AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation95747510.1155/2009/957475957475Research ArticleExistence and Exponential Stability of Periodic Solution for a Class of Generalized Neural Networks with Arbitrary DelaysZhangYimin1LiYongkun2YeKuohui2MawhinJean1Department of MathematicsZhaotong Teacher's CollegeZhaotong Yunnan 657000China2Department of MathematicsYunnan UniversityKunming Yunnan 650091Chinaynu.edu.cn200913122009200904082009011220092009Copyright © 2009This 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.

By the continuation theorem of coincidence degree and M-matrix theory, we obtain some sufficient conditions for the existence and exponential stability of periodic solutions for a class of generalized neural networks with arbitrary delays, which are milder and less restrictive than those of previous known criteria. Moreover our results generalize and improve many existing ones.

1. Introduction

Consider the following generalized neural networks with arbitrary delays:

x(t)=A(t,x(t))[B(t,x(t))+F(t,xt)], where A(t,x(t))=diag(a1(t,x1(t)),a2(t,x2(t)),,an(t,xn(t)),B(t,x(t))=b1(t,x1(t)),    b2(t,x2(t)),,bn(t,xn(t)))T,   F(t,xt)  =(f1(t,xt),f2(t,xt),,fn(t,xt))T, fi(t,xt)=fi(t,x1t, x2t,,xnt),   xt=(x1t,x2t,,xnt)T is defined by xt(θ)=x(t+θ) = (x1(t+θ),x2(t+θ), ,xn(t+θ))T,  θE, and E is a subset of R-=(-,0].

System (1.1) contains many neural networks, for examples, the higher-order Cohen-Grossberg type neural networks with delays (see )

xi(t)=-ai(xi(t))[bi(xi(t))-j=1naij(t)gj(xj(t))-j=1nbij(t)gj(xj(t-τj(t)))-j=1nl=1nbijl(t)gj(xj(t-τj(t)))gl(xl(t-τl(t)))+Ii(t)],i=1,2,,n, the Cohen-Grossberg neural networks with bounded and unbounded delays (see )

xi(t)=-ai(t,xi(t))[bi(t,xi(t))-j=1ncij(t)fj(xj(t-τij(t)))-j=1ndij(t)gj(0Kij(u)xj(t-u)du)+Ii(t)],i=1,2,,n, the Cohen-Grossberg neural networks with time-varying delays (see )

xi(t)=-ai(t,xi(t))[bi(t,xi(t))-j=1ncij(t)fj(xj(t))-j=1ndij(t)fj(xj(t-τij(t)))+Ii(t)],i=1,2,,n, the celluar neural networks (see [4, Page 193]):

xi(t)=-ri(t)xi(t)+j=1naij(t)gj(xj(t))+j=1nbij(t)gj(xj(t-τij(t)))+Ii(t),i=1,2,,n, and so on.

Since the model of Cohen-Grossberg neural networks was first introduced by Cohen and Grossberg in , the dynamical characteristics (including stable, unstable, and periodic oscillatory) of Cohen-Grossberg neural networks have been widely investigated for the sake of theoretical interest as well as application considerations. Many good results have already been obtained by some authors in  and the references cited therein. Moreover, the existing results are based on the assumption that demanding either the activation functions, the behaved functions, or delays is bounded in the above-mentioned literature. However, to the best of our knowledge, few authors have discussed the existence and exponential stability of periodic solutions of (1.1). In this paper, by using the continuation theorem of coincidence degree and M-matrix theory, we study model (1.1), and get some sufficient conditions for the existence and exponential stability of the periodic solution of system (1.1); our results generalize and improve many existing ones.

Let A=(aij)n×n,B=(bij)n×nRn×n be two matrices, u=(u1,u2,,un)TRn,v=(v1,v2,,vn)TRn be two vectors. For convenience, we introduce the following notations.

A0(A>0) means that each element aij is nonnegative (positive) respectively,

AB(>B) means A-B0(>0),

u0(u>0) means each element ui0(ui>0),

uv(u<v) means v-u0    (v-u>0),

|u|=(|u1|,|u2|,,|un|)T.

