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

Authors: Kaiyu Liu; Zhengqiu Zhang; Liping Wang
Journal: Abstract and Applied Analysis
 (2012)

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## 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.


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## Body

## 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:(1.1)dxi(t)dt=-ai(xi(t))[bi(t,xi(t))-∑j=1mpij(t)∫0∞Kji(u)×fj(t,λjyj(t-u))du-Ii(t)],i=1,2,…,n,dyj(t)dt=-cj(yj(t))[dj(t,yj(t))-∑i=1nqji(t)∫0∞Lij(u)×gi(t,μixi(t-u))du-Jj(t)],j=1,2,…,m.
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:
(1.2)dxi(t)dt=-ai(xi(t))[bi(xi(t))-∑j=1mpij(t)fj(λjyj(t-τij(t)))-Ii(t)],i=1,2,…,n,dyj(t)dt=-cj(yj(t))[dj(yj(t))-∑i=1nqji(t)gi(μixi(t-σji(t)))-Jj(t)],j=1,2,…,m,
where n,m≥2 are the number of neurons in the networks with initial value conditions:
(1.3)xi(θ)=ϕi(θ),θ∈[-r1,0],yj(θ)=ϕj(θ),θ∈[-r2,0],
where r1=max1≤i≤n,1≤j≤m,0≤t≤ω{σji(t)}, r2=max1≤i≤n,1≤j≤m,0≤t≤ω{τij(t)}, ai(xi(t)), bi(xi(t)), cj(yj(t)), dj(yj(t)) are continuous functions, fj(λjyj(t-τji(t))),gi(μixi(t-δij(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:
(1.4)dxi(t)dt=-ai(xi(t))[bi(xi(t))-∑j=1maij(t)fj(yj(t-τij(t)))-∑j=1mbij(t)fj(yj(t-σij(t)))-Ii(t)],i=1,2,…,n,dyj(t)dt=-cj(yj(t))[dj(yj(t))-∑i=1ncji(t)gi(xi(t-pji(t)))-∑i=1ndji(t)gi(xi(t-qji(t)))-Jj(t)],j=1,2,…,m.
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.LetX, 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 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(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).
LetL 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¯=max0≤t≤ω|f(t)|,f̲=min0≤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)
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 that(2.1)li≤ai≤li*,kj≤cj≤kj*;(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,y≠x∈R,(2.2)sign⁡(x-y)[bi(x)-bi(y)]≥Mi|x-y|,sign⁡(x-y)[dj(x)-dj(y)]≥Nj|x-y|;(iv)
there exist positive constantsAj,Bi(i=1,…,n,j=1,2,…,m) such that for all x,y∈R,(2.3)|fj(x)-fj(y)|≤Aj|x-y|,|gi(x)-gi(y)|≤Bi|x-y|;(v)
there exist two positive constantsri>1,i=1,2 with τij′<min{1,1-r1-1}<1 and σji′<min{1,1-r2-1}<1 such that for i=1,…,n;j=1,…,m,(2.4)liMi>∑j=1mli*pij¯Ajλjr1,kjNj>∑i=1nkj*qji¯Biμir2.
Then, system (1.2) has at least one ω-periodic solution.Proof.
In order to apply Lemma2.1 to system (1.2), let
(2.5)X={u=(x1,x2,…,xn,y1,y2,…,ym)T∈C(R,Rm+n):u(t+ω)=u(t)},Z={z∈C(R,Rm+n):z(t+ω)=z(t)}.
Define
(2.6)‖u‖=max⁡t∈[0,ω]∑i=1n|xi(t)|+max⁡t∈[0,ω]∑j=1m|yj(t)|,u∈XorZ.
Equipped with the above norm ∥·∥,X and Z are Banach spaces.
