# Periodic Solutions for a Numerical Discretization Neural Network

Authors: Chun Lu
Journal: Discrete Dynamics in Nature and Society
 (2010)

---

## Abstract

The existence and global exponential stability of periodic solutions for a class of numerical discretization neural networks are considered. Using coincidence degree theory and Lyapunov method, sufficient conditions for the existence and global exponential stability of periodic solutions are obtained. Numerical simulations are given to illustrate the results.


---

## Body

## 1. Introduction

In this paper, we investigate the existence and stability of periodic solutions for a numerical discretization neural network, which results from theθ-method for neural networks with finite delays and distributed delays:(1.1)ẋi(t)=-bi(t)xi(t)+fi(∫0+∞dt,x1(t-τi1(t)),…,xm(t-τim(t));∫0+∞ki1(s)x1(t-s)ds,…,∫0+∞kim(s)xm(t-s)ds)+Ii(t),t≥0,τij(t)≥0,i,j=1,…,m.
System (1.1) is a more general form of neural networks with delay. Many authors have discussed other neural networks with delays [1–8], and most of their systems can be deduced from (1.1).For system (1.1), we make the following assumptions. (H1)
For eachi∈{1,…,m}, bi(t) is bounded and continuous with bi(t)>0 and bi(t+ω¯)=bi(t) for t∈R+, Ii(t) is bounded and continuous, and Ii(t+ω¯)=Ii(t) for t∈R+, where ω¯ is a positive constant.(H2)
For eachi,j∈{1,…,m},kij:R+→R+ satisfies(1.2)∫0+∞kij(s)ds=1,∫0+∞skij(s)ds<+∞.(H3)
For eachi∈{1,…,m},fi:R+×Rm×Rm→R is bounded and continuous and there exists nonnegative, bounded, and continuous functions αij(t),βij(t) defined on R+ such that
(1.3)|fi(t,u1,…,um;v1,…,vm)-fi(t,u¯1,…,u¯m;v¯1,…,v¯m)|≤∑j=1m[αij(t)|uj-u¯j|+βij(t)|vj-v¯j|],i=1,…,m.
for any (u1,…,um),(u¯1,…,u¯m),(v1,…,vm),(v¯1,…,v¯m)∈Rm.For system (1.1), we consider initial conditions of the form(1.4)xi(t)=φi(t),t∈(-∞,0],i=1,…,m,
where φi(t) is bounded and continuous on (-∞,0].For convenience in our study, we adopt the following notations. LetZ denote the set of all integers, Z0+={0,1,2,…},[a,b]Z={a,a+1,…,b-1,b} where a,b∈Z,a≤b, and [a,∞)Z={a,a+1,a+2,…} where a∈Z. We begin approximating the continuous-time network (1.1) by replacing the integral terms with discrete sums of the form(1.5)∫0+∞kij(s)xj(t-s)ds≈∑[s/h]=1+∞ωij(h)kij([sh]h)xj([th]h-[sh]h)
for t∈[nh,(n+1)h),s∈[ph,(p+1)h),n∈Z0+,p∈Z+, where [r] denotes the integer part of a real number r, h>0 is a fixed number denoting a uniform discretization step size satisfying h=ω̅/ω, ω∈N, ωij(h)>0, for h>0 and ωij(h)≈h+O(h2) for small h>0. We note that ωij(h) are chosen so that the analogue kernels(1.6)ℋij([sh]h)=ωij(h)kij([sh]h)
satisfy the following properties:(H4)
