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,θ∈E, 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 [1])

xi′(t)=-ai(xi(t))[bi(xi(t))-∑j=1naij(t)gj(xj(t))-∑j=1nbij(t)gj(xj(t-τj(t)))-∑j=1n∑l=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 [2])

xi′(t)=-ai(t,xi(t))[bi(t,xi(t))-∑j=1ncij(t)fj(xj(t-τij(t)))-∑j=1ndij(t)gj(∫0∞Kij(u)xj(t-u)du)+Ii(t)],i=1,2,…,n,
the Cohen-Grossberg neural networks with time-varying delays (see [3])

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 [5], 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 [6–15] 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×n∈Rn×n be two matrices, u=(u1,u2,…,un)T∈Rn,v=(v1,v2,…,vn)T∈Rn be two vectors. For convenience, we introduce the following notations.

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

A≥B(>B) means A-B≥0(>0),

u≥0(u>0) means each element ui≥0(ui>0),

u≤v(u<v) means v-u≥0(v-u>0),

|u|=(|u1|,|u2|,…,|un|)T.

For continuous ω-periodic function g:R→R, we denote |g|¯=max0≤t≤ω|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∥=max0≤i≤n|ϕi|, where |ϕi|=supθ∈E|ϕi(θ)|. The initial conditions of system (1.1) are of the form

x0=ϕ,thatis,xi(θ)=ϕi(θ),θ∈E,i=1,2,…,n,
where ϕ=(ϕ1,ϕ2,…,ϕn)T∈CE. For V(t)∈C((a,+∞),R), let

For i=1,2,…,n,ai,bi∈C[R2,R],fi∈C[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 t∈R,u∈Rn,ϕ∈CE.

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×n∈Rn×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)T∈CE,|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 ϕ̃∈CE, 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)|≤M∥ϕ-ϕ̃∥e-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×n∈Rn×n is said to be an M-matrix if wij≤0,i,j=1,2,…,n,i≠j, and W-1≥0.

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 Au≤d,u,d∈Rn, then u≤A-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 wij≤0,i,j=1,2,…,n,i≠j, 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 A≥0 be an n×n matrix and ρ(A)<1, then (En-A)-1≥0, 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 Ω⊂X is an open bounded set and N:X→Z is a continuous operator which is L-compact on Ω¯. Then Lx=Nx has at least one solution in Dom L∩Ω¯, if the following conditions are satisfied:

Lx≠λNx, for all (x,λ)∈(Dom L∩∂Ω)×(0,1),

QNx≠0, for all x∈∂Ω∩Ker L,

deg{JQN|KerL∩Ω̅,Ω∩Ker L,0}≠0.

Let

X=Y={x=(x1,x2,…,xn)T∈C(R,Rn):x(t+ω)=x(t),t∈R}
with the norm defined by ∥x∥=max1≤i≤n|xi|¯, where |xi|¯=maxt∈[0,ω]|xi(t)|. Clearly, X and Y are two Banach spaces. Let x=(x1,x2,…,xn)T∈X=Y, we define the linear operator L:DomL⊂X→Y as

(Lx)(t)=x′(t)=(x1′(t),x2′(t),…,xn′(t))T,Dom L={x∈X:x′∈Y},
and the operators N:X→X,P:X→X,Q:Y→Y as

(Nx)(t)=A(t,x(t))[B(t,x(t))+F(t,xt)]:=Δ(t,xt),Px=Qx=1ω∫0ωx(t)dt=(1ω∫0ωx1(t)dt,1ω∫0ωx2(t)dt,…,1ω∫0ωxn(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={y∈Y=X:∫0ωy(t)dt=0}=KerQ=Im(I-Q),ImLisclosedinY,dimKerL=n=codimImL.
Thus, the mapping L is a Fredholm mapping of index zero and the isomorphism J:ImQ→KerL is the identity operator; the generalized inverse (of L|DomL∩KerP)KP:ImL→KerP∩DomL exists, which has the form

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 Ω⊂X,QN(Ω¯),KP(I-Q)N(Ω¯), and (KP(I-Q)Nx)′,x∈Ω¯ 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))T∈X 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|+Di≤∑j=1ncij|xj|¯+Di,i=1,2,…,n.
Thus
|xi|¯≤bi-1∑j=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={x∈X:|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)T∈∂Ω∩KerL, 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,sothatQNx≠0.
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-1xi∫0ωai(t,xi(t))[bi(t,xi(t))+fi(t,xt)]dt>0.
Thus (3.11) is valid.