Let foru∈X(2.7)Nu=(Hi(t)Kj(t))=(-ai(xi(t))[bi(xi(t))-∑j=1mpij(t)fj(λjyj(t-τij(t)))-Ii(t)]-cj(yj(t))[dj(yj(t))-∑i=1nqji(t)gi(μixi(t-σji(t)))-Jj(t)]),Lu=u′=du(t)dt,Pu=1ω∫0ωu(t)dt,u∈X,Qz=1ω∫0ωz(t)dt,z∈Z.
Then, it follows that KerL=R(m+n), ImL={z∈Z:∫0ωz(t)dt=0} is closed in Z, dimKerL=m+n=codimImL, and P,Q are continuous projectors such that
(2.8)Im⁡P=Ker⁡L,Ker⁡Q=Im⁡L=Im⁡(I-Q).
Hence, L is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to L) Kp:ImL→KerP∩DomL is given by
(2.9)Kp(z)=∫0tz(s)ds-1ω∫0ω∫0sz(t)dtds.
Then,
(2.10)QNu=(1ω∫0ωH1(s)ds1ω∫0ωH2(s)ds⋮1ω∫0ωHn(s)ds1ω∫0ωK1(s)ds1ω∫0ωK2(s)ds⋮1ω∫0ωKm(s)ds),Kp(I-Q)Nu=(f0tH1(s)ds-1ω∫0ω∫0tH1(s)dsdt+(12-tω)∫0ωH1(s)dsf0tH2(s)ds-1ω∫0ω∫0tH2(s)dsdt+(12-tω)∫0ωH2(s)ds⋮f0tHn(s)ds-1ω∫0ω∫0tHn(s)dsdt+(12-tω)∫0ωHn(s)dsf0tK1(s)ds-1ω∫0ω∫0tK1(s)dsdt+(12-tω)∫0ωK1(s)ds⋮f0tKm(s)ds-1ω∫0ω∫0tKm(s)dsdt+(12-tω)∫0ωKm(s)ds).
Obviously, QN and KP(I-Q)N are continuous. It is not difficult to show that Kp(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 Theorem2.2 implies that for all x∈R(2.11)sign⁡xbi(x)≥Mi|x|+sign⁡xbi(0),sign⁡xdj(x)≥Nj|x|+sign⁡xdj(0).
Condition (iv) in Theorem 2.2 implies that for allx∈R(2.12)|fj(x)|≤Aj|x|+|fj(0)|,|gi(x)|≤Bi|x|+|gj(0)|.
Corresponding to the operator equation Lx=λNx,λ∈(0,1), we have for i=1,2,…,n,j=1,…,m(2.13)dxi(t)dt=λHi(t),dyj(t)dt=λKj(t).
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 xi(t) and integrating over [0,ω], we have
(2.14)∫0ωxi(t)sign⁡xi(t)sign⁡xi(t)×{ai(xi(t))[bi(xi(t))-∑j=1mpij(t)fj(λjyj(t-τij(t)))-Ii(t)]}dt=0.
Multiplying the second equation of system (2.13) by yj(t) and integrating over [0,ω], we have
(2.15)∫0ωyj(t)sign⁡yj(t)sign⁡yj(t)×{cj(yj(t))[dj(yj(t))-∑i=1nqji(t)gi(μixi(t-σji(t)))-Jj(t)]}dt=0.
From (2.14) and (2.15), we obtain
(2.16)liMi∫0ω|xi(t)|2dt≤li*∫0ω|xi(t)|{-aisign⁡xi(t)bi(0)+∑j=1mpij¯(Ajλj|yj(t-τij(t))|+|fj(0)|)+Ii¯}dt,(2.17)kiNj∫0ω|yj(t)|2dt≤kj*∫0ω|yj(t)|{-cjsign⁡yj(t)dj(0)+∑i=1nqji¯(Biμi|xi(t-σji(t))|+|gi(0)|)+Jj¯}dt.