For continuous ω-periodic function g:RR, we denote |g|¯=max0tω|g(t)|,CE=C[E,Rn] is the family of continuous functions ϕ=(ϕ1,ϕ2,,ϕn)T from E(-,0] to Rn. Clearly, it is a Banach space with the norm ϕ=max0in|ϕi|, where |ϕi|=supθE|ϕi(θ)|. The initial conditions of system (1.1) are of the form

x0=ϕ,that  is,xi(θ)=ϕi(θ),θE,i=1,2,,n, where ϕ=(ϕ1,ϕ2,,ϕn)TCE. For V(t)C((a,+),R), let

D-V(t)=limsuph0-D(t+h)-D(t)h,D-V(t)=liminfh0-D(t+h)-D(t)h,t(a,+).

Throughout this paper, we assume the following:

For i=1,2,,n,ai,biC[R2,R],fiC[R×CE,R] and are ω-periodic for their first arguments, respectively, that is, ai(t+ω,u)=ai(t,u),bi(t+ω,u)=bi(t,u),fi(t+ω,ϕ)=fi(t,ϕ) so A(t+ω,u)=A(t,u),B(t+ω,u)=B(t,u),F(t+ω,ϕ)=F(t,ϕ), for all tR,  uRn,ϕCE.

There exists a positive diagonal matrix A=diag(a1,a2,,an) such that A(t,u)A, for all (t,u)Rn+1.

There is a positive diagonal matrix B=diag(b1,b2,,bn) such that |B(t,u)|B|u|, and bi(t,ui)ui>0 or bi(t,ui)ui<0 for all (t,u)Rn+1,  i=1,2,,n.

There exist a nonnegative matrix C=(cij)n×nRn×n and a nonnegative vector D=(D1,D2,,Dn)T such that |F(t,ϕ)|C|ϕ|+D, for all (t,ϕ)R×CE, where ϕ=(ϕ1,ϕ2,,ϕn)TCE,|ϕ|=(|ϕ1|,|ϕ2|,,|ϕn|)T.

2. Preliminaries

In this section, we first introduce some definitions and lemmas which play an important role in the proof of our main results in this paper.

Definition 2.1.

Let x̃(t)=(x̃1(t),x̃2(t),,x̃n(t))T be an ω-periodic solution of system (1.1) with initial value ϕ̃CE, if there exist two constants α>0 and M>1 such that for every solution x(t)=(x1(t),x2(t),,xn)T of system (1.1) with initial value (1.6), |xi(t)-x̃i(t)|Mϕ-ϕ̃e-αt,t>0,i=1,2,,n. Then x̃(t) is said to be globally exponential stable.

Definition 2.2.

A real matrix W=(wij)n×nRn×n is said to be an M-matrix if wij0,i,j=1,2,,n,ij, and W-10.

Lemma 2.3 (see [<xref ref-type="bibr" rid="B15">15</xref>, <xref ref-type="bibr" rid="B16">16</xref>]).

Assume that A is an M-matrix and Aud,u,dRn, then uA-1d.

Lemma 2.4 (see [<xref ref-type="bibr" rid="B15">15</xref>, <xref ref-type="bibr" rid="B16">16</xref>]).

Let W=(wij)n×n with wij0,i,j=1,2,,n,ij, then the following statements are equivalent.

W is an M-matrix.

There exists a positive vector η=(η1,η2,,ηn)>0 such that ηW>0.

There exists a positive vector ξ=(ξ1,ξ2,,ξn)T>0 such that Wξ>0.

Lemma 2.5 (see [<xref ref-type="bibr" rid="B15">15</xref>, <xref ref-type="bibr" rid="B16">16</xref>]).

Let A0 be an n×n matrix and ρ(A)<1, then (En-A)-10, where En denotes the identity matrix of size n, so En-A is an M-matrix.

Now we introduce Mawhin's continuation theorem which will be fundamental in this paper.

Lemma 2.6 (see [<xref ref-type="bibr" rid="B17">17</xref>]).