ℋij:Z+→[0,+∞),∑p=1+∞ℋij(p)=1,∑p=1+∞ℋij(p)p<∞.This analogue has been employed elsewhere (see, e.g., [9–11] in the formulation of discrete-time analogues of the distributed delay). With (1.5) and (1.6), we approximate (1.1) by differential equations with piecewise constant arguments of the form(1.7)ẋi(t)=-bi(t)xi(t)+fi(t,x1(t-τi1(t)),…,xm(t-τim(t));∑[s/h]=1+∞ℋi1([sh]h)x1([th]h-[sh]h),…,∑[s/h]=1+∞ℋim([sh]h)xm([th]h-[sh]h))+Ii(t)
for t∈[nh,(n+1)h),s∈[ph,(p+1)h), and n∈Z0+,p∈Z+. Noting that [t/h]=n, and [s/h]=p and adopting the notation u(n)=u(nh), we rewrite (1.7) as(1.8)ẋi(t)=-bi(t)xi(t)+fi(∑p=1+∞ft,x1(t-τi1(t)),…,xm(t-τim(t));∑p=1+∞ℋi1(p)x1(n-p),…,∑p=1+∞ℋim(p)xm(n-p))+Ii(t),t≥0,i=1,…,m.
The initial values of (1.8) will be given below in (1.14).The application ofθ-method to the differential equation (1.9)ẋ(t)=f(t,x(t))
gives(1.10)xn+1=xn+hθf(tn,xn)+h(1-θ)f(tn+1,xn+1),0≤θ≤1,
where h is step size.Applyingθ-method (1.10) to (1.8) over the interval [nh,t], where t<(n+1)h and adopting the notation u(n)=u(nh), we have(1.11)xi(t)=xi(n)+(t-nh)θ×(∑p=1+∞ℋi1-bi(n)xi(n)+fi(∑p=1+∞ℋi1n,x1(n-τi1(n)),…,xm(n-τim(n));∑p=1+∞ℋi1(p)x1(n-p),…,∑p=1+∞ℋim(p)xm(n-p))+Ii(n)∑p=1+∞ℋi1)+(t-nh)(1-θ)×(∑p=1+∞ℋi1-bi(t)xi(t)+fi(∑p=1+∞ℋi1t,x1(t-τi1(t)),…,xm(t-τim(t));∑p=1+∞ℋi1(p)x1(n-p),…,∑p=1+∞ℋim(p)xm(n-p))+Ii(t)∑p=1+∞ℋi1),i=1,…,m,n∈Z0+,here0≤θ≤1,
and by allowing t→(n+1)h in the above, we obtain(1.12)xi(n+1)=xi(n)+hθ×(∑p=1+∞ℋi1-bi(n)xi(n)+fi(∑p=1+∞ℋi1n,x1(n-τi1(n)),…,xm(n-τim(n));∑p=1+∞ℋi1(p)x1(n-p),…,∑p=1+∞ℋim(p)xm(n-p))+Ii(n)∑p=1+∞ℋi1)+h(1-θ)×(∑p=1+∞ℋi1-bi(n+1)xi(n+1)+fi(∑p=1+∞ℋi1n+1,x1(n+1-τi1(n+1)),…,xm(n+1-τim(n+1));∑p=1+∞ℋi1(p)x1(n-p),…,∑p=1+∞ℋim(p)xm(n-p))+Ii(n+1)),i=1,…,m,n∈Z0+,here0≤θ≤1.
Then we have(1.13)xi(n+1)=11+(1-θ)hbi(n+1)×(∑p=1+∞ℋi1(1-θhbi(n))xi(n)+θh(∑p=1+∞ℋi1fi(∑p=1+∞ℋi1n,x1(n-τi1(n)),…,xm(n-τim(n));∑p=1+∞ℋi1(p)x1(n-p),…,∑p=1+∞ℋim(p)xm(n-p))+Ii(n)∑p=1+∞ℋim)+(1-θ)h(fi(∑p=1+∞ℋi1n+1,x1(n+1-τi1(n+1)),…,xm(n+1-τim(n+1));∑p=1+∞ℋi1(p)x1(n-p),…,∑p=1+∞ℋim(p)xm(n-p))+Ii(n+1))),i=1,…,m,n∈Z0+,here0≤θ≤1.
One can show that (1.13) converges towards (1.1) when h→0+. In studying the discrete-time analogue (1.13), we assume that(H5)
h∈(0,∞),bi:Z→(0,∞),τij:Z→Z0+,i,j=1,2,…,m, and the function fi satisfies (H3). The system (1.13) is supplemented with initial values given by(1.14)xi(s)=φi(s),s∈Z0-={0,-1,-2,…}.
For convenience, we will use the following notations (1.15)Iω={0,1,…,ω-1},u¯=1ω∑k=0ω-1u(k),
where {u(k)} is ω-periodic sequence of real numbers defined for k∈Z and the notations: (1.16)αijM=maxn∈Zαij(n),βijM=maxn∈Zβij(n),bi̲=minn∈Iω{bi(n)},bi¯=maxn∈Iω{bi(n)},i=1,2,…,m,M=supu∈R+×Rm×Rm{|fi(u)|,i=1,2,…,m},JM=maxn∈Iω{|Ii(n)|,i=1,2,…,m}.


