# Existence and Global Exponential Stability of  Periodic Solutions for General Neural Networks  with Time-Varying Delays

Authors: Xinsong Yang
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

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## Abstract

By using the coincidence degree theorem and differential inequality
techniques, sufficient conditions are obtained for the existence and global exponential
stability of periodic solutions for general neural networks with time-varying (including
bounded and unbounded) delays. Some known results are improved and some new
results are obtained. An example is employed to illustrate our feasible results.


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## Body

## 1. Introduction

In recent
years, the delayed cellular neural networks (DCNNs) have been
extensively studied because of their immense potentials of
application perspective in different areas such as pattern
recognition, optimization, and signal and image processing [1–3]. Hence, they have been the object of
intensive analysis by numerous authors, and some interesting results
on the existence and stability of periodic and almost periodic
solutions have been obtained [4–12]. To our knowledge, few
authors have considered global stability of periodic solutions for
the neural networks with bounded and  unbounded time-varying
delays. In theory and application, global stability of periodic
solutions of DCNNs is of great importance since the
global stability of equilibrium points can be considered as a
special case of periodic solution with zero period [8]. Hence, in this paper, we will study the
existence and global exponential stability of periodic solutions of
the following general neural networks with time-varying
delays:(1.1)xi′(t)=−ai(t,xi(t))+∑j=1n[aij(t)fj(xj(t))+bij(t)fj(xj(t−τij(t)))+cij(t)∫−∞tkij(t−s)gj(xj(s))ds]+Ii(t),i=1,2,…,n,where xi(t) is the state of
neuron, i=1,2,…,n,A=(aij)n×n,B=(bij)n×n, and C=(cij)n×n are connection
matrices, I=(I1,I2,…,In)T is the input
function, and f(x)=(f1(x1),f2(x2),…,fn(xn))T is the
activation function of the neurons.DCNNs in [4–12] and the references cited therein are
special cases of (1.1). In particular, when ai(t,xi(t))=aixi(t),aij(t)=aij,bij(t)=bij,cij(t)=cij,Ii(t)=Ii,ai,aij,bij,cij,Ii are constants,
the authors of [9] considered the
existence and global exponential stability for (1.1) with periodic
impulses. The methods used in [9] are
Mawhin's coincidence degree theorem [13] and Lyapunov functions. In [14], by using Mawhin's coincidence degree theorem [13], the authors investigated the global
existence of positive periodic solutions of mutualism systems with
bounded and unbounded time-varying delays, and some sufficient
conditions are obtained. In [12], the
authors considered (1.1) when ai,aij,bij,cij are constants.We assume what follows.(H1)
aij(t),bij(t),cij(t),τij(t),Ii(t) are continuous ω-periodic
functions, and ai(t,xi(t)) are continuous ω-periodic
in the first variable. τij(t)≥0,i,j=1,2,…,n.(H2)
There exist
positive constantsa¯i and a¯i such that (ai(t,u)−ai(t,v))(u−v)>0, and a¯i|u|≤|ai(t,u)|≤a¯i|u|, for all t,u,v∈R,i=1,2,…,n.(H3)
fj,gj∈C(R,R),j=1,2,…,n. There are positive constants Ljf>0, Ljg>0, such that |gj(x)−gj(y)|≤Ljg|x−y|,|fj(x)−fj(y)|≤Ljf|x−y|, for   all x,y∈R.(H4)
The delay
kernelskij:[0,+∞)→R are continuous,
integrable, and satisfy(1.2)∫0+∞|kij(s)|ds≤βij,i,j=1,2,…,n.(H5)
There exists a
constantα0>0 such
that(1.3)∫0+∞|kij(s)|eα0sds≤+∞,i,j=1,2,…,n.The organization of this paper is as follows. In
Section2, we introduce some notations and definitions, and state
some preliminary results needed in later sections. We then study, in
Section 3, the existence of periodic solutions of system (1.1) by
using the continuation theorem of coincidence degree theorem
proposed by Gains and Mawhin [13]. In
Section 4, by constructing Lyapunov function we will derive
sufficient conditions for the global exponential stability of the
periodic solution of system (1.1). At last, an example is employed
to illustrate the feasible results of this paper.


