Employing fixed point theorem, we make a further investigation of a class of
neural networks with delays in this paper. A family of sufficient conditions is given for
checking global exponential stability. These results have important leading significance in
the design and applications of globally stable neural networks with delays. Our results
extend and improve some earlier publications.

1. Introduction

The stability of dynamical neural networks with time delay which have been used in many applications such as optimization, control, and image processing has received much attention recently (see, e.g., [1–15]). Particularly, the authors [3, 8, 9, 14, 16] have studied the stability of neural networks with time-varying delays.

As pointed out in [8], Global dissipativity is also an important concept in dynamical neural networks. The concept of global dissipativity in dynamical systems is a more general concept, and it has found applications in areas such as stability theory, chaos and synchronization theory, system norm estimation, and robust control [8]. Global dissipativity of several classes of neural networks was discussed, and some sufficient conditions for the global dissipativity of neural networks with constant delays are derived in [8].

In this paper, without assuming the boundedness, monotonicity, and differentiability of activation functions, we consider the following delay differential equations:
xi′(t)=-di(t)xi(t)+∑j=1naij(t)fj(xj(t))+∑j=1nbij(t)fj(xj(t-τij(t)))+∑j=1ncij(t)∫-∞tHij(t-s)fj(xj(s))ds+Ji(t),i=1,2,…,n,
where n denotes the number of the neurons in the network, xi(t) is the state of the ith neuron at time t, x(t)=[x1(t),x2(t),…,xn(t)]T∈Rn,f(x(t))=[f1(x1(t)),f2(x2(t)),…,fn(xn(t))]T∈Rn denote the activation functions of the jth neuron at time t, and the kernels Hij:[0,+∞)→[0,+∞) are piece continuous functions with ∫0+∞Hij(s)ds=hij<∞ for i,j=1,2,…,n. Moreover, we consider model (1.1) with τij(t), di(t), aij(t), bij(t), cij(t), and Ji(t) satisfying the following assumptions:

the time delays τij(t)∈C(R,[0,∞)) are periodic functions with a common period ω(>0) for i,j=1,2,…,n;

cij(t)∈C(R,[0,∞)), aij(t),bij(t),cij(t),Ji(t)∈C(R,R) are periodic functions with a common period ω(>0) and fi∈C(R,R),i,j=1,2,…,n.

The organization of this paper is as follows. In Section 2, problem formulation and preliminaries are given. In Section 3, some new results are given to ascertain the global robust dissipativity of the neural networks with time-varying delays. Section 4 gives an example to illustrate the effectiveness of our results.

2. Preliminaries and Lemmas

For the sake of convenience, two of the standing assumptions are formulated below as follows.

|fj(u)|≤pj|u|+qj for all u∈R, j=1,2,…,n, where pj, qj are nonnegative constants.

There exist nonnegative constants pj, j=1,2,…,n, such that |fj(u)-fj(v)|≤pj|u-v| for any u,v∈R.

Let
τ=max1≤i,j≤nsupt≥0{τij(t)}.

The initial conditions associated with system (1.1) are of the form
xi(s)=ϕi(s),s∈[-τ,0],i=1,2,…,n,
in which ϕi(s) is continuous for s∈[-τ,0].

For continuous functions ϕi defined on [-τ,0], i=1,2,…,n, we set ϕ=(ϕ1,ϕ2,…,ϕn)T. If x0=(x10,x10,…,xn0)T is an equilibrium of system (1.1), then we denote
∥ϕ-x0∥=∑i=1n(sup-τ≤t≤0|ϕi(t)-xi0|).

Definition 2.1.

The equilibrium x0=(x10,x10,…,xn0)T is said to be globally exponentially stable, if there exist constants λ>0 and m≥1 such that for any solution x(t)=(x1(t),x2(t),…,xn(t))T of (1.1), we have
|xi(t)-xi0|≤m∥ϕ-x0∥e-λt
for t≥0, where λ is called to be globally exponentially convergent rate.

If ρ(K)<1 for matrix K=(kij)n×n≥0, then (E-K)-1≥0, where E denotes the identity matrix of size n.

3. Periodic Solutions and Exponential Stability

We will use the coincidence degree theory to obtain the existence of a ω-periodic solution to systems (1.1). For the sake of convenience, we briefly summarize the theory as follows.