## 2. Existence of Periodic Solutions

In this section, based on Mawhin's continuation theorem (see [12–14]), we will study the existence of at least one periodic solution of (1.13).LetX, Z be normed vector spaces, L:DomL⊂X→Z a linear mapping, and N:X→Z a continuous mapping. This mapping L will be called a Fredholm mapping of index zero if dim Ker L = codim Im L<∞ and Im L is closed in Z. If L is a Fredholm mapping of index zero and there exist continuous projectors P:X→X and Q:Z→Z such that Im P = Ker L, Ker Q = Im L = Im(I-Q), it follows that L∣DomL∩KerP:(I-P)X→ Im L is invertible; we denote the inverse of that map 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 exist isomorphisms J:ImQ→KerL.Lemma 2.1.
LetL be a Fredholm mapping of index zero and let N be L-compact on Ω¯. Suppose that(a)
for eachλ∈(0,1) every solution x of Lx=λNx is such that x∉∂Ω,(b)
QNx≠0 for each x∈∂Ω∩KerL and deg{JQN,Ω∩KerL, 0}≠0.Then the equationLx=Nx has at least one solution lying in DomL∩Ω¯.Theorem 2.2.
Assume that(H1)–(H5) and bi¯h<1 hold, Then system (1.13) has at least one ω-periodic solution.Proof.
Define(2.1)lm={x=x(k):x(k)∈Rm,k∈Z},|x|=(∑i=1mxi2)1/2,∀x∈Rm.
Let lω⊂lm denote the subspace of all ω-periodic sequences equipped with the norm ∥·∥, that is, ∥x∥=maxk∈Iω|x(k)|, for any x(k)={(x1(k),x2(k),…,xm(k))T,k∈Z}∈lω.
It is easy to prove thatlω is a finite-dimensional Banach space.
Let(2.2)l0ω={y={y(k)}∈lω:1ω∑k=0ω-1y(k)=0},lcω={y={y(k)}∈lω:y(k)=c∈Rm,k∈Z},
then it follows that l0ω and lcω are both closed linear subspaces of lω and
(2.3)lω=l0ω⊕lcω,dimlcω=m.
Now let us define X=Y=lω,(Lx)(k)=x(k+1)-x(k),x∈X,k∈Z, and
(2.4)(Nxi)(n)=-h(θbi(n)+(1-θ)bi(n+1))1+(1-θ)hbi(n+1)xi(n)+θh1+(1-θ)hbi(n+1)×(fi(n,x1(n-τi1(n)),…,xm(n-τim(n));∑p=1+∞ℋi1(p)x1(n-p),…,∑p=1+∞ℋim(p)xm(n-p))+Ii(n))+(1-θ)h1+(1-θ)hbi(n+1)×(fi(∑p=1+∞ℋi1n+1,x1(n+1-τi1(n+1)),…,xm(n+1-τim(n+1));∑p=1+∞ℋi1(p)x1(n-p),…,∑p=1+∞ℋim(p)xm(n-p))+Ii(n+1)),i=1,…,m,n∈Z0+.
It is easy to see that L is a bounded linear operator and
(2.5)KerL=lcω,ImL=l0ω,
as well as
(2.6)dimKerL=m=codimImL;
then it follows that L is a Fredholm mapping of index zero.
Define(2.7)Px=1ω∑k=0ω-1x(k),x∈X,Qy=1ω∑k=0ω-1y(k),y∈Y.
It is not difficult to show that P and Q are continuous projectors such that
(2.8)ImP=KerL,KerQ=ImL=Im(I-Q).
Furthermore, the generalized inverse(to L) kP:ImL→KerP∩DomL is given by
(2.9)(kPy)(n)=∑ν=0n-1y(ν)-1ω∑ν=1ω∑s=0ν-1y(s),n∈[0,ω-1]Z.