## 2. Preliminaries

For convenience, we usea¯ij,b¯ij,c¯ij,I¯i to denote the
maximums of |aij(t)|,|bij(t)|,|cij(t)|,|Ii(t)|, respectively.
We also use symbols
τ=max{τij(t):t∈[0,ω],i,j=1,2,…,n},x=(x1,x2,…,xn)T to denote a
column vector, in which the symbol (T) denotes the
transpose of a vector. En denotes the
identity matrix of size n. A matrix or vector D≥0 means that all
entries of D are greater
than or equal to zero (resp., D>0).
For matrices or vectors D and E, D≥E (resp., D>E)
means that D−E≥0 (resp., D−E>0).The initial conditionϕ=(ϕ1,…,ϕn)T of (1.1) is of
the form(2.1)xi(s)=ϕi(s),s∈(−∞,0],i=1,…,n,where ϕi(s),i=1,2,…,n, are continuous ω-periodic
solutions.Definition 2.1.
Letx*(t) be an ω-periodic
solution of (1.1) with initial value ϕ*=(ϕ1*,…,ϕn*)T∈C((−∞,0];Rn). If there exist constants α>0 and P>1 such that for
every solution x(t) of (1.1) with
initial value ϕ∈C((−∞,0];Rn),(2.2)|xi(t)−xi*(t)|≤P∥ϕ−ϕ*∥e−αt,∀t>0,i=1,2,…,n,where ∥ϕ−ϕ*∥=sups≤0max1≤i≤n|ϕi(s)−ϕi*(s)|, then x*(t) is said to be
globally exponentially stable.Definition 2.2. (See [15, 16]).
A realn×n matrix F=(fij)n×n is said to be a
nonsingular M-matrix
if fij≤0,i,j=1,2,…,n,i≠j, and F−1≥0, where F−1 denotes the
inverse of F.Lemma 2.3 (See [15, 16]).
LetF=(fij)n×n with fij≤0,i,j=1,2,…,n,i≠j. Then the following statements are equivalent:
(1)
F is a
nonsingular M-matrix,(2)
there exists a
vectorηT=(η1,η2,…,ηn)>0 such that ηF>0,(3)
there exists a
vectorξ=(ξ1,ξ2,…,ξn)T>0 such that Fξ>0.Lemma 2.4 (See [16]).
LetA≥0 be an n×n matrix and ρ(A)<1, then there exists a vector ξ=(ξ1,ξ2,…,ξn)T>0 such that (En−A)ξ>0, where ρ(A) denotes the
spectral radius of A.To end this section, we introduce Mawhin's
continuation theorem [13, page 40] as follows.Consider an abstract equation in a Banach spaceX, (2.3)Lx=λNx,λ∈(0,1),where L:DomL∩X→X is a Fredholm
operator with index-zero, and λ is a parameter.
Let P and Q denote two
projectors, P:DomL∩X→KerL and Q:X→X/ImL such that ImP=KerL,KerQ=ImL.Lemma 2.5.
LetX be a Banach
space. Suppose that L:DomL∩X→X is a Fredholm
operator with index-zero, let Ω⊂X be an open
bounded set, and let N:X→X be a continuous
operator which is L-compact
on Ω¯. Moreover, assume that the following conditions are
satisfied:
(a)
for eachλ∈(0,1),x∈∂Ω∩DomL,Lx≠λNx,(b)
for eachx∈∂Ω∩KerL,QNx≠0,(c)
deg(QN,Ω∩KerL,0)≠0.
Then, Lx=Nx has at least
one solution in Ω¯∩DomL.