Let X and Z be normed spaces, and let L:DomL⊂X↦Z be a linear mapping and be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dimKer L=codimIm L<∞ and Im L is closed in Z. If L is a Fredholm mapping of index zero, then there exist continuous projectors P:X↦X and Q:Z↦Z such that Im P=Ker L and Im L=Ker Q=Im(I-Q). It follows that L∣Dom L∩Ker P:(I-P)X↦Im L is invertible. We denote the inverse of this map by Kp. If Ω is a bounded open subset of X, the mapping N is called L-compact on Ω¯ if QN(Ω¯) is bounded and Kp(I-Q)N:Ω¯↦X is compact. Because ImQ is isomorphic to Ker L, there exists an isomorphism J:ImQ↦Ker L.

Let Ω⊂Rn be open and bounded, f∈C1(Ω,Rn)∩C(Ω¯,Rn) and y∈Rn∖f(∂Ω∪Sf), that is, y is a regular value of f. Here, Sf={x∈Ω:Jf(x)=0}, the critical set of f, and Jf is the Jacobian of f at x. Then the degree deg{f,Ω,y} is defined by
deg{f,Ω,y}=∑x∈f-1(y)sgnJf(x)
with the agreement that the above sum is zero if f-1(y)=∅. For more details about the degree theory, we refer to the book of Deimling [18].

Let L be a Fredholm mapping of index zero, and let N be L-compact on Ω¯. Suppose that

for each λ∈(0,1), every solution x of Lx=λNx is such that x∈∂Ω;

QNx≠0 for each x∈∂Ω∩KerL and
deg{JQN,Ω∩KerL,0}≠0.

Then the equation Lx=Nx has at least one solution lying in DomL∩Ω¯.

For the simplicity of presentation, in the remaining part of this paper, for a continuous function g:[0,ω]↦R, we denote
g*=maxt∈[0,ω]g(t),g*=mint∈[0,ω]g(t),g¯=1ω∫0ωg(t)dt.

Theorem 3.2.

Let (A1)–(A3) hold, kij=(1/di¯+ω)(|aij|¯+|bij|¯+|cijhij|¯)pj and K=(kij)n×n. If ρ(K)<1, then system (1.1) has at least a ω-periodic solution.

Proof.

Take X=Z={x(t)=(x1(t),x2(t),…,xn(t))T∈C(R,Rn):x(t)=x(t+ω),forallt∈R}, and denote
|xi|=maxt∈[0,ω]|xi(t)|,i=1,2,…,n,∥x∥=max1≤i≤n|xi|.
Equipped with the norms ∥·∥, both X and Z are Banach spaces. Denote
Δ(xi,t):=-di(t)xi(t)+∑j=1naij(t)fj(xj(t))+∑j=1nbij(t)fj(xj(t-τij(t)))+∑j=1ncij(t)∫-∞tHij(t-s)fj(xj(s))ds+Ji(t).
Since
∑j=1ncij(t)∫-∞tHij(t-s)fj(xj(s))ds=∑j=1ncij(t)∫0∞Hij(s)fj(xj(t-s))ds,
then, for any x(t)∈X, because of the periodicity, it is easy to check that
Δ(xi,t)=-di(t)xi(t)+∑j=1naij(t)fj(xj(t))+∑j=1nbij(t)fj(xj(t-τij(t)))+∑j=1ncij(t)∫0∞Hij(s)fj(xj(t-s))ds+Ji(t)∈Z.
Let
L:DomL={x∈X:x∈C1(R,Rn)}∋x↦x′(·)∈Z,P:X∋x↦1ω∫0ωx(t)dt∈X,Q:Z∋z↦1ω∫0ωz(t)dt∈Z,N:X∋x↦Δ(xi,t)∈Z.
Here, for any W=(w1,w2,…,wn)T∈Rn, we identify it as the constant function in X or Z with the value vector W=(w1,w2,…,wn)T. Then system (1.1) can be reduced to the operator equation Lx=Nx. It is easy to see that
KerL=Rn,ImL={z∈Z:1ω∫0ωz(t)dt=0},which is closed inZ,dimKerL=codimImL=n<∞,
and P, Q are continuous projectors such that
ImP=kerL,KerQ=ImL=Im(I-Q).
It follows that L is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to L) Kp:ImL↦KerP∩DomL is given by
(Kp(z))i(t)=∫0tzi(s)ds-1ω∫0ω∫0szi(v)dvds.
Then,
(QNx)i(t)=1ω∫0ωΔ(xi,s)ds,(Kp(I-Q)Nx)i(t)=∫0tΔ(xi,s)ds-1ω∫0ω∫0tΔ(xi,s)dsdt+(12-tω)∫0ωΔ(xi,s)ds.
Clearly, QN and Kp(I-Q)N are continuous. For any bounded open subset Ω⊂X, QN(Ω¯) is obviously bounded. Moreover, applying the ArzelaCAscoli theorem, one can easily show that Kp(I-Q)N(Ω¯)¯ is compact. Therefore, N is L-compact on with any bounded open subset Ω∈X. Since ImQ=KerL, we take the isomorphism J of ImQ onto KerL to be the identity mapping.