Clearly, QN and kP(I-Q)N are continuous, since X is a finite-dimensional Banach space. Using the Arzela-Ascoli theorem, it is not difficult to show that kP(I-Q)N(Ω¯)¯ is compact for any open bounded set Ω⊂X. Moreover, QN(Ω¯) is bounded. Thus, N is L-compact on Ω¯. Since Im Q=KerL, the isomorphic mapping J from Im Q to Ker L is I. We now are in position to search for an appropriate open, bounded subset Ω⊂X for the continuation theorem.
Corresponding to operator equationLx=λNx,λ∈(0,1), we have(2.10)xi(n+1)-xi(n)=λ(-h(θbi(n)+(1-θ)bi(n+1))1+(1-θ)hbi(n+1)xi(n)+θh1+(1-θ)hbi(n+1)×(fi(∑p=1+∞ℋi1n,x1(n-τi1(n)),…,xm(n-τim(n));∑p=1+∞ℋi1(p)x1(n-p),…,∑p=1+∞ℋim(p)xm(n-p))+Ii(n))+(1-θ)h1+(1-θ)hbi(n+1)×(fi(∑p=1+∞ℋimn+1,x1(n+1-τi1(n+1)),…,xm(n+1-τim(n+1));∑p=1+∞ℋi1(p)x1(n-p),…,∑p=1+∞ℋim(p)xm(n-p))+Ii(n+1))),i=1,…,m,n∈Z0+.
Suppose that {(x1(n),x2(n),…,xm(n))T}∈X is a solution of system (2.10) for a certain λ∈(0,1). In view of (2.10) and condition bi¯h<1 in Theorem 2.2, we have
(2.11)maxn∈Iω|xi(n)|=maxn∈Iω|xi(n+1)|≤maxn∈Iω(∑p=1+∞ℋi1(1-λh(θbi(n)+(1-θ)bi(n+1))1+(1-θ)hbi(n+1))|xi(n)|+λθh1+(1-θ)hbi(n+1)×|fi(∑p=1+∞ℋi1n,x1(n-τi1(n)),…,xm(n-τim(n));∑p=1+∞ℋi1(p)x1(n-p),…,∑p=1+∞ℋim(p)xm(n-p))+Ii(n)|+λ(1-θ)h1+(1-θ)hbi(n+1)×|fi(∑p=1+∞ℋi1n+1,x1(n+1-τi1(n+1)),…,xm(n+1-τim(n+1));∑p=1+∞ℋi1(p)x1(n-p),…,∑p=1+∞ℋim(p)xm(n-p))+Ii(n+1)|)≤(1-λhbi̲1+hbi¯)maxn∈Iω|xi(n)|+λh(M+JM).
Therefore,
(2.12)bi̲1+hbi¯maxn∈Iω|xi(n)|≤(M+JM),i=1,2,…,m,
that is,
(2.13)maxn∈Iω|xi(n)|≤(M+JM)(1+hbi¯)bi̲:=Ai,i=1,2,…,m.
Denote A=(∑i=1mAi2)1/2+B, where B>0 is a constant. Clearly, A is independent of λ. Now we take Ω={x∈X:∥x∥<A}. This Ω satisfies condition (i) in Lemma 2.1. When x∈∂Ω∩KerL=∂Ω∩Rm, x is a constant vector in Rm with ∥x∥=A. Then, if necessary, we can let A be greater such that
(2.14)xTQNx=∑i=1m(-xi2h∑k=0ω-1θbi(k)+(1-θ)bi(k+1)ω(1+(1-θ)hbi(k+1))+xi∑k=0ω-1(θhω(1+(1-θ)hbi(k+1))(fi(k,x1…,xm)+Ii(k))+(1-θ)hω(1+(1-θ)hbi(k+1))(fi(k+1,x1…,xm)+Ii(k+1)))∑k=0ω-1)≤∑i=1m[-bi̲h1+hb¯i|xi|2+|xi|h(M+JM)]≤-min1≤i≤m{bi̲h1+hbi¯}∥x∥2+mh(M+JM)∥x∥<0.
Therefore, QNx≠0 for any x∈∂Ω∩kerL. Let ψ(γ;x)=-γx+(1-γ)QNx, γ∈[0,1], then for any x∈∂Ω∩kerL, xTψ(γ;x)<0. From the homotopy invariance of Brower degree, it follows that
(2.15)deg{JQN,Ω∩KerL,0}=deg{-x,Ω∩KerL,0}≠0.
Condition (b) of Lemma 2.1 is also satisfied. Thus, by Lemma 2.1 we conclude that Lx=Nx has at least one solution in X, that is, (1.13) has at least one ω-periodic solution. The proof is complete.