## 3. Existence of Periodic Solutions

Theorem 3.1.
Let(H1)–(H4) hold. Assume
that the following condition is satisfied:
(H6) there exists a
vector ηT=(η1,…,ηn)>0 such that(3.1)η¯T=(η¯1,…,η¯n)=ηT(En−Γ)>0,where Γ=(γij)n×n,γij=a¯i−1(a¯ijLjf+b¯ijLjf+c¯ijβijLjg),i,j=1,2,…,n.
Then, (1.1) has at least one ω-periodic
solution.Proof.
For convenience, we introduce the
following notations:(3.2)Gi(t)=−ai(t,xi(t))+∑j=1n[aij(t)fj(xj(t))+bij(t)fj(xj(t−τij(t)))+cij(t)∫−∞tkij(t−s)gj(xj(s))ds]+Ii(t),i=1,2,…,n. In order to use Lemma 2.5, we take X={x(t)∈C(R,Rn):x(t+ω)=x(t),t∈R}; then X is a Banach
space with the norm(3.3)∥x∥=max{|x1|0,|x2|0,…,|xn|0},|xi|0=supt∈[0,ω]|xi(t)|,i=1,2,…,n. Set(3.4)Lx=x′,Px=Qx=1ω∫0ωx(t)dt,x∈X,Nx=(G1(t),G2(t),…,Gn(t))T,x∈X.
Obviously, KerL={x∣x∈X,x=h,h∈Rn},ImL={x∣x∈X,∫0ωx(s)ds=0}, and dimKerL=n=codimImL. So, ImL is closed in X. It is easy to show that P and Q are continuous
projectors satisfying ImP=KerL,ImL=KerQ=Im(I−Q). Hence, L is a Fredholm
mapping of index-zero. Furthermore, through an easy computation, we
find that the generalized inverse KP−1:ImL→KerP∩domL of L has the
form(3.5)KP−1(x)=∫0tx(s)ds−1ω∫0ω∫0tx(s)dsdt.Thus,(3.6)QNx=(1ω∫0ωG1(t)dt,…,1ω∫0ωGn(t)dt)T,x∈X,Kp−1(I−Q)Nx=(∫0tG1(s)ds⋮∫0tGj(s)ds⋮∫0tGn(s)ds)−(1ω∫0ω∫0tG1(s)dsdt⋮1ω∫0ω∫0tGj(s)dsdt⋮1ω∫0ω∫0tGn(s)dsdt)−((tω−12)∫0ωG1(s)ds⋮(tω−12)∫0ωGj(s)ds⋮(tω−12)∫0ωGn(s)ds).
Clearly, QN and KP−1(I−Q)N are continuous.
Using the Arzela-Ascoli theorem, it is not difficult to show that
QN(Ω¯), KP−1(I−Q)N(Ω¯) are relatively
compact for any open bounded set Ω⊂X. Therefore, N is L-compact
on Ω¯ for any open
bounded set Ω⊂X.
Now, we reach the position to search for an
appropriate open bounded subsetΩ for the
application of Lemma 2.5. Corresponding to the operator equation
(2.3), we have(3.7)xi′(t)=λ{−ai(t,xi(t))+∑j=1n[aij(t)fj(xj(t))+bij(t)fj(xj(t−τij(t)))+cij(t)∫−∞tkij(t−s)gj(xj(s))ds]+Ii(t)},i=1,2,…,n,λ∈(0,1).Let x(t)=(x1(t),x2(t),…,xn(t))T∈X be a solution
of system (3.7) for some λ∈(0,1). Then, for any i=1,2,…,n,xi are all
continuously differentiable. Thus, there exist ti∈[0,ω],i=1,2,…,n, such that |xi(ti)|=maxt∈[0,ω]|xi(t)|. Hence, xi′(ti)=0,i=1,2,…,n. From (3.7), we