Now, we reach the point to search for an appropriate open bounded set Ω for the application of the continuation theorem corresponding to the operator equation Lx=λNx, λ∈(0,1), and we have
xi′(t)=λΔ(xi,t)for1=1,2,…,n.
Assume that x=x(t)∈X is a solution of system (1.1) for some λ∈(0,1). Integrating both sides of (3.13) over the interval [0,ω], we obtain
0=∫0ωxi′(t)dt=λ∫0ωΔ(xi,t)dt.
Then
∫0ωdi(t)xi(t)dt=∫0ω{∑j=1naij(t)fj(xj(t))+∑j=1nbij(t)fj(xj(t-τij(t)))+∑j=1ncij(t)∫0∞Hij(s)fj(xj(t-s))ds+Ji(t)}dt.
Noting that
|fj(u)|≤pj|u|+qj∀u∈R,j=1,2,…,n,
we get
|xi|*di¯≤∑j=1n(|aij|¯+|bij|¯+|cijhij|¯)pj|xj|*+∑j=1n(|aij|¯+|bij|¯+|cijhij|¯)qj+|Ji|¯.
It follows that
|xi|*≤1di¯∑j=1n(|aij|¯+|bij|¯+|cijhij|¯)pj|xj|*+1di¯{∑j=1n(|aij|¯+|bij|¯+|cijhij|¯)qj+|Ji|¯}.
Note that each xi(t) is continuously differentiable for i=1,2,…,n, and it is certain that there exists ti∈[0,ω] such that |xi(ti)|=|xi(t)|*. Set
D=(D1,D2,…,Dn)T,Di=(1di¯+ω){∑j=1n(|aij|¯+|bij|¯+|cijhij|¯)qj+|Ji|¯}.
In view of ρ(K)<1 and Lemma 2.2, we have (E-K)-1D=l=(l1,l2,…,ln)T≥0, where li is given by
li=∑j=1nkijlj+Di,i=1,2,…,n.
Let
Ω={(x1,x2,…,xn)T∈Rn;|xi|≤li,i=1,2,…,n}.
Then, for t∈[ti,ti+ω], we have
|xi(t)|≤|xi(ti)|+∫titD+|xi(t)|dt≤|xi(t)|*+∫titi+ωD+|xi(t)|dt≤1di¯∑j=1n(|aij|¯+|bij|¯+|cijhij|¯)pj|xj|*+1di¯{∑j=1n(|aij|¯+|bij|¯+|cijhij|¯)qj+|Ji|¯}+∫titi+ωD+|xi(t)|dt≤(1di¯+ω)∑j=1n(|aij|¯+|bij|¯+|cijhij|¯)pj|xj|*+(1di¯+ω){∑j=1n(|aij|¯+|bij|¯+|cijhij|¯)qj+|Ji|¯}≤∑j=1nkijlj+Di=li,
where D+ denotes the right derivative. Clearly, li, i=1,2,…,n, are independent of λ. Then there are no λ∈(0,1) and x∈Ω such that Lx=λNx. When u=(x1,x2,…,xn)T∈∂Ω∩KerL=∂Ω∩Rn, u is a constant vector in Rn with |xi|=li, i=1,2,…,n. Note that QNu=JQNu; when u∈KerL, it must be
(QNu)i=-di¯+∑j=1n(aij¯+bij¯+cijhij¯)fj(xj)+Ji¯.
We claim that
|(QNu)i|>0fori=1,2,…,n.
On the contrary, suppose that there exists some i such that |(QNu)i|=0, that is,
di¯xi=∑j=1n(aij¯+bij¯+cijhij¯)fj(xj)+Ji¯.
Then, we have
li=|xi|=1di¯|∑j=1n(aij¯+bij¯+cijhij¯)fj(xj)+Ji¯|≤1di¯∑j=1n(|aij|¯+|bij|¯+|cijhij|¯)pjlj+1di¯{∑j=1n(|aij|¯+|bij|¯+|cijhij|¯)qj+|Ji|¯}≤(1di¯+ω)∑j=1n(|aij|¯+|bij|¯+|cijhij|¯)pjlj+(1di¯+ω){∑j=1n(|aij|¯+|bij|¯+|cijhij|¯)qj+|Ji|¯}=∑j=1nkijlj+Di=li,
which is a contradiction. Therefore,
QNu≠0foranyu∈∂Ω∩KerL=∂Ω∩Rn.
Consider the homotopy F:(Ω∩KerL)×[0,1]↦Ω∩KerL defined by
F(u,μ)=μdiag(-d1¯,-d2¯,…,-dn¯)u+(1-μ)QNu,(u,μ)∈(Ω∩KerL)×[0,1]. Note that F(·,0)=JQN; if F(u,μ)=0, then, as before, we have
|xi|=1-μdi¯|∑j=1n(aij¯+bij¯+cijhij¯)fj(xj)+Ji¯|≤1di¯∑j=1n(|aij|¯+|bij|¯+|cijhij|¯)pj|xj|+1di¯{∑j=1n(|aij|¯+|bij|¯+|cijhij|¯)qj+|Ji|¯}≤1di¯∑j=1n(|aij|¯+|bij|¯+|cijhij|¯)pjlj+1di¯{∑j=1n(|aij|¯+|bij|¯+|cijhij|¯)qj+|Ji|¯}<∑j=1nkijlj+Di=li,
Hence
F(u,μ)≠0,for(u,μ)∈(∂Ω∩KerL)×[0,1].
It follows from the property of invariance under a homotopy that
deg{JQN,Ω∩KerL,0}=deg{F(·,0),Ω∩KerL,0}=deg{F(·,1),Ω∩KerL,0}=deg{diag(-d1¯,-d2¯,…,-dn¯)}≠0.
Thus, we have shown that Ω satisfies all the assumptions of Lemma 3.1. Hence, Lu=Nu has at least one ω-periodic solution on DomL∩Ω¯. This completes the proof.

