𝑗(𝑠)||||d𝑠_𝑛_𝑚𝑖=1_𝑗=1𝑝_𝑖𝑗𝜆_𝑗𝐴_𝑗^{𝑡𝑡−𝜏}_{𝑖𝑗(𝑡)}exp𝜃𝜎+𝜏_𝑖𝑗𝑔⁻¹(𝜎)||𝑦^𝑗(𝜎)−𝑦^∗𝑗(𝜎)||+1−𝜏^𝑖𝑗𝑔⁻¹(𝜎)d𝜎_𝑛_𝑚𝑖=1_𝑗=1𝑞_𝑗𝑖𝜇_𝑖𝐵_𝑖^{𝑡𝑡−𝜎}_{𝑗𝑖(𝑡)}exp𝜃𝜎+𝜎_𝑗𝑖ℎ⁻¹(𝜎)||𝑥^𝑖(𝜎)−𝑥^∗𝑖(𝜎)||1−𝜎^𝑗𝑖ℎ⁻¹(𝜎)d𝜎,(3.3) where 𝑔(𝑡)=𝑡−𝜏_𝑖𝑗(𝑡),ℎ(𝑡)=𝑡−𝜎_𝑗𝑖(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)^𝑛_𝑖=1𝜃||||^{𝑥𝑖𝑥(𝑡)∗𝑖}_(𝑡)1𝑎_𝑖(𝑠)d𝑠||||−𝑀^𝑖+||𝑥_𝑖(𝑡)−𝑥^∗𝑖(𝑡)_𝑚||_𝑗=1+exp(𝜃𝑡)_𝑚𝑝_𝑖𝑗𝜆_𝑗𝐴_𝑗||𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−𝑦^∗𝑗𝑡−𝜏_𝑖𝑗(𝑡)||_𝑗=1𝜃||||^{𝑦𝑗𝑦(𝑡)∗𝑗}_(𝑡)1𝑐_𝑗(𝑠)d𝑠||||−𝑁^𝑗+||𝑦_𝑗(𝑡)−𝑦^∗𝑗(𝑡)_𝑛||_𝑖=1+exp(𝜃𝑡)_𝑛𝑞_𝑗𝑖𝜇_𝑖𝐵_𝑖||𝑥_𝑖𝑡−𝜎_𝑗𝑖(𝑡)−𝑥^∗𝑖𝑡−𝜏_𝑗𝑖(𝑡)||_𝑚𝑖=1_𝑗=1𝑝_𝑖𝑗𝜆_𝑗𝐴_𝑗||𝑦^𝑗(𝑡)−𝑦^∗𝑗(𝑡)||exp𝜃𝜏_𝑖𝑗𝑠⁻¹(𝑡)−||𝑦1−𝜏^𝑖𝑗𝑠⁻¹(𝑡)_𝑗+exp(𝜃𝑡)_𝑛𝑡−𝜏_𝑖𝑗(𝑡)−𝑦^∗𝑗𝑡−𝜏_𝑖𝑗(𝑡)||_𝑚𝑖=1_𝑗=1𝑞_𝑗𝑖𝜇_𝑖𝐵_𝑖||𝑥^𝑖(𝑡)−𝑥^∗𝑖(𝑡)||exp𝜃𝜎_𝑗𝑖ℎ⁻¹(𝑡)−||𝑥1−𝜎^𝑗𝑖ℎ⁻¹(𝑡)_𝑖𝑡−𝜎_𝑗𝑖(𝑡)−𝑥^∗𝑖𝑡−𝜎_𝑗𝑖(𝑡)||.(3.4) Since there exist points 𝜉_𝑖,𝜂_𝑗 such that ||||^𝑥_𝑖^𝑥(𝑡)_{∗𝑖(𝑡)}1𝑎_𝑖(𝑠)d𝑠||||=𝑎_𝑖1𝜉_𝑖||𝑥^𝑖||||(𝑡)−𝑥^∗𝑖(𝑡)_{𝑦𝑗𝑦(𝑡)∗𝑗(𝑡)}||,1𝑐_j(𝑠)d𝑠||||=𝑐_𝑗1𝜂_𝑗||𝑦^𝑗(𝑡)−𝑦^∗𝑗(𝑡)||,(3.5) from (3.4), we have 𝐷⁺𝑉(𝑡)≤−exp(𝜃𝑡)^𝑚_𝑗=1𝑁^𝑗−𝜃𝑘_𝑗−^𝑛_𝑖=1−exp(𝜃𝑡)_𝑛𝑝_𝑖𝑗𝜆_𝑗𝐴_𝑗𝑟₁exp𝜃𝜎_𝑗𝑖||𝑦_𝑗(𝑡)−𝑦^∗𝑗(𝑡)||_𝑖=1𝑀^𝑖−𝜃𝑙_𝑖−^𝑚_𝑗=1𝑞_𝑗𝑖𝜇_𝑖𝐵_𝑖𝑟₂exp𝜃𝜏_𝑖𝑗||𝑥_𝑖(𝑡)−𝑥^∗𝑖(𝑡)||.(3.6) In view of (3.2), it follows that 𝑉(𝑡)<𝑉(0). Therefore, exp(𝜃𝑡)^𝑛_𝑖=1||||^𝑥_𝑖^𝑥(𝑡)_{∗𝑖(𝑡)}1𝑎_𝑖(𝑠)d𝑠||||+^𝑚_𝑗=1||||^𝑦_𝑗^𝑦(𝑡)_{∗𝑗(𝑡)}1𝑐_𝑗(𝑠)d𝑠||||<𝑉(𝑡)<𝑉(0).(3.7) Equation (3.3) implies that 𝑉(0)<^𝑛_𝑖=1𝑙1_𝑖+^𝑚_𝑗=1𝑤_𝑗𝑖𝜇_𝑖𝐵_𝑖𝑟₂^0−𝜎_𝑗𝑖(0)+exp𝜃𝜎+𝜎_𝑗𝑖ℎ⁻¹(𝜎)d𝜎sup_0≤≤𝜔||𝑥_𝑖(𝑠)−𝑥^∗𝑖(𝑠)_𝑚||_𝑗=1𝑘1_𝑗+^𝑛_𝑖=1ℎ_𝑖𝑗𝜆_𝑗𝐴_𝑗𝑟₁^0−𝜏_𝑖𝑗exp𝜃𝜎+𝜏_𝑖𝑗𝑔⁻¹(𝜎)d𝜎sup_{0≤𝑠≤𝜔}||𝑦_𝑗(𝑠)−𝑦^∗𝑗(𝑠)||.(3.8) Substituting (3.8) into (3.7) gives ^𝑛_𝑖=1||𝑥_𝑖(𝑡)−𝑥^∗𝑖(𝑡)||+^𝑚_𝑗=1<𝑀||𝑦_𝑗(𝑡)−𝑦^∗𝑗(𝑡)||𝑁exp(−𝜃𝑡)^𝑛_𝑖=1sup_{0≤𝑠≤𝜔}||𝑥_𝑖(𝑠)−𝑥^∗𝑖(𝑠)||+^𝑚_𝑗=1sup_{0≤𝑠≤𝜔}||𝑦_𝑗(𝑠)−𝑦^∗𝑗(𝑠)||,(3.9) where 𝑀=max_{1≤𝑖≤𝑛,1≤𝑗≤𝑚}𝑙1_𝑖+^𝑛_𝑖=1ℎ_𝑖𝑗𝜆_𝑗𝐴_𝑗𝑟₁^0−𝜏_𝑖𝑗(0)1exp𝜃+𝜏_𝑖𝑗𝑔⁻¹(𝜎)d𝜎,𝑘_𝑗+^𝑚_𝑗=1𝑤_𝑗𝑖𝜇_𝑖𝐵_𝑖𝑟₂^0−𝜎_𝑗𝑖(0)𝑁=min1exp𝜃+𝜎_𝑗𝑖ℎ⁻¹(𝜎)d𝜎,𝑙^∗𝑖,1𝑘^∗𝑗.(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+sin𝑥¹200𝑥₁(𝑡)+100sin𝑥₁(𝑡)−(2+sin𝑡)|||𝑦₁𝑡−1+sin𝑡d𝑦₁2|||−sin𝑡,(𝑡)d𝑡=−3+cos𝑦¹200𝑦₁(𝑡)+100sin𝑦₁(𝑡)−(2+cos𝑡)|||𝑥₁𝑡−1+sin𝑡3|||−cos𝑡.(4.1)