Let X and Y be two Banach spaces and L a Fredholm mapping of index zero. Assume that ΩX is an open bounded set and N:XZ is a continuous operator which is L-compact on Ω¯. Then Lx=Nx has at least one solution in Dom LΩ¯, if the following conditions are satisfied:

LxλNx, for all (x,λ)(Dom LΩ)×(0,1),

QNx0, for all xΩKer L,

deg{JQN|KerLΩ̅,  ΩKer L,0}0.

Let

X=Y={x=(x1,x2,,xn)TC(R,Rn):x(t+ω)=x(t),tR}   with the norm defined by x=max1in|xi|¯, where |xi|¯=maxt[0,ω]|xi(t)|. Clearly, X and Y are two Banach spaces. Let x=(x1,x2,,xn)TX=Y, we define the linear operator L:DomLXY as

(Lx)(t)=x(t)=(x1(t),x2(t),,xn(t))T,Dom L={xX:xY}, and the operators N:XX,P:XX,Q:YY as

(Nx)(t)=A(t,x(t))[B(t,x(t))+F(t,xt)]:=Δ(t,xt),Px=Qx=1ω0ωx(t)dt=(1ω0ωx1(t)dt,1ω0ωx2(t)dt,,1ω0ωxn(t)dt)T. It is not difficult to show that P and Q are continuous projectors and the following conditions are satisfied:

KerL=Rn=ImP=ImQ,ImL={yY=X:0ωy(t)dt=0}=KerQ=Im(I-Q),ImL  is  closed  in  Y,dimKerL=n=codimImL. Thus, the mapping L is a Fredholm mapping of index zero and the isomorphism J:ImQKerL is the identity operator; the generalized inverse (of L|DomLKerP)  KP:ImLKerPDomL exists, which has the form

(KPx)(t)=0tx(s)  ds-1ω0ω0tx(s)dsdt,xImL. Therefore

(QNx)(t)=1ω0ωΔ(t,xt)dt,(KP(I-Q)Nx)(t)=0tΔ(s,xs)ds-1ω0ω0tΔ(s,xs)dsdt+(12-tω)0ωΔ(s,xs)ds.

3. Existence of Periodic Solutions

In this section, we shall use Lemma 2.6 to study the existence of at least one periodic solution of system (1.1).

Theorem 3.1.

Let (H1)–(H4) hold. Moveover, suppose that

E-K is a M-matrix, where the matrix K=(kij)n×n=B-1C.

then

system (1.1) has at least one ω-periodic solution;

there exists a nonnegative constant δ such that for all ω-periodic solution x(t)=(x1(t),x2(t),,xn(t))T of system (1.1), |xi(t)|δ,i=1,2,,n.

Proof.

Clearly, QN and KP(I-Q)N are continuous functions and for every bounded subset ΩX,  QN(Ω¯),  KP(I-Q)N(Ω¯), and (KP(I-Q)Nx),  xΩ¯ are bounded. By using the Arzela-Ascoli theorem, QN(Ω¯) and KP(I-Q)N(Ω¯) are compact, therefore N is L-compact on Ω¯. Consider the following operator equation: Lx=λNx,λ(0,1). That is, x(t)=λA(t,x(t))[B(t,x(t))+F(t,xt)], or xi(t)=λai(t,xi(t))[bi(t,xi(t))+fi(t,xt)],i=1,2,,n. Assume that x(t)=(x1(t),x2(t),,xn(t))TX is a solution of (3.3) for some λ(0,1). Then, for any i=1,2,,n,xi(t) are all continuous ω-periodic functions, and there exist ti[0,ω], such that |xi(ti)|=maxt[0,  ω]|xi(t)|=|xi|¯,xi(ti)=0,i=1,2,,n, from (H2), we have bi(ti,xi(ti))+fi(ti,xti)=0,i=1,2,,n. It follows from (H3) that bi|xi|¯|bi(ti,xi(ti))|=|fi(ti,xti)|j=1ncij|xjti|+Dij=1ncij|xj|¯+Di,i=1,2,,n. Thus |xi|¯bi-1j=1ncij|xj|¯+bi-1Di,i=1,2,,n, we denote the vector d=(d1,d2,,dn)T,|x|¯=(|x1|¯,|x2|¯,,|xn|¯)T, where di=bi-1(Di+1)>0,i=1,2,,n. It follows from (3.7) that (E-K)|x|¯<d. Since (H5), and application of Lemma 2.3 yields |x|¯<(E-K)-1d=(h1,h2,,hn)T=h, where h satisfies the equation h=Kh+d, that is, hi=j=1nkijhj+di>0.