## 3. Stability of Periodic Solution

In this section, we shall construct appropriate Lyapunov functions to study the stability of periodic solutions of (1.13).Theorem 3.1.
Assume that(H1)–(H5) holds. Furthermore, assume that τij(n)≡τij≥2,τij∈Z+,n∈Z,i,j=1,2,…,m are constants and
(3.1)bi¯h<1,θbi̲>∑j=1m(αjiM+βjiM),∑p=1+∞pλpℋji(p)<+∞,
then the ω-periodic solution x*(n)={(x1*(n),x2*(n),…,xm*(n))T} of (1.13) is unique and global exponential stability in the sense that there exist constants λ>1 and δ≥1 such that
(3.2)∑i=1m|xi(n)-xi*(n)|≤δ(1λ)n∑i=1m{sups∈Z0-|xi(s)-xi*(s)|},n∈Z0+.Proof.
Letx(n)={(x1(n),x2(n),…,xm(n))⊤} be an arbitrary solution of (1.12), and x*(n)={(x1*(n),x2*(n),…,xm*(n))⊤} a ω-periodic solution of (1.12). Then
(3.3)|xi(n+1)-xi*(n+1)|≤|xi(n)-xi*(n)|(1-θbi̲h)+θh∑j=1mαijM|xj(n-τij)-xj*(n-τij)|+h∑j=1mβijM∑p=1+∞ℋij(p)|xj(n-p)-xj*(n-p)|+(1-θ)h∑j=1mαijM|xj(n+1-τij)-xj*(n+1-τij)|i=1,2,…,m.
Now we consider functions βi(·,·),i=1,2,…,m, defined by
(3.4)βi(υi,n)=1-υi(1-θbi(n)h)-h∑j=1mαjiMυiτji+1-h∑j=1mβjiM∑p=1+∞υip+1ℋji(p),
where υi∈[1,∞),n∈Iω,i=1,2,…,m, since
(3.5)βi(1,n)=θbi(n)h-h∑j=1mαjiM-h∑j=1mβjiM=h(θbi(n)-∑j=1mαjiM-∑j=1mβjiM)≥hη>0,i=1,2,…,m,
where
(3.6)η=min1≤i≤m{bi̲θ-∑j=1mαjiM-∑j=1mβjiM};
using the continuity of βi(υi,n) on [1,+∞) with respect to υi and the fact that βi(υi,n)→-∞ as υi→∞ uniformly in n∈Iω,i=1,2,…,m, we see that there exists υi*(n)∈(1,∞) such that βi(υi*(n),n)=0 for n∈Iω,i=1,2,…,m. By choosing λ=min{υi*(n),n∈Iω,i=1,2,…,m} where λ>1, we obtain βi(λ,n)≥0 for all n∈Iω,i=1,2,…,m with the implications