In Theorem 3.1, 𝐴₁=1,𝐵₁𝑁=1,𝑙₁=1,𝑙^∗1=3,𝑘₁=2,𝑘^∗1=4,𝑀₁=100,₁=100,𝜏𝑝₁₁=3,𝑞₁₁=3,𝜆₁=𝜇₁=1,_11=cos𝑡2,𝜎^11=cos𝑡3.(4.2) Since 1−cos𝑡2≥1−|co2s𝑡|≥21,1−co3s𝑡≥1−|co3s𝑡|≥32,(4.3) then 𝑟₁=2,𝑟₂=3/2.

Since 𝑀₁=100>𝑞₁₁𝜇₁𝐵₁𝑟₂=92,𝑁¹𝑙=100>𝑝₁₁𝜆₁𝐴₁𝑟₁=6,₁𝑀₁=100>𝑙^∗1𝑝₁₁𝐴₁√𝑟₁=9√2,𝑘₁𝑁₁=200>𝑞₁₁𝐵₁𝜇₁√𝑟₂=1232,(4.4) then conditions (H₁), (H₂), and (v) are satisfied. It is easy to prove that the rest of the conditions in Theorem 3.1 are satisfied. By Theorem (3.2), system (4.1) has a unique 𝜔 periodic solution that is globally exponentially stable.

5. Conclusion

We investigate first the existence of the 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. In our result, the hypotheses for monotonicity in [27, 28] on the behaved functions are replaced with sign conditions and the assumption for boundedness on activation functions is removed. We just require that the behaved functions satisfy sign conditions and activation functions are globally Lipschitz continuous.

Acknowledgment

This paper is supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China (no. 20091341, 2011508) and the Fund of National Natural Science of China (no. 61065008).