Hence,
(2.18)liMi∫0ω|xi(t)|2dt≤li*(∫0ω|xi(t)|2dt)1/2×{∑j=1mpij¯[Ajλj(∫0ω|yj(t-τij(t))|2dt)1/2+ω|fj(0)|]+li*|bi(0)|+ωIi¯},(2.19)kjNj∫0ω|yj(t)|2dt≤kj*(∫0ω|yj(t)|2dt)1/2×{∑i=1nqji¯[Biμi(∫0ω|xi(t-σji(t))|2dt)1/2+ω|gi(0)|]+kj*|dj(0)|+ωJj¯}
Denoting s=t-τij(t)=g(t),σ=t-σji(t)=h(t), then
(2.20)(∫0ω|yj(t-τij(t))|2dt)1/2=(∫0ω|yj(s)|21-τij′(g-1(s))ds)1/2,(2.21)(∫0ω|xi(t-σji(t))|2dt)1/2=(∫0ω|xi(σ)|21-σji′(h-1(σ))dσ)1/2.
Substituting (2.20) into (2.18) and substituting (2.21) into (2.19) give for i=1,…,n,j=1,…,m(2.22)liMi∫0ω|xi(t)|2dt≤li*(∫0ω|xi(t)|2dt)1/2×{∑j=1mpij¯Ajλjr1(∫0ω|yj(t)|2dt)1/2+ω(∑j=1mpij¯|fj(0)|+|bi(0)|+Ii¯)},(2.23)kjNj∫0ω|yj(t)|2dt≤kj*(∫0ω|yj(t)|2)1/2×{∑i=1nqji¯Biμir2(∫0ω|xi(t)|2dt)1/2+ω(∑i=1nqji¯|gi(0)|+|dj(0)|+Jj¯)}.
Denoting for the sake of convenience
(2.24)max⁡1≤i≤n{(∫0ω|xi(t)|2dt)1/2}=(∫0ω|xi0(t)|2dt)1/2,max⁡1≤j≤m{(∫0ω|yj(t)|2dt)1/2}=(∫0ω|yj0(t)|2dt)1/2,
where, i0∈{1,2,…,n},j0∈{1,2,…,m}, and from (2.22) and (2.23), we obtain
(2.25)li0Mi0(∫0ω|xi0(t)|2dt)1/2≤li0*∑j=1mpi0j¯Ajλjr1(∫0ω|yj0(t)|2dt)1/2+li0*ω(∑j=1mpi0j¯|fj(0)|+|bi0(0)|+Ii0¯),(2.26)kj0Nj0(∫0ω|yj0(t)|2dt)1/2≤kj0*∑i=1nqj0i¯Biμir2(∫0ω|xi0(t)|2dt)1/2+kj0*ω(∑i=1nqj0i¯|gi(0)|+|dj0(0)|+Jj0¯).
Now we consider two possible cases for (2.26) and (2.25):
(2.27)(i)(∫0ω|yj0(t)|2dt)1/2≤(∫0ω|xi0(t)|2dt)1/2,(ii)(∫0ω|yj0(t)|2dt)1/2>(∫0ω|xi0(t)|2dt)1/2.
When (∫0ω|yj0(t)|2dt)1/2≤(∫0ω|xi0(t)|2dt)1/2, from (2.25), we have
(2.28)(li0Mi0-li0*∑j=1mpi0j¯Ajλjr1)(∫0ω|xi0(t)|2dt)1/2≤li0*ω(∑j=1mpi0j¯|fj(0)|+|bi0(0)|+Ii0¯).
Thus,
(2.29)(∫0ω|xi0(t)|2dt)1/2≤li0*ω(∑j=1mpi0j¯|fj(0)|+|bi0(0)|+Ii0¯)li0Mi0-li0*∑j=1mpi0j¯Ajλjr1≤max⁡1≤i≤n{li*ω(∑j=1mpij¯|fj(0)|+|bi(0)|+Ii¯)liMi-li*∑j=1mpij¯Ajλjr1}=defd1.