(3.7)λ(1-θbi(n)h)+h∑j=1mαjiMλτji(n)+1+h∑j=1mβjiM∑p=1+∞λp+1ℋji(p)≤1,n∈Iω,i=1,2,…,m.
Hence,(3.8)λ(1-θbi̲h)+h∑j=1mαjiMλτji(n)+1+h∑j=1mβjiM∑p=1+∞λp+1ℋji(p)≤1,i=1,2,…,m.
Now let us consider
(3.9)ui(n)=λn|xi(n)-xi*(n)|h,n∈Z,i=1,2,…,m.
Using (3.3) and (3.9), we derive that
(3.10)ui(n+1)≤λ(1-θbi̲h)ui(n)+∑j=1mθhαijMλτij+1uj(n-τij)+∑j=1mhβijM∑p=1+∞λp+1ℋij(p)uj(n-p)+∑j=1m(1-θ)hαijMλτij+1uj(n+1-τij).
We consider the Lyapunov function
(3.11)V(n)=∑i=1m(ui(n)+∑j=1mθhαijMλτij+1∑s=n-τijn-1uj(s)+∑j=1mhβijM∑p=1+∞ℋij(p)λp+1∑s=n-pn-1uj(s)+∑j=1m(1-θ)hαijMλτij+1∑s=n+1-τijn-1uj(s)).
Consider the difference ΔV(n)=V(n+1)-V(n) along (3.11), we obtain
(3.12)ΔV(n)=∑i=1m(ui(n+1)-ui(n)+∑j=1mθhαijMλτij+1(uj(n)-uj(n-τij))+∑j=1mhβijM∑p=1+∞ℋij(p)λp+1(uj(n)-uj(n-p))+∑j=1m(1-θ)hαijMλτij+1(uj(n)-uj(n+1-τij)))≤∑i=1m((λ(1-θbi̲h)-1)ui(n)+∑j=1mhαijMλτij+1uj(n)+∑j=1mhβijM∑p=1+∞λp+1ℋij(p)uj(n))≤-∑i=1m(1-λ(1-θbi̲h)-h∑j=1mαjiMλτji+1-h∑j=1mβjiM∑p=1+∞λp+1ℋji(p))ui(n)
and by using (3.8) above, we deduce that ΔV(n)≤0 for n∈Z0+. From this result and (3.9), we have
(3.13)∑i=1mui≤V(n)≤V(0)forn∈Z+.
Thus,(3.14)∑i=1mui=λn∑i=1m|xi(n)-xi*(n)|h≤∑i=1m(ui(0)+∑j=1mθhαijMλτij+1∑s=-τij-1uj(s)+∑j=1mhβijM∑p=1+∞ℋij(p)λp+1∑s=-p-1uj(s)+∑j=1m(1-θ)hαijMλτij+1∑s=1-τij-1uj(s))≤∑i=1m1h(1+h∑j=1mαjiMλτji+1τji+h∑j=1mβjiM∑p=1+∞ℋji(p)λp+1p)sups∈Z0-{|xi(s)-xi*(s)|}.
Therefore, we obtain the assertion in Theorem 3.1:
(3.15)δ=max1≤i≤m{1+h∑j=1mαjiMλτji+1τji+h∑j=1mβjiM∑p=1∞ℋji(p)λp+1p}≥1.
We conclude from (3.2) that the unique periodic solution of (1.13) is global exponential stability. This completes the proof.