Take Ω={xX:|xi|¯<hi,i=1,2,,n}. It is easy to see that Ω satisfies condition (1) in Lemma 2.6.

For all x=(x1,x2,,xn)TΩKerL, x is a constant vector in Rn and there exists some i{1,2,,n} such that |xi|=|xi|¯=hi, we claim that |(QNx)i|>0,so  that  QNx0. We firstly claim that

if bi(t,ui)ui>0, then xi(QNx)i>0,

if bi(t,ui)ui<0, then xi(QNx)i<0.

We only prove (1), since the proof of (2) is similar. If bi(t,ui)ui>0, we have xi[bi(t,xi(t))+fi(t,xt)]bixi2-|xi|[j=1ncij|xj|¯+Di]>bihi[hi-(j=1nbi-1cijhj+di)]=bihi[hi-(j=1nkijhj+di)]=0. Therefore xi(QNx)i=ω-1xi0ωai(t,xi(t))[bi(t,xi(t))+fi(t,xt)]dt>0. Thus (3.11) is valid.

Next, we define continuous functions Hi:(ΩKerL)×[0,1]ΩKerL,i=1,2, by H1(x,t)=tx+(1-t)QNx,(x,t)(ΩKerL)×[0,1],H2(x,t)=-tx+(1-t)QNx,(x,t)(ΩKerL)×[0,1], respectively. If bi(t,ui)ui>0, from (i) we have H1(x,t)  0,(x,t)KerLΩ×[0,1], If bi(t,ui)ui<0, from (2) we can get H2(x,t)0,(x,t)KerLΩ×[0,1]. Using the homotopy invariance theorem, we obtain if bi(t,ui)ui>0, deg{JQN|Ω¯KerL,ΩKerL,0}=deg{H1(·,0),ΩKerL,0}=deg{H1(·,1),ΩKerL,0}=deg{x,ΩKerL,0}=1, or if bi(t,ui)ui<0, deg{JQN|Ω̅KerL,ΩKerL,0}=deg{H2(·,0),ΩKerL,0}=deg{H2(·,1),ΩKerL,0}=deg{-x,ΩKerL,0}=(-1)n. To summarize, Ω satisfies all the conditions of Lemma 2.6. This completes the proof of (i).

For all ω-periodic solution x(t)=(x1(t),x2(t),,xn(t))T of system (1.1), from (3.3)–(3.7) we have |xi|¯bi-1j=1ncij|xj|¯+bi-1Di,|x|¯(E-K)-1υ=ν=(ν1,ν2,,νn)T, where υ=(υ1,υ2,,υn)T,|x|¯=(|x1|¯,|x2|¯,,|xn|¯)T, υi=bi-1Di0,i=1,2,,n. Notes δ=max1inνi0, thus |xi(t)|δ, for all i=1,2,,n. This completes the proof of (ii).

From the proof of Theorem 3.1, we can easily obtain the following corollary.

Corollary 3.2.

Suppose that (H1)–(H5) hold, and D=0 in (H4), then system (1.1) has only one ω-periodic solution x(t)=0.

Some special cases of Theorem 3.1 are in what follows.

Corollary 3.3.

Equation (1.3) has at least one ω-periodic solution, if the following conditions are satisfied.

For i,j=1,2,,n,ai,bi,aij,bij,τj,Ii:RR are continuous ω-periodic (ω>0) functions.

For i=1,2,,n, ai(x) are positive, and there exist ai>0 such that ai(x)ai>0.

For i=1,2,,n, there exist bi>0 such that |bi(x)|bi|x|,bi(x)x>0,orbi(x)x<0,xR.