have(3.8)ai(ti,xi(ti))=∑j=1n[aij(ti)fj(xj(ti))+bij(ti)fj(xj(ti−τij(ti)))+cij(ti)∫−∞tikij(ti−s)gj(xj(s))ds]+Ii(ti),i=1,2,…,n.In view of (H1)–(H4), we have(3.9)a¯i|xi(ti)|≤|ai(ti,xi(ti))|≤∑j=1n[|aij(ti)|(|fj(xj(ti))−fj(0)|+|fj(0)|)+|bij(ti)|(|fj(xj(ti−τij(ti)))−fj(0)|+|fj(0)|)+|cij(ti)|∫−∞ti|kij(ti−s)|(|gj(xj(s))−gj(0)|+|gj(0)|)ds]+|Ii(ti)|≤∑j=1n[a¯ijLjf|xj(ti)|+b¯ijLjf|xj(ti−τij(ti))|+c¯ij∫−∞ti|kij(ti−s)|(Ljg|xj(s)|+|gj(0)|)ds+(a¯ij+b¯ij)|fj(0)|]+I¯i≤∑j=1n(a¯ijLjf+b¯ijLjf+c¯ijβijLjg)|xj(tj)|+∑j=1n(a¯ij|fj(0)|+b¯ij|fj(0)|+c¯ijβij|gj(0)|)+I¯i,i=1,2,…,n.
Set Hi=∑j=1na¯i−1(a¯ij|fj(0)|+b¯ij|fj(0)|+c¯ijβij|gj(0)|)+a¯i−1I¯i. Clearly, (3.9) implies that(3.10)|xi(ti)|≤∑j=1nγij|xj(tj)|+Hi,i=1,2,…,n.Thus,(3.11)(En−Γ)(|x1(t1)|,|x2(t2)|,…,|xn(tn)|)T≤(H1,H2,…,Hn)T:=H;together with (H6), we have(3.12)min{η¯1,η¯2,…,η¯n}(|x1(t1)|+|x2(t2)|+⋯+|xn(tn)|)≤η¯1|x1(t1)|+η¯2|x2(t2)|+⋯+η¯n|xn(tn)|=ηT(En−Γ)(|x1(t1)|,|x2(t2)|,…,|xn(tn)|)T≤ηTH.Therefore,(3.13)|xi|0=|xi(ti)|≤ηTHmin{η¯1,η¯2,…,η¯n}:ν,i=1,2,…,n.Again from (H6), it follows from Lemma 2.3 that En−Γ is a
nonsingular M-matrix,
and there exists a vector ξ=(ξ1,ξ2,…,ξn)T>0 such that (En−Γ)ξ>0, which implies that we can choose a constant d>1 such that ξ¯=(ξ¯1,ξ¯2,…,ξ¯n)T=dξ and ξ¯i=dξi>ν,i=1,2,…,n, and(3.14)(En−Γ)ξ¯=d(En−Γ)ξ>H. We take(3.15)Ω={x(t)∈X:−ξ¯<x(t)<ξ¯,∀t∈R},which satisfies Condition (a) of Lemma 2.5.
If x(t)∈∂Ω∩kerL, then x(t) is a constant
vector in Rn, and there exists some i∈{1,2,…,n} such that |xi|=ξ¯i. We claim that(3.16)|(QNx)i|>0,x∈∂Ω∩kerL.By way of contradiction, suppose
that |(QNx)i|=0; then there
exists some t*∈[0,ω] such
that(3.17)−ai(t*,xi)+∑j=1n[aij(t*)fj(xj)+bij(t*)fj(xj)+cij(t*)∫−∞t*kij(t*−s)gj(xj(s))ds]+Ii(t*)=0,which implies
that(3.18)a¯iξ¯i=a¯i|xi|≤|ai(t*,xi)|=|∑j=1n[aij(t*)fj(xj)+bij(t*)fj(xj)+cij(t*)∫−∞t*kij(t*−s)gj(xj(s))ds]+Ii(t*)|≤∑j=1n(a¯ijLjf+b¯ijLjf+c¯ijβijLjg)|xj|+∑j=1n(a¯ij|fj(0)|+b¯ij|fj(0)|+c¯ijβij|gj(0)|)+I¯i≤∑j=1n(a¯ijLjf+b¯ijLjf+c¯ijβijLjg)ξ¯j+∑j=1n(a¯ij|fj(0)|+b¯ij|fj(0)|+c¯ijβij|gj(0)|)+I¯i;hence(3.19)ξ¯i≤∑j=1nγijξ¯j+Hi.This implies that ((En−Γ)ξ¯)i≤Hi, which contradicts (3.14). Therefore, (3.16) holds, and
hence, Condition (b) of Lemma 2.5 is
satisfied.
Furthermore, we define a continuous functionW:Ω∩KerL×[0,1]→X by(3.20)W(x,μ)=μdiag(−a¯1,−a¯2,…,−a¯n)x+(1−μ)QNx,for all x∈Ω∩KerL=Ω∩Rn and μ∈[0,1]. It follows that(3.21)(W(x,μ))i=−μa¯ixi+(1−μ)1ω∫0ωGi(t)dt. If x(t)∈∂Ω∩KerL, then x(t) is a constant