When cij=0, (1.1) turns into the following system:
xi′(t)=-di(t)xi(t)+∑j=1naij(t)fj(xj(t))+∑j=1nbij(t)fj(xj(t-τ(t)))+Ji(t),i=1,2,…,n.

Corollary 3.3.

Let (A1)–(A3) hold, kij=(1/di¯+ω)(|aij|¯+|bij|¯)pj, and K=(kij)n×n. If ρ(K)<1, then system (P) has at least a ω-periodic solution.

Theorem 3.4.

Let (A1), (A2), and (A4) hold, kij=(1/di¯+ω)(|aij|¯+|bij|¯+|cijhij|¯)pj, and K=(kij)n×n. If ρ(K)<1, and that
di¯-∑j=1n(|aij|¯+|bij|¯+|cijhij|¯)pjedi*τ>0,
then system (1.1) has exactly one ω-periodic solution. Moreover, it is globally exponentially stable.

Proof.

Let C=C([-τ,0],Rn) with the supnorm ∥φ∥=sups∈[-τ,0];1≤i≤n|φi(s)|, φ∈C. As usual, if (-∞≤)a≤b(≤∞) and ψ∈C([-τ+a,b],Rn), then for t∈[a,b] we define ψt∈C by ψt(θ)=ψ(t+θ), θ∈[-τ,0]. From (A4), we can get |fj(u)|≤pj|u|+|fj(0)|, j=1,2,…,n. Hence, all the hypotheses in Theorem 3.2 hold with qj=|fj(0)|, j=1,2,…,n. Thus, system (1.1) has at least one ω-periodic solution, say x̃(t)=(x̃1(t),x̃2(t),…,x̃n(t))T. Let x(t)=(x1(t),x2(t),…,xn(t))T be an arbitrary solution of system (1.1). For t≥0, a direct calculation of the right derivative D+|xi(t)-x̃i(t)| of |xi(t)-x̃i(t)| along the solutions of system (1.1) leads to
D+|xi(t)-x̃i(t)|=D+{sgn(xi(t)-x̃i(t)}(xi(t)-x̃i(t))≤-di(t)|xi(t)-x̃i(t)|+∑j=1n(|aij(t)[fj(xj(t))-fj(x̃j(t))]|+∑j=1n|bij(t)[fj(xj(t-τij(t)))-fj(x̃j(t-τij(t)))]|+∑j=1n|cij(t)||∫0+∞kij(s)[fj(xj(t-s))-fj(x̃j(t-s))]ds|≤-di(t)|xi(t)-x̃i(t)|+∑j=1n|aij(t)|pj|xj(t)-x̃j(t)|+∑j=1n|bij(t)|pj|xj(t-τij(t))-x̃j(t-τij(t))|+∑j=1n|cij(t)hij|pjsup-τ≤s≤t|xj(s)-x̃j(s)|≤-di(t)|xi(t)-x̃i(t)|+∑j=1n(|aij(t)|+|bij(t)|+|cij(t)hij|)pjsup-τ≤s≤t|xj(s)-x̃j(t)|.
Let zi(t)=|xi(t)-x̃i(t)|. Then (3.33) can be transformed into