Therefore,
(2.30)(∫0ω|yj0(t)|2dt)1/2≤(∫0ω|xi0(t)|2dt)1/2≤d1.
(ii) When(∫0ω|yj0(t)|2dt)1/2>(∫0ω|xi0(t)|2dt)1/2, from (2.26), we have
(2.31)(kj0Nj0-kj0*∑i=1nqj0i¯Biμir2)(∫0ω|yj0(t)|2dt)1/2≤kj0*ω(∑i=1nqj0i¯|gi(0)|+|dj0(0)|+Jj0¯).
Thus,
(2.32)(∫0ω|yj0(t)|2dt)1/2≤kj0*ω(∑i=1nqj0i¯|gi(0)|+|dj0(0)|+Jj0¯)kj0Nj0-kj0*∑i=1nqj0i¯Biμir2≤max⁡1≤j≤m{kj*ω(∑i=1nqji¯|gi(0)|+|dj(0)|+Jj¯)kjNj-kj*∑i=1nqji¯Biμir2}=defd2.
Therefore,
(2.33)(∫0ω|xi0(t)|2dt)1/2≤(∫0ω|yj0(t)|2dt)1/2≤d2.
Hence, from (2.30) and (2.33), we have for i=1,2,…,n,j=1,2,…,m,t∈[0,ω](2.34)(∫0ω|xi(t)|2dt)1/2<max⁡{d1,d2}=defd,(2.35)(∫0ω|yj(t)|2dt)1/2<max⁡{d1,d2}=d.
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
(2.36)∫0ωai(xi(t))bi(xi(t))xi′(t)dt=0,
it follows that
(2.37)(∫0ω|xi′(t)|2dt)1/2≤li*∑j=1mpij¯Ajλj(∫0ω|yj(t-τij(t))|dt)1/2+li*ω(∑j=1mpij¯|fj(0)|+Ii¯)≤li*∑j=1mpij¯Ajλjr1(∫0ω|yj(t)|2dt)1/2+li*ω(∑j=1mpij¯|fj(0)|+Ii¯)<max⁡1≤i≤n{li*∑j=1mpij¯Ajλjr1d+li*ω(∑j=1mpij¯|fj(0)|+Ii¯)}=defc1.
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
(2.38)∫0ωcj(yj(t))dj(yj(t))yj′(t)dt=0,
it follows that there exists a positive constant c2 such that
(2.39)(∫0ω|yj′(t)|2dt)1/2<c2.
From (2.34) and (2.35), it follows that there exist points ti and tj¯ such that
(2.40)|xi(ti)|<dω,(2.41)|yj(tj¯)|<dω.
Since for allt∈[0,ω],
(2.42)|xi(t)|≤|xi(ti)|+∫0ω|xi′(t)|dt≤|xi(ti)|+ω(∫0ω|xi′(t)|2)1/2,(2.43)|yj(t)|≤|yj(ti)|+∫0ω|yj′(t)|dt≤|yj(t)|+ω(∫0ω|yj′(t)|2)1/2,
then from (2.40)–(2.43), we have for t∈[0,ω],i=1,…,n,j=1,…,m(2.44)|xi(t)|≤dω+ωc1,|yj(t)|≤dω+ωc2.
Obviously,d/ω,ωc1, and ωc2 are all independent of λ. Now let
(2.45)Ω={(dω+r2+ωc2)u=(x1,x2,…,xn;y1,y2,…,ym)T∈X:‖u‖<n(dω+r1+ωc1)+m(dω+r2+ωc2)},
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
(2.46)QNu=(1ω∫0ωH1(t)dt,1ω∫0ωH2(t)dt,…,1ω∫0ωHn(t)dt;1ω∫0ωK1(t)dt,…,1ω∫0ωKm(t)dt)T=(0,…,0)T.