vector in Rn, and there exists some i∈{1,2,…,n} such that |xi|=ξ¯i. We claim that(3.22)|(W(x,μ))i|>0,x∈∂Ω∩kerL,μ∈[0,1].By way of contradiction, suppose
that |(W(x,μ))i|=0; then there
exists some t˜∈[0,ω] such
that(3.23)−μa¯ixi+(1−μ){−ai(t˜,xi)+∑j=1n[aij(t˜)fj(xj)+bij(t˜)fj(xj)+cij(t˜)∫−∞t˜kij(t˜−s)gj(xj(s))ds]+Ii(t˜)}=0,that is,(3.24)μ(ai(t˜,xi)−a¯ixi)−ai(t˜,xi)=−(1−μ){∑j=1n[aij(t˜)fj(xj)+bij(t˜)fj(xj)+cij(t˜)∫−∞t˜kij(t˜−s)gj(xj(s))ds]+Ii(t˜)}.Now, we will consider the
following two cases.
Case 1.
Ifxi>0, from (H2), we have(3.25)ai(t˜,xi)>0,ai(t˜,xi)−a¯ixi≤0.Then, from (3.24), we
have(3.26)ai(t˜,xi)≤(1−μ){∑j=1n[aij(t˜)fj(xj)+bij(t˜)fj(xj)+cij(t˜)∫−∞t˜kij(t˜−s)gj(xj(s))ds]+Ii(t˜)},which implies
that(3.27)a¯iξ¯i=a¯i|xi|≤|ai(t˜,xi)|≤|∑j=1n[aij(t˜)fj(xj)+bij(t˜)fj(xj)+cij(t˜)∫−∞t˜kij(t˜−s)gj(xj(s))ds]+Ii(t˜)|≤∑j=1n(a¯ijLjf+b¯ijLjf+c¯ijβijLjg)|xj|+∑j=1n(a¯ij|fj(0)|+b¯ij|fj(0)|+c¯ijβij|gj(0)|)+I¯i≤∑j=1n(a¯ijLjf+b¯ijLjf+c¯ijβijLjg)ξ¯j+∑j=1n(a¯ij|fj(0)|+b¯ij|fj(0)|+c¯ijβij|gj(0)|)+I¯i.
Hence,(3.28)ξ¯i≤∑j=1nγijξ¯j+Hi;this implies that ((En−Γ)ξ¯)i≤Hi, which contradicts (3.14). Therefore, (3.22) holds.
Case 2.
Ifxi<0, from (H2), we have(3.29)ai(t˜,xi)<0,ai(t˜,xi)−a¯ixi≥0.Then, from (3.24), we
have(3.30)ai(t˜,xi)≥(1−μ){∑j=1n[aij(t˜)fj(xj)+bij(t˜)fj(xj)+cij(t˜)∫−∞t˜kij(t˜−s)gj(xj)ds]+Ii(t˜)}.The later proof is similar to
that of Case 1. We can also show that (3.22) holds. It follows that W(x,μ)≠0 for x∈∂Ω∩KerL, μ∈[0,1]. Hence, by
homotopy invariance theorem and a¯i>0, i=1,2,…,n, we obtain deg{QN,Ω∩kerL,0}=deg{(−a¯1x1,−a¯2x2,…,−a¯nxn)T,Ω∩kerL,0}≠0 for x∈∂Ω∩KerL. Till now, we have proved that Ω satisfies all
conditions of Lemma 2.5. Therefore, (1.1) has a periodic solution
x*(t). This completes the proof.Corollary 3.2.
Let(H1)–(H4) hold. Assume
that En−Γ is a
nonsingular M-matrix,
then (1.1) has at least one ω-periodic
solution, where Γ is defined as
above.Corollary 3.3.
Let(H1)–(H4) hold. Assume
that ρ(Γ)<1, then (1.1) has at least one ω-periodic
solution, where Γ is defined as
above.Proof.
Obviously,Γ≥0. By ρ(Γ)<1 and from Lemma 2.4,
there exists a vector ξ=(ξ1,ξ2,…,ξn)T>0 such that (En−A)ξ>0. The remaining part of the proof is the same as that
of Theorem 3.1.