D+zi(t)≤-di(t)zi(t)+∑j=1n(|aij(t)|+|bij(t)|+|cij(t)hij|)pjsup-τ≤s≤tzj(s).
Thus, for t>t0 we have
D+{zi(t)e∫t0tdi(s)ds}≤∑j=1n(|aij(t)|+|bij(t)|+|cij(t)hij|)pj∥zt∥e∫t0tdi(s)ds,
It follows that
zi(t)e∫t0tdi(s)ds≤|zi(t0)|+∫t0t{∑j=1n(|aij(u)|+|bij(u)|+|cij(u)hij|)pj∥zu∥e∫t0udi(s)ds}du.
Thus, for any t>0 and θ∈[-min(τ,t),0], we have
e∫t0t+θdi(s)ds=e(∫t0t+∫tt+θ)di(s)ds≥e∫t0tdi(s)ds-di*τ.
Therefore,
e∫t0tdi(s)ds-di*τzi(t+θ)≤e∫t0t+θdi(s)dszi(t+θ)≤∥zt0∥+∫t0t+θ{∑j=1n(|aij(u)|+|bij(u)|+|cij(u)hij|)pj∥zu∥e∫t0udi(s)ds}du.
It follows that
e∫t0tdi(s)ds∥zt∥≤edi*τ∥zt0∥+∫t0tedi*τ{∑j=1n(|aij(u)|+|bij(u)|+|cij(u)hij|)pj∥zu∥e∫t0udi(s)ds}du.
By Gronwall's inequality, we obtain
∥zt∥≤edi*τ∥zt0∥e∫t0tedi*τ∑j=1n(|aij(u)|+|bij(u)|+|cij(u)hij|)pjdue∫t0t-di(s)ds,t≥t0.
Without loss of generality, we let t0=0. For t≥0, [t/ω] denotes the largest integer less than or equal to t/ω. Noting [t/ω]≥t/ω-1, and di¯>∑j=1n(|aij¯|+|bij¯|+|cij¯hij|)pjedi*τ, we get
∥zt∥≤edi*τ∥z0∥e∫0tedi*τ∑j=1n(|aij(u)|+|bij(u)|+|cij(u)hij|)pjdue∫0t-di(s)ds=edi*τ∥z0∥e(∫0ω[t/ω]+∫ω[t/ω]t){edi*τ∑j=1n(|aij(u)|+|bij(u)|+|cij(u)hij|)pj-di(u)}du≤edi*τ+(-di¯+∑j=1n(|aij|¯+|bij|¯+|cijhij|¯)pjedi*τ)ω[t/ω]×∥z0∥e∫ω[t/ω]t{-di(s)+∑j=1n(|aij(s)|+|bij(s)|+|cij(s)hij|)pjedi*τ}ds≤edi*τ+(-di¯+∑j=1n(|aij|¯+|bij|¯+|cijhij|¯)pjedi*τ)ω[t/ω]×∥z0∥e∫0ω{-di(s)+∑j=1n(|aij(s)|+|bij(s)|+|cij(s)hij|)pjedi*τ}ds≤edi*τ∥z0∥e-{di¯-∑j=1n(|aij|¯+|bij|¯+|cijhij|¯)pjedi*τ}t≤m∥z0∥e-λt,t≥0,
where m=max1≤i≤n{edi*τ} and λ=min1≤i≤n{di¯-∑j=1n(|aij|¯+|bij|¯+|cijhij|¯)pjedi*τ} are positive constants. From (3.41), it is obvious that the periodic solution is global exponentially stable, and this completes the proof of Theorem 3.4.