Therefore, there exist points ξi(i=1,2,…,n) and ηj(j=1,2,…,m) such that
(2.47)Hi(ξi)=0,Kj(ηj)=0.
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](2.48)|xi(t)|<dω,|yj(t)|<dω.
Hence,
(2.49)‖u‖<ndω+mdω.
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
(2.50)deg⁡{-JQNu,Ω∩Ker⁡L,(0,0,…,0)T}=deg⁡{(l1M1x1,l2M2x2,…,lnMnxn;k1N1y1,…,kmNmym)T,Ω∩Ker⁡L,(0,…,0)T}.
To this end, we define a mapping ϕ:DomL×[0,1]→X by
(2.51)ϕ(x1,x2,…,xn;y1,y2,…,ym,μ)=-μω(∫0ωH1(t)dt,∫0ωH2(t)dt,…,∫0ωHn(t)dt,∫0ωK1(t)dt,…,∫0ωKm(t)dt)+(1-μ)(l1M1x1,l2M2x2,…,lnMnxn;k1N1y1,…,kmNmym),
where μ∈[0,1] is a parameter. We show that when u∈∂Ω∩KerL=∂Ω∩R(m+n), ϕ(x1,x2,…,xn;y1,…,ym,μ)≠(0,0,…,0)T. If it is not true, then when u∈∂Ω∩KerL=∂Ω∩R(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,
(2.52)μω∫0ω{ai(xi)[bi(xi)-ai(xi)∑j=1mpij(t)fj(λjyj)-Ii(t)]}dt+(1-μ)liMixi=0,μω∫0ω{cj(yj)[dj(yj)-cj(yj)∑i=1nqji(t)gi(μiui)-Jj(t)]}dt+(1-μ)kjNjyj=0.
That is,
(2.53)μω∫0ωsign⁡xi{ai(xi)(bi(xi)-bi(0))+ai(xi)bi(0)-ai(xi)[∑j=1mpij(t)fj(λjyj)-Ii(t)]}dt+(1-μ)liMi|xi|=0,(2.54)μω∫0ωsign⁡yj{cj(yj)(dj(yj)-dj(0))+cj(yj)dj(0)-cj(yj)[∑i=1nqji(t)gi(μiui)-Jj(t)]}dt+(1-μ)kjNj|yj|=0.
Denote |yj0|=max1≤j≤m{|yj|},|xi0|=max1≤i≤n{|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
(2.55)μω∫0ωsign⁡xi0{ai(xi0)(bi(xi0)-bi(0))+ai(xi0)[bi(0)-∑j=1mpij(t)fj(λjyj)-Ii(t)]}dt+(1-μ)liMi|xi0|≥μliMi|xi0|-li*[|bi(0)|+∑j=1mpij¯(λjAj|yj|+|fj(0)|)+Ii¯]+(1-μ)liMi|xi0|≥liMi|xi0|-li*[|bi(0)|+∑j=1mpij¯(λjAj|yj0|+|fj(0)|)+Ii¯]≥(liMi-li*∑j=1mAjλjpij¯)|xi0|-li*[|bi(0)|+∑j=1mpij¯(λjAj|yj0|+|fj(0)|)+Ii¯]≥(liMi-li*∑j=1mAjλjpij¯)(d1ω+ωc1+r1)-li*[|bi(0)|+∑j=1mpij¯(λjAj|yj0|+|fj(0)|)+Ii¯]>(liMi-li*∑j=1mAjλjpij¯)r1>0,which contradicts (2.53).(ii)
When|yj0|>|xi0|, we have
(2.56)μω∫0ωsign⁡yj0{cj(yj0)(dj(yj0)-dj(0))+cj(yj0)[dj(0)-∑i=1nqji(t)gi(μixi)-Jj(t)]}dt+(1-μ)kjNj|yj0|≥μkjNj|yj0|-kj*[|dj(0)|+∑i=1nqji¯(μiBi|xi|+|gi(0)|)+Jj¯]+(1-μ)kjNj|yj0|≥kjNj|yj0|-kj*[|dj(0)|+∑i=1nqji¯(μiBi|xi0|+|gi(0)|)+Jj¯]≥(kjNj-kj*∑i=1nBiμiqji¯)|yj0|-kj*[|dj(0)|+∑i=1nqji¯(μiBi|xi0|+|gi(0)|)+Jj¯]≥(kjNj-kj*∑i=1nBiμiqji¯)(d2ω+ωc1+r1)-kj*[|dj(0)|+∑i=1nqji¯(μiBi|xi0|+|gi(0)|)+Jj¯]>(kjNj-kj*∑i=1nBiμiqji¯)r1>0,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 Claim1, respectively, therefore Claim 2 holds.Thus,|xi|<(d1/ω)+c1ω+r1 and |yj|<(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