## 4. Global Exponential Stability of Periodic Solutions

In this section, we will construct some suitable
Lyapunov functionals to derive sufficient conditions which ensure
that (1.1) has a unique ω-periodic
solution, and all solutions of (1.1) exponentially converge to its
unique ω-periodic
solution.Theorem 4.1.
Assume that(H1)–(H6) hold and
(H7)|ai(t,u)−ai(t,v)|≥a¯i|u−v|, for all t,u,v∈R,i=1,2,…,n, where a¯i,i=1,2,…,n, are the
positive constants of Hypothesis (H2). Then, (1.1) has exactly one ω-periodic
solution, which is globally exponentially stable.Proof.
By
Theorem3.1, there exists an ω-periodic
solution x*(t)=(x1*(t),x2*(t),…,xn*(t))T of (1.1).
Suppose that x(t) is an arbitrary
solution of (1.1). Set z(t)=(z1(t),z2(t),…,zn(t))T=x(t)−x*(t). Then,(4.1)zi′(t)=−di(t,zi(t))+∑j=1n[aij(t)wj(zj(t))+bij(t)wj(zj(t−τij(t)))+cij(t)∫−∞tkij(t−s)uj(zj(s))ds],i=1,2,…,n,where(4.2)di(t,zi(t))=ai(t,zi(t)+xi*(t))−ai(t,xi*(t)),wj(zj(t))=fj(zj(t)+xj*(t))−fj(xj*(t)),wj(zj(t−τij(t)))=fj(zj(t−τij(t))+xj*(t−τij(t)))−fj(xj*(t−τij(t))),uj(zj(t))=gj(zj(t)+xj*(t))−gj(xj*(t)). From (H6), it follows from Lemma 2.3 that En−Γ is a
nonsingular M-matrix,
and there exists a vector ξ=(ξ1,ξ2,…,ξn)T>0 such that (En−Γ)ξ>0. Then,(4.3)−a¯iξi+∑j=1nξj(a¯ijLjf+b¯ijLjf+c¯ijβijLjg)<0,i=1,2,…,n.Set(4.4)hi(λ)=(λ−a¯i)ξi+∑j=1nξj(a¯ijLjf+eλτb¯ijLjf+c¯ijLjg∫0+∞|kij(s)|eλsds),i=1,2,…,n.Clearly, hi(λ),i=1,2,…,n, are continuous functions on [0,α0], where α0 is the positive
constant of Hypothesis (H5). Since(4.5)hi(0)=−a¯iξi+∑j=1nξj(a¯ijLjf+b¯ijLjf+c¯ijLjg∫0+∞|kij(s)|ds)≤−a¯iξi+∑j=1nξj(a¯ijLjf+b¯ijLjf+c¯ijβijLjg)<0,we can choose a positive
constant α∈(0,α0] such
that(4.6)hi(α)=(α−a¯i)ξi+∑j=1nξj(a¯ijLjf+eατb¯ijLjf+c¯ijLjg∫0+∞|kij(s)|eαsds)<0,i=1,2,…,n. Now, we choose a positive constant d such
that(4.7)dξie−αt>1,fort∈(−∞,0],i=1,2,…,n. Define a Lyapunov function V=(V1,V2,…,Vn)T by Vi=eαt|zi(t)|,i=1,2,…,n. In view of
(4.1), we obtain(4.8)d+Vi(t)dt=eαtsgnzi{−di(t,zi(t))+∑j=1n[aij(t)wj(zj(t))+bij(t)wj(zj(t−τij(t)))+cij(t)∫−∞tkij(t−s)uj(zj(s))ds]}+αeαt|zi(t)|≤eαt{(α−a¯i)|zi(t)|+∑j=1n[a¯ijLjf|zj(t)|+b¯ijLjf|zj(t−τij(t))|+c¯ijLjg∫−∞t|kij(t−s)||zj(s)|ds]}=(α−a¯i)eαt|zi(t)|+∑j=1n[a¯ijLjfeαt|zj(t)|+eατij(t)b¯ijLjfeα(t−τij(t))|zj(t−τij(t))|+c¯ijLjg∫−∞t|kij(t−s)|eα(t−s)eαs|zj(s)|ds]≤(α−a¯i)Vi(t)+∑j=1n[a¯ijLjfVj(t)+eατb¯ijLjfVj(t−τij(t))+c¯ijLjg∫−∞t|kij(t−s)|eα(t−s)Vj(s)ds],i=1,2,…,n.From (4.7), we
have(4.9)dξi∥ϕ−ϕ*∥e−αt>∥ϕ−ϕ*∥≥|zi(t)|,fort∈(−∞,0],i=1,2,…,n,which implies
that(4.10)|zi(t)|<dξi∥ϕ−ϕ*∥e−αt,fort∈(−∞,0],i=1,2,…,n,where ∥ϕ−ϕ*∥ is defined as
that in Definition 2.1.