Corollary 3.5.

Let (A1), (A2), and (A4) hold, kij=(1/di¯+ω)(|aij|¯+|bij|¯)pj and K=(kij)n×n. If ρ(K)<1, and that
di¯-∑j=1n(|aij|¯+|bij|¯)pjedi*τ>0,
then system (P) has exactly one ω-periodic solution. Moreover, it is globally exponentially stable.

Remark 3.6.

To the best of our knowledge, few authors have considered the existence of periodic solution and global exponential stability for model (1.1) with coefficients and delays all periodically varying in time. We only find model (P) in [20, 21]; however, it is assumed in [20] that τij(t)≥0 are constants and in [21] that aij(t), bij(t), Ji(t) are continuous ω-periodic functions, and di are positive constants. Especially, the authors of [21] suppose that τij(t)≥0 are continuously differentiable ω-periodic functions and 0≤τij′(t)<1, clearly, which implies that τij(t) are also constants. Obviously, our model is more general. Furthermore, in [20, 21] fi, i=1,2,…,n, are assumed to be strictly monotone, and the explicit presence of the maximum value of the coefficients functions in Theorems 3.2 and 3.4 (see [20, 21]) may impose a very strict constraint on the model (e.g., when some of the maximum value of the coefficients functions are very large). Therefore, our results are more convenient when designing a cellular neural network.

4. An Example

In this section, an example is used to demonstrate that the method presented in this paper is effective.

Example 4.1.

Consider the following two state neural networks:
(x1′(t)x2′(t))=-(d1(t)00d2(t))(x1(t)x2(t))+(a11(t)a12(t)a21(t)a22(t))(f1(x1(t))f2(x2(t)))+(b11(t)b12(t)b21(t)b22(t))(f1(x1(t-τ1(t)))f2(x2(t-τ2(t))))+(c11(t)c12(t)c21(t)c22(t))∫-∞t(H11(t-s)H12(t-s)H21(t-s)H22(t-s))(f1(x1(s))f2(x2(s)))ds+(3cost2sint),
where, all di(t)>0, aij(t), bij(t), cij(t), τi(t) are 2π-periodic continuous functions. The activation function f1(x)=cos((1/3)x)+(1/3)x, f2(x)=sin((1/2)x)+(1/4)x. τ=0.6, d1¯=4, d2¯=3; |a11|¯+|b11|¯+|c11h11|¯=3/80; |a12|¯+|b12|¯+|c12h12|¯=1/6; |a21|¯+|b21|¯+|c21h21|¯=3/40; |a22|¯+|b22|¯+|c22h22|¯=2/21; d1*=5, d2*=4. Clearly, fi satisfies the hypothesis with p1=2/3, p2=3/4. By some simple calculations, we have
di¯-∑j=1n(|aij|¯+|bij|¯+|cijhij|¯)pjedi*τ>0,i=1,2,K=(1+8π1601+8π321+6π601+6π42),ρ(K)≈0.860<1.
Therefore, by Theorem 3.4, the system (1.1) has an exponentially stable 2π-periodic solution.

Acknowledgment

The first author was partially supported financially by the National Natural Science Foundation of China (10801088).

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