For i=1,2,,n, there exist Gi,pi,qi0 such that |gi(x)|Gi,|gi(x)|pi|x|+qi.

ρ(K)<1,K=(kij)n×n,and    kij=bi-1(|aij|¯+|bij|¯+j=1n|bijl|¯Gl)pj,i,j=1,2,,n.

Proof.

It is clear that A(t,x)=diag(a1(x1),a2(x2),,an(xn))diag(a1,a2,,an)=A,|B(t,x)|=(|b1(x1)|,|b2(x2)|,,|bn(xn)|)T(b1|x1|,b2|x2|,,bn|xn|)T=diag(b1,b2,,bn)(|x1|,|x2|,,|xn|)T=B|x|,|fi(t,ϕ)|=|j=1naij(t)gj(ϕj(0))+j=1nbij(t)gj(ϕj(-τj(t)))+j=1nl=1nbijl(t)gj(ϕj(-τj(t)))gl(ϕl(-τl(t)))-Ii(t)|j=1n|aij|̅[pj|ϕj|+qj]+j=1n|bij|¯[pj|ϕj|+qj]+j=1nl=1nGl|bijl|¯(pj|ϕj|+qj)+|Ii|¯=j=1n[|aij|¯+|bij|¯+l=1n|bijl|¯Gl]pj|ϕj|+j=1n[|aij|¯+|bij|¯+l=1n|bijl|¯Gl]qj+|Ii|¯=j=1ncij|ϕj|+Di,i=1,2,,n. Thus |f(t,ϕ)|C|ϕ|+D, where C=(cij)n×nRn×n,D=(D1,D2,,Dn)T,cij=[|aij|¯+|bij|¯+l=1n|bijl|¯Gl]pj0,Di=j=1n[|aij|¯+|bij|¯+l=1n|bijl|¯Gl]qj+|Ii|¯0,i,j=1,2,,n. Therefore, by using Lemma 2.5 and Theorem 3.1, we know that (1.3) has an ω-periodic solution. The proof is complete.

Remark 3.4.

For [1, Equation (1.2)], τj(t),j=1,2,,n are continuous differentiable ω-periodic solutions and 0τj(t)1, this implies that τj(t),j=1,2,,n are constant functions, thus ξj=1,j=1,2,,n. It is not difficult to verify that all of conditions of Corollary 3.3 are satisfied under the conditions of [1, Theorem 1] moreover the other requirements of [1, Theorem 1] are more restrictive than ours. Therefore, Corollary 3.3 improves the corresponding result obtained in .

Corollary 3.5.

If the following conditions are satisfied:

for i,j=1,2,,n,cij,dij,τij,Ii:RR are continuous ω-periodic (ω>0) functions, ai,bi are continuous functions on R2, and are ω-periodic for their first arguments, respectively,

for i=1,2,,n, there exist positive constants ai such that ai(t,u)ai, for all t,uR,

for i=1,2,,n, there exist positive constants bi such that |bi(t,u)|bi|u|,bi(u)u>0    or    bi(u)u<0, for all t,uR,

there exist nonnegative constants pjf,qjf,pjg,qjg such that |fj(u)|pjf|u|+qjf,|gj(u)|pjg|u|+qjg,uR,j=1,2,,n,

the delay kernels Kij:[0,]R satisfy 0|Kij(s)|dskij,i,j=1,2,,n,

ρ(K)<1,K=(kij)n×nRn×n, where kij=bi-1(|cij|¯pjf+|dij|¯kijpjg),i,j=1,2,,n.

then (1.3) has at least one ω-periodic solution.

Remark 3.6.

In [2, Theorem 3.1], the activation functions fj(u),gj(u),j=1,2,,n, are required to be Lipschitzian, which implies that condition (B3) in Corollary 3.5 holds. Therefore, Corollary 3.5 improves Theorem 3. In 2.

Corollary 3.7.