(2.57)deg⁡{-JQNu,Ω∩Ker⁡L,(0,0)T}=deg⁡{ϕ(u1,u2,…,un;v1,v2,…,vm,1),Ω∩Ker⁡L,(0,0)T}=deg⁡{ϕ(u1,u2,…,un;v1,v2,…,vm,0),Ω∩Ker⁡L,(0,0)T}=deg⁡{(l1M1x1,l2M2x2,…,lnMnxn;k1N1y1,…,kmNmym)T,Ω∩Ker⁡L,(0,…,0)T}≠0.
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 Theorem2.2, one assumes further that the following conditions hold:(H1)
there exists two positive constantsri≥1(i=1,2) with Mi>∑j=1mqji¯μiBir2 and Nj>∑i=1npij¯λjAjr1 such that τij′<min{1,1-r1-1}<1 and σji′<min{1,1-r2-1}<1;(H2)
there exist constantsτij and σji,i=1,2,…,n,j=1,2,…,m, such that(3.1)0<τij(t)<τij,0<σji(t)<σji.
Then, the ω periodic solution of system (1.2) is globally exponentially stable.Proof.
By Theorem2.2, 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
(3.2)Mi>θli+∑j=1mqji¯μiBir2exp⁡(θτij),Nj>θkj+∑i=1npij¯λjAjriexp⁡(θσji).
We define a Lyapunov functional as follows for t>0,i=1,2,…,n,j=1,2,…,m:(3.3)V(t)=exp⁡(θt){∑i=1n|∫xi*(t)xi(t)1ai(s)ds|+∑j=1m|∫yj*(t)yj(t)1cj(s)|ds}+∑i=1n∑j=1mpij¯λjAj∫t-τij(t)texp⁡[θ(σ+τij(g-1(σ)))]|yj(σ)-yj*(σ)|1-τij′(g-1(σ))dσ+∑i=1n∑j=1mqji¯μiBi∫t-σji(t)texp⁡[θ(σ+σji(h-1(σ)))]|xi(σ)-xi*(σ)|1-σji′(h-1(σ))dσ,
where g(t)=t-τij(t),h(t)=t-σji(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
(3.4)D+V(t)≤exp⁡(θt)∑i=1n{θ|∫xi*(t)xi(t)1ai(s)ds|-Mi|xi(t)-xi*(t)|+∑j=1mpij¯λjAj|yj(t-τij(t))-yj*(t-τij(t))|}+exp⁡(θt)∑j=1m{θ|∫yj*(t)yj(t)1cj(s)ds|-Nj|yj(t)-yj*(t)|+∑i=1nqji¯μiBi|xi(t-σji(t))-xi*(t-τji(t))|}+exp⁡(θt)∑i=1n∑j=1mpij¯λjAj{|yj(t)-yj*(t)|exp⁡⁡[θτij(s-1(t))]1-τij′(s-1(t))-|yj(t-τij(t))-yj*(t-τij(t))||yj(t)-yj*(t)|exp⁡⁡[θτij(s-1(t))]1-τij′(s-1(t)}+exp⁡(θt)∑i=1n∑j=1mqji¯μiBi{|xi(t)-xi*(t)|exp⁡⁡[θσji(h-1(t))]1-σji′(h-1(t))-|xi(t-σji(t))-xi*(t-σji(t))||xi(t)-xi*(t)|exp⁡⁡[θσji(h-1(t)))]1-σji′(h-1(t)}.