We claim that(4.11)Vi(t)=|zi(t)|eαt<dξi∥ϕ−ϕ*∥,i=1,2,…,n,∀t>0.Contrarily, there must exist i0∈{1,2,…,n} and t˜>0 such
that
(4.12)Vi0(t˜)=dξi0∥ϕ−ϕ*∥,Vθ(t˜)≤dξθ∥ϕ−ϕ*∥,θ=1,2,…,i0−1,i0+1,…,n,Vi(t)<dξi∥ϕ−ϕ*∥,∀t∈(−∞,t˜),i=1,2,…,n.Together with (4.8) and (4.12),
we obtain(4.13)0≤d+(Vi0(t˜)−dξi0∥ϕ−ϕ*∥)dt=d+Vi0(t˜)dt≤(α−a¯i0)Vi0(t˜)+∑j=1n[a¯i0jLjfVj(t˜)+eατb¯i0jLjfVj(t˜−τi0j(t˜))+c¯i0jLjg∫−∞t˜|ki0j(t˜−s)|eα(t˜−s)Vj(s)ds]≤(α−a¯i0)dξi0∥ϕ−ϕ*∥+∑j=1ndξj∥ϕ−ϕ*∥(a¯i0jLjf+eατb¯i0jLjf+c¯i0jLjg∫0+∞|ki0j(s)|eαsds).Hence,(4.14)0≤(α−a¯i0)ξi0+∑j=1nξj(a¯i0jLjf+eατb¯i0jLjf+c¯i0jLjg∫0+∞|ki0j(s)|eαsds),which contradicts (4.6).
Therefore, (4.11) holds. It follows that(4.15)|zi(t)|<dξi∥ϕ−ϕ*∥e−αt,t>0,i=1,2,…,n.Let M=max1≤i≤n{dξi+1}. Then, from (4.15), we get(4.16)|xi(t)−xi*(t)|=|zi(t)|≤M∥ϕ−ϕ*∥e−αt,∀t>0,i=1,2,…,n.In view of Definition 2.1, the ω-periodic
solution x*(t) of system (1.1)
is globally exponentially stable. This completes the proof.Corollary 4.2.
Let(H1)–(H7) hold. Assume
that En−Γ is a
nonsingular M-matrix
or ρ(Γ)<1. Then, system (1.1) has exactly one ω-periodic
solution, which is globally exponentially stable, where Γ is defined as
that in (H6).Remark 4.3.
As a special case,ai(t,xi(t))=ai(t)xi(t) and ai(t)>0. Let a¯i=min{|ai(t)|:t∈[0,ω]},a¯i=max{|ai(t)|:t∈[0,ω]}, then Lia=a¯i,i=1,2,…,n. Obviously, (H2) and (H7) hold.Remark 4.4.
Whenai(t,xi(t))=cixi(t),ci are constants,
and cij≡0, i,j=1,2,…,n, one can easily
know that [11, Theorems 1 and 2]
are direct corollaries of Theorems 3.1 and 4.1 of this paper,
respectively. Moreover, we need not the following assumption:
(H0)fj(0)=0,gj(0)=0,0≤τij′(t)<1,i,j=1,2,…,n.Very recently, Zhou and Hu [12] considered the global
exponential periodicity and stability of the following cellular
neural networks:(4.17)xi′(t)=−dixi(t)+∑j=1n[aijfj(xj(t))+bijfj(xj(t−τij(t)))+cij∫−∞tkij(t−s)gj(xj(s))ds]+Ii(t),i=1,2,…,n,where di>0,aij,bij,cij,i,j=1,2,…,n, are constants, Ii(t+ω)=Ii(t),i,j=1,2,…,n,
τij′(t)<1, and 0≤τij(t)≤τ. The assumptions in [12] on the delay kernels kij of (4.17) are
as follows.(F1)
The delay
kernelskij:R+→R+ are real-valued
nonnegative continuous functions and ∫0+∞kij(s)ds=1,i,j=1,2,…,n.(F2)
∫0+∞skij(s)ds≤+∞,i,j=1,2,…,n.(F3)
There exists a
positive numberμ such that ∫0+∞eμskij(s)ds≤+∞,i,j=1,2,…,n.
Obviously,
(4.17) is a special case of (1.1). Moreover, (3.1) of Theorem 3.1 and (3.15) of Theorem 3.2 in [12] are
special cases of (4.3) in this paper, that is, ξ=(1,1,…,1)T,Ljf=Ljg, and βij=1. So, [12, Theorems 3.1 and 3.2] are special cases
of Theorem 4.1 in this paper. To summarize, the results of this
paper are completely new and generalize the results of [4–12] and
the references cited therein.