## 4. Numerical Simulations

Remark 4.1.
Ifθ=1/2 or θ=1, then (1.13) reduces to a discrete neural network by Trapezoidal method or Euler method (see Examples 4.2 or 4.3).Example 4.2.
One has(4.1)x(n+1)=22+h(cos(7(n+1)π)+5)×((1-2h(cos(7nπ)+5))x(n)+y(n-3)h6+(y(n-3))2+x(n-1)h1+(x(n-1))2+z(n-1)h4+(z(n-1))2+∑p=1100(eh-1)e-phx(n-p)h1+(∑p=1100(eh-1)e-phx(n-p))2+5sin(nπ)+5sin((n+1)π)y(n-3)h6+(y(n-3))2),y(n+1)=23+h(4sin(5(n+1)π)+2)×((1-2h(cos(5nπ)+5))y(n)+x(n-4)h4+(x(n-4))2+y(n-1)h1+(y(n-1))2+z(n-6)h1+(z(n-6))2+∑p=1100(eh-1)e-phx(n-p)h2+(∑p=1100(eh-1)e-phx(n-p))2+5cos(nπ)+4cos((n+1)π)x(n-4)h4+(x(n-4))2),z(n+1)=22+h(4cos(3(n+1)π)+1)×((1-2h(cos(3nπ)+5))z(n)+x(n-3)h6+(x(n-3))2+y(n-1)h1+(y(n-1))2+z(n-7)h1+(z(n-7))2+∑p=1100(eh-1)e-phx(n-p)h4+(∑p=1100(eh-1)e-phx(n-p))2+7cos(nπ)+6cos((n+1)π)x(n-3)h6+(x(n-3))2),
with initial conditions(1)
φ1(n)=sin(πnh)+7,φ2(n)=4cos(3πnh)+1,φ3(n)=2sin(2πnh)-15,(2)
φ1(n)=0.5sin(2πnh)-6,φ2(n)=0.8cos(2πnh)+15,φ3(n)=sin(3πnh)-6,(3)
φ1(n)=1,φ2(n)=3cos(πnh)-15,φ3(n)=2sin(2πnh)+9,respectively.It is easy to verify that(4.2)bi¯h<1,bi̲2>∑j=1m(αjiM+βjiM),
and the function ℋij(p) satisfies (4.3)∑p=1100(eh-1)e-ph≈1,∑p=1100p(eh-1)e-ph<∑p=1+∞p(eh-1)e-ph<+∞.
So condition (H4) is satisfied. Furthermore, using the same method as Theorem 3.1, we can choose a constant λ such that (4.4)∑p=1100pλpℋji(p)=∑p=1100pλp(eh-1)e-ph<∑p=1+∞pλp(eh-1)e-ph=λ(eh-1)e-h(1-λe-h)2<+∞.
Therefore, by using Theorems 2.2 and 3.1, the system (4.1) has a unique positive 2-periodic solution which is global exponential stability (see Figure 1).θ=1/2, h = 0.025.
(a)(b)(c)Example 4.3.
One has(4.5)x(n+1)=x(n)-4h(sin(7nπ)+5)x(n)+x(n)h1+[x(n)]2+y(n)h2+[y(n)]2+z(n)h3+[z(n)]2+x(n-1)h1+[x(n-1)]2+∑p=1200(eh-1)e-phx(n-p)h1+[∑p=1200(eh-1)e-phx(n-p)]2+5sin(nπ),y(n+1)=y(n)-4h[sin(5nπ)+7]y(n)+x(n)h1+[x(n)]2+y(n)h2+[y(n)]2+z(n)h3+[z(n)]2+x(n-1)h1+[x(n-1)]2+∑p=1200(eh-1)e-phx(n-p)h2+[∑p=1200(eh-1)e-phx(n-p)]2+6cos(nπ),z(n+1)=z(n)-4h[sin(3nπ)+7]z(n)+x(n)h1+[x(n)]2+y(n)h2+[y(n)]2+z(n)h3+[z(n)]2+x(n-1)h1+[x(n-1)]2+∑p=1200(eh-1)e-phx(n-p)h3+[∑p=1200(eh-1)e-phx(n-p)]2+7sin(nπ),
with initial conditions  (1)
φ1(n)=40,φ2(n)=1,φ3(n)=-20,(2)
φ1(n)=50,φ2(n)=20,φ3(n)=-8,(3)
φ1(n)=-10,φ2(n)=-15,φ3(n)=-25,respectively.Similarly, it is straight forward to check that all the conditions needed in Theorems2.2 and 3.1 are satisfied. Therefore, system (4.5) has exactly one 2-periodic solution which is global exponentially stability (see Figure 2).θ=1, h = 0.025.
(a)(b)(c)(d)



---

*Source: 239787-2010-05-26.xml*