Since there exist points ξi,ηj such that
(3.5)|∫xi*(t)xi(t)1ai(s)ds|=1ai(ξi)|xi(t)-xi*(t)|,|∫yj*(t)yj(t)1cj(s)ds|=1cj(ηj)|yj(t)-yj*(t)|,
from (3.4), we have
(3.6)D+V(t)≤-exp⁡(θt)∑j=1m{Nj-θkj-∑i=1npij¯λjAjr1exp⁡(θσji)}|yj(t)-yj*(t)|-exp⁡(θt)∑i=1n{Mi-θli-∑j=1mqji¯μiBir2exp⁡(θτij)}|xi(t)-xi*(t)|.
In view of (3.2), it follows that V(t)<V(0). Therefore,
(3.7)exp⁡(θt){∑i=1n|∫xi*(t)xi(t)1ai(s)ds|+∑j=1m|∫yj*(t)yj(t)1cj(s)ds|}<V(t)<V(0).
Equation (3.3) implies that
(3.8)V(0)<∑i=1n{1li+∑j=1mwji¯μiBir2∫-σji(0)0exp⁡[θ(σ+σji(h-1(σ)))]dσ}sup⁡0≤≤ω|xi(s)-xi*(s)|+∑j=1m{1kj+∑i=1nhij¯λjAjr1∫-τij0exp⁡⁡[θ(σ+τij(g-1(σ)))]dσ}sup⁡0≤s≤ω|yj(s)-yj*(s)|.
Substituting (3.8) into (3.7) gives
(3.9)∑i=1n|xi(t)-xi*(t)|+∑j=1m|yj(t)-yj*(t)|<MNexp⁡(-θt){∑i=1nsup⁡0≤s≤ω|xi(s)-xi*(s)|+∑j=1msup⁡0≤s≤ω|yj(s)-yj*(s)|},
where
(3.10)M=max⁡1≤i≤n,1≤j≤m{∑j=1mwji¯μiBir2∫-σji(0)0exp⁡[θ+σji(h-1(σ))]dσ1li+∑i=1nhij¯λjAjr1∫-τij(0)0exp⁡[θ+τij(g-1(σ))]dσ,1kj+∑j=1mwji¯μiBir2∫-σji(0)0exp⁡[θ+σji(h-1(σ))]dσ},N=min⁡{1li*,1kj*}.
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:(4.1)dx1(t)dt=-(2+sin⁡x1){200x1(t)+100sin⁡x1(t)-(2+sin⁡t)|y1[t-(1+sin⁡t2)]|-sin⁡t},dy1(t)dt=-(3+cos⁡y1){200y1(t)+100sin⁡y1(t)-(2+cos⁡t)|x1[t-(1+sin⁡t3)]|-cos⁡t}.In Theorem3.1,
(4.2)A1=1,B1=1,l1=1,l1*=3,k1=2,k1*=4,M1=100,N1=100,p11¯=3,q11¯=3,λ1=μ1=1,τ11′=cos⁡t2,σ11′=cos⁡t3.
Since
(4.3)1-cos⁡t2≥1-|cos⁡t|2≥12,1-cos⁡t3≥1-|cos⁡t|3≥23,
then r1=2,r2=3/2.Since(4.4)M1=100>q11¯μ1B1r2=92,N1=100>p11¯λ1A1r1=6,l1M1=100>l1*p11¯A1r1=92,k1N1=200>q11¯B1μ1r2=1232,
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.



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*Source: 805846-2012-05-14.xml*