Assume that the following conditions are satisfied:

cij,dij,τij,Ii:RR are continuous ω-periodic (ω>0) functions, ai,bi are continuous functions on R2, and are ω-periodic in the first variable,

there exist positive constants ai such that ai(t,u)ai,t,uR,i=1,2,,n,

there exist positive constants bi such that |bi(t,u)|bi|u|,bi(u)u>0orbi(u)u<0,t,uR,i=1,2,,n,

There exist nonnegative constants pi,qi such that |fi(u)|pi|u|+qi,uR,i=1,2,,n,

ρ(K)<1, where K=(kij)n×nRn×n and kij=bi-1(|cij|¯+|dij|¯)pj,i,j=1,2,,n.

Then (1.4) has at least one ω-periodic solution.

Remark 3.8.

In [3, Theorem 3.1], the activation functions fj(u),j=1,2,,n, are Lipschitzian (which also implies that condition (C4) in Corollary 3.7 holds) and the behaved functions bi(t,u) are required to satisfy that there exist positive constants b̲i,  b¯i such that 0ubi(t,u),b̲i|u||bi(t,u)|b¯i|u| for all t,uR,i=1,2,,n, which are more restrictive than that of Corollary 3.7.

Corollary 3.9.

Assume that the following conditions are satisfied

For i,j=1,2,,n,Ii,aij,bij,τij:RR are continuous ω-periodic solution (ω>0) functions.

For j=1,2,,n,gj:RR are continuous functions and there exist nonnegative constants pj,qj such that |gj(v)|pj|v|+qj,vR,j=1,2,,n,

ρ(K)<1,K=(kij)n×nRn×n and kij=ri-1(|aij|¯+|bij|¯)pj,i,j=1,2,,n,

then (1.5) has at least one ω-periodic solution.

The proofs of Corollaries 3.53.9 are the same as that of Corollary 3.3.

4. Uniqueness and Exponential Stability of Periodic Solution

In this section, we establish some results for the uniqueness and exponential stability of the ω-periodic solution of (1.1).

Theorem 4.1.

Assume that E is a bounded subset of R-, and (H1)–(H3) and (H5) hold. Suppose also the following conditions are satisfied.

There exists a nonnegative matrix C=(cij)n×nRn×n such that |F(t,ϕ)-F(t,φ)|C|ϕ-φ|,(t,ϕ),(t,φ)R×CE, where ϕ=(ϕ1,ϕ2,,ϕn)T,φ=(φ1,φ2,,φn)TCE,|ϕ-φ|=(|ϕ1-φ1|,|ϕ2-φ2|, ,|ϕn-φn|)T.

ai,i=1,2,,n, are Lipschitzian with Lipschitz constants Lia>0, and there exist a̅i such that ai(t,u)a̅i,|ai(t,u)-ai(t,v)|Lia|u-v|,(t,u),(t,v)R2,i=1,2,,n.

For all t,u,vR,i=1,2,,n, there exist positive constants Liab such that [ai(t,u)bi(t,u)-ai(t,v)bi(t,v)](u-v)0,i=1,2,,n,|ai(t,u)bi(t,u)-ai(t,v)bi(t,v)|Liab|u-v|,  i=1,2,,n.

For i=1,2,,n, set Δi=max0tω|fi(t,0)|, and assume that En-W is an M-matrix, where W=(wij)n×nRn×n, and

wij=(Liab-LiaΔi)-1(a̅i+Liaδ)cij,Liab-LiaΔi>0,i,j=1,2,,n.

Proof.