Next, we define continuous functions Hi:(Ω∩KerL)×[0,1]→Ω∩KerL,i=1,2, by
H1(x,t)=tx+(1-t)QNx,∀(x,t)∈(Ω∩KerL)×[0,1],H2(x,t)=-tx+(1-t)QNx,∀(x,t)∈(Ω∩KerL)×[0,1],
respectively. If bi(t,ui)ui>0, from (i) we have
H1(x,t)≠0,∀(x,t)∈KerL⋂∂Ω×[0,1],
If bi(t,ui)ui<0, from (2) we can get
H2(x,t)≠0,∀(x,t)∈KerL⋂∂Ω×[0,1].
Using the homotopy invariance theorem, we obtain if bi(t,ui)ui>0,
deg{JQN|Ω¯∩KerL,Ω∩KerL,0}=deg{H1(·,0),Ω∩KerL,0}=deg{H1(·,1),Ω∩KerL,0}=deg{x,Ω∩KerL,0}=1,
or if bi(t,ui)ui<0,
deg{JQN|Ω̅∩KerL,Ω∩KerL,0}=deg{H2(·,0),Ω∩KerL,0}=deg{H2(·,1),Ω∩KerL,0}=deg{-x,Ω∩KerL,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-1∑j=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-1Di≥0,i=1,2,…,n. Notes δ=max1≤i≤nνi≥0, 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:R→R 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,∀x∈R.

For i=1,2,…,n, there exist Gi,pi,qi≥0 such that
|gi(x)|≤Gi,|gi(x)|≤pi|x|+qi.

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=1n∑l=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=1n∑l=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×n∈Rn×n,D=(D1,D2,…,Dn)T,cij=[|aij|¯+|bij|¯+∑l=1n|bijl|¯Gl]pj≥0,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 0≤τj′(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 [1].

Corollary 3.5.

If the following conditions are satisfied:

for i,j=1,2,…,n,cij,dij,τij,Ii:R→R 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,u∈R,

for i=1,2,…,n, there exist positive constants bi such that |bi(t,u)|≥bi|u|,bi(u)u>0orbi(u)u<0, for all t,u∈R,

there exist nonnegative constants pjf,qjf,pjg,qjg such that
|fj(u)|≤pjf|u|+qjf,|gj(u)|≤pjg|u|+qjg,∀u∈R,j=1,2,…,n,

the delay kernels Kij:[0,∞]→R satisfy
∫0∞|Kij(s)|ds≤kij,i,j=1,2,…,n,

ρ(K)<1,K=(kij)n×n∈Rn×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:R→R 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,u∈R,i=1,2,…,n,

there exist positive constants bi such that
|bi(t,u)|≥bi|u|,bi(u)u>0orbi(u)u<0,∀t,u∈R,i=1,2,…,n,

There exist nonnegative constants pi,qi such that
|fi(u)|≤pi|u|+qi,∀u∈R,i=1,2,…,n,

ρ(K)<1, where K=(kij)n×n∈Rn×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 0≤ubi(t,u),b̲i|u|≤|bi(t,u)|≤b¯i|u| for all t,u∈R,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:R→R are continuous ω-periodic solution (ω>0) functions.

For j=1,2,…,n,gj:R→R are continuous functions and there exist nonnegative constants pj,qj such that
|gj(v)|≤pj|v|+qj,∀v∈R,j=1,2,…,n,

ρ(K)<1,K=(kij)n×n∈Rn×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 3.5–3.9 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×n∈Rn×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)T∈CE,|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,v∈R,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=max0≤t≤ω|fi(t,0)|, and assume that En-W is an M-matrix, where W=(wij)n×n∈Rn×n, and

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)T∈CE. 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)|+a̅i∑j=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 ∀θ∈Eαξi+[-(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,∀θ∈E,i=1,2,…,n.
Set, for all ε>0, for all t∈E,
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],∀θ∈E.
Thus
D-Zi(t)≥-(Liab-LiaΔi)Zi(t)+(a̅i+Liaδ)∑j=1ncij|Zjt|,
where |Zjt|=supθ∈EZj(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)|,∀t∈E.
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
0≤D-(|yi(ti)|-Zi(ti))=lim suph→0-[|yi(ti+h)|-Zi(ti+h)]-[|yi(ti)|-Zi(ti)]h≤lim suph→0-|yi(ti+h)|-|yi(ti)|h-liminfh→0-Zi(ti+h)-Zi(ti)h≤D-|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 ε→0+ and M=n max1≤i≤n{βξi+1}>1, from (4.17), we have
|xi(t)-x̃i(t)|=|yi(t)|≤βξi∑j=1n|yj0|e-αt≤βξin∥ϕ-ϕ̃∥e-αt≤M∥ϕ-ϕ̃∥e-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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