## 5. Application

In this section, we give an example to illustrate that
our results are feasible. Consider the following simple DCNNs with
time-varying delays:(5.1)xi′(t)=−ai(t,xi(t))+∑j=12[aij(t)fj(xj(t))+bij(t)fj(xj(t−τij(t)))+cij(t)∫−∞tkij(t−s)gj(xj(s))ds]+Ii(t),i=1,2,where fj(x)=gj(x)=|x|,ai(t,x)=2x+sinx,Ii(t)=sin40πt,τij(t)=(cos20πt)2,kij(t)=sinte−t, and aij(t)=bij(t)=cij(t)=(1/12)cos40πt,i,j=1,2. Then, we have
(5.2)ω=120,Ljf=Ljg=1,a¯i=1,a¯i=3,βij=1,a¯ij=b¯ij=c¯ij=112,i,j=1,2. Hence,(5.3)Γ=(14141414).
We take ηT=(η1,η2)=(1,1), then(5.4)η¯=(11)(34−14−1434)=(1212)>(00).It is easy to check that all the
conditions needed in Theorem 4.1 are satisfied. Therefore, (5.1) has
a unique global exponential 1/20-periodic
solution.Remark 5.1.
Becauseai(t,xi(t)) is not linear
about xi(t), thus none of the results in [4–12, 14] can be applied
to (5.1). This implies that the results of this paper are new.



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*Source: 843695-2008-05-15.xml*