Obviously, (H4) implies (H4), since (H1)–(H5) hold, it follows from Theorem 3.1 that system (1.1) has at least one ω-periodic solution x̃(t)=(x̃1(t),x̃2(t),,x̃n(t))T with the initial value ϕ̃=(ϕ̃1,ϕ̃2,,ϕ̃n)TCE. Let x(t)=(x1(t),x2(t),,xn(t))T be an arbitrary solution of system (1.1) with the initial value (1.6), set y(t)=x(t)-x̃(t). Then for i=1,2,,n,yi(t)=ai(t,yi(t)+x̃i(t))bi(t,yi(t)+x̃i(t))-ai(t,x̃i(t))bi(t,x̃i(t))+ai(t,yi(t)+x̃i(t))[fi(t,yt+x̃t)-fi(t,x̃t)]+fi(t,x̃t)[ai(t,xi(t))-ai(t,x̃i(t))]. Thus, for i=1,2,,n, D-|yi(t)|-Liab|yi(t)|+a̅ij=1ncij|yit|+Lia|yi(t)|[j=1ncij|x̃it|+|fi(t,0)|]-(Liab-LiaΔi)|yi(t)|+(a̅i+Liaδ)j=1ncij|yit|, for (H8) and Lemma 2.4, there exist a positive constant σ>0 and a positive constant vector ξ=(ξ1,ξ2,,ξn)T>0 such that (En-W)ξ>(σ,σ,,σ)T. Hence ξi-j=1nwijξj>σ, where wij=(Liab-LiaΔi)-1(a̅i+Liaδ)cij,i,j=1,2,,n. Moreover for all i=1,2,,n, -(Liab-LiaΔi)ξi+(a̅i+Liaδ)j=1ncijξj<(Liab-LiaΔi)σ. Since, E is a bounded subset of R-, we can choose a positive constant α<1, such that θEαξi+[-(Liab-LiaΔi)ξi+(a̅i+Liaδ)j=1ncijξje-αθ]<0,i=1,2,,n, and also can choose a positive constant β>1 such that βξie-αθ>1,θE,i=1,2,,n. Set, for all ε>0, for all tE, Zi(t)=βξi[j=1n|yj0|+ε]e-αt,i=1,2,,n. It follows from (4.11) and (4.13) that D-Zi(t)=-αβξi[j=1n|yj0|+ε]e-αt>[-(Liab-LiaΔi)ξi+(a̅i+Liaδ)j=1ncijξje-αθ]β[j=1n|yj0|+ε]e-αt=-(Liab-LiaΔi)ξiβ[j=1n|yj0|+ε]e-αt+(a̅i+Liaδ)j=1ncijξje-α(θ+t)β[j=1n|yj0|+ε],θE. Thus D-Zi(t)-(Liab-LiaΔi)Zi(t)+(a̅i+Liaδ)j=1ncij|Zjt|, where |Zjt|=supθEZj(t+θ), from (4.12) and (4.13), we can get Zi(t)=βξi[j=1n|yj0|+ε]e-αt>j=1n|yj0|+ε>|yi(t)|,tE. We claim that |yi(t)|<Zi(t),t>0,i=1,2,,n. Suppose that it is not true, then there exits some i{1,2,,n} and ti>0 such that |yi(ti)|=Zi(ti),|yj(t)|Zj(t),t<ti,j=1,2,,n. Thus 0D-(|yi(ti)|-Zi(ti))=lim suph0-[|yi(ti+h)|-Zi(ti+h)]-[|yi(ti)|-Zi(ti)]hlim suph0-|yi(ti+h)|-|yi(ti)|h-liminfh0-Zi(ti+h)-Zi(ti)hD-|yi(ti)|-D-Zi(ti). It follows from (4.8), (4.15), and (4.18) that D-|yi(ti)|-(Liab-LiaΔi)|yi(ti)|+(a̅i+Liaδ)j=1ncij|yjti|-(Liab-LiaΔi)|Zi(ti)|+(a̅i+Liaδ)j=1ncij|Zjti|<D-Zi(ti), which contradicts to (4.19), thus (4.17) holds. Set ε0+ and M=n max1in{βξi+1}>1, from (4.17), we have |xi(t)-x̃i(t)|=|yi(t)|βξij=1n|yj0|e-αtβξinϕ-ϕ̃e-αtMϕ-ϕ̃e-αt, where i=1,2,,n. This completes the proof of Theorem 4.1.

5. Conclusion

In this paper, a class of generalized neural networks with arbitrary delays have been studied. Some sufficient conditions for the existence and exponential stability of the periodic solutions have been established. These obtained results are new and they improve and complement previously known results.

Acknowledgment

This work is supported by the National Natural Sciences Foundation of China under Grant 10971183.

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