# Global Stability of Almost Periodic Solution of a Class of Neutral-Type BAM Neural Networks

Authors: Tetie Pan; Bao Shi; Jian Yuan
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

A class of BAM neural networks with variable coefficients and neutral delays are investigated. By employing fixed-point theorem, the exponential dichotomy, and differential inequality techniques, we obtain some sufficient conditions to insure the existence and globally exponential stability of almost periodic solution. This is the first time to investigate the almost periodic solution of the BAM neutral neural network and the results of this paper are new, and they extend previously known results.


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## Body

## 1. Introduction

Neural networks have been extensively investigated by experts of many areas such as pattern recognition, associative memory, and combinatorial optimization, recently, see [1–10]. Up to now, many results about stability of bidirectional associative memory (BAM) neural networks have been derived. For these BAM systems, periodic oscillatory behavior, almost periodic oscillatory properties, chaos, and bifurcation are their research contents; generally speaking, almost periodic oscillatory property is a common phenomenon in the real world, and in some aspects, it is more actual than other properties, see [11–21].Time delays cannot be avoided in the hardware implementation of neural networks because of the finite switching speed of amplifiers and the finite signal propagation time in biological networks. The existence of time delay may lead to a system’s instability or oscillation, so delay cannot be neglected in modeling. It is known to all that many practical delay systems can be modelled as differential systems of neutral type, whose differential expression concludes not only the derivative term of the current state, but also concludes the derivative of the past state. It means that state’s changing at the past time may affect the current state. Practically, such phenomenon always appears in the study of automatic control, population dynamics, and so forth, and it is natural and important that systems will contain some information about the derivative of the past state to further describe and model the dynamics for such complex neural reactions [22]. Authors in [18–29] added neutral delay into the neural networks. In these papers, only [18–20] studied the almost periodic solution of the neutral neural networks. For example, in [19] the following network was studied:
(1.1)x˙i(t)=-ci(t)xi(t)+∑j=1n‍aij(t)fj(xj(t-τij(t)))+∑j=1n‍bij(t)gj(x˙j(t-σij(t)))+Ii(t).
Some sufficient conditions are obtained for the existence and globally exponential stability of almost periodic solution by employing fixed-point theorem and differential inequality techniques. References [21–26] studied the global asymptotic stability of equilibrium point, where [22] investigated the equilibrium point of the following BAM neutral neural network with constant coefficients:
(1.2)u˙i(t)=-aiui(t)+∑j=1m‍w1jigj(vj(t-d))+∑j=1n‍w2iju˙j(t-h)+Ii,v˙j(t)=-bjvj(t)+∑i=1n‍r1ijgi(ui(t-h))+∑i=1m‍r2jiv˙i(t-d)+Ji.
By using the Lyapunov method and linear matrix inequality techniques, a new stability criterion was derived. References [27–29] studied the exponential stability of equilibrium point.It is obviously that men always studied the stability of the equilibrium point of the neutral neural networks, and there is little result for the almost periodic solution of neutral neural networks, especially, for the BAM neutral type neural networks. Besides, in papers [11, 23, 27, 28], time delay must be differentiable, and its derivative is bounded, which we think is a strict condition.Motivated by the above discussions, in this paper, we consider the almost periodic solution of a class of BAM neural networks with variable coefficients and neutral delays. By fixed-point theorem and differential inequality techniques, we obtain some sufficient conditions to insure the existence and globally exponential stability of almost periodic solution. To the best of the authors’ knowledge, this is the first time to investigate the almost periodic solution of the BAM neutral neural network, and we can remove delay’s derivable condition, so the results of this paper are new, and they extend previously known results.


## 2. Preliminaries

In this paper, we consider the following system:(2.1)x˙i(t)=-ci(t)xi(t)+∑j=1m‍aij(t)f1j(yj(t-τij(t)))+∑j=1n‍bji(t)f2j(x˙j(t-δ-ji(t)))+Ii(t),y˙j(t)=-dj(t)yj(t)+∑i=1n‍pji(t)g1i(xi(t-δji(t)))+∑i=1m‍qij(t)g2i(y˙j(t-τ-ij(t)))+Jj(t),
where i=1,2,…,n; j=1,2,…,m. xi(t), yj(t) are the states of the ith neuron of X layer and the jth neuron of Y layer, respectively; aij(t), pji(t) and bji(t), qij(t) are the delayed strengths of connectivity and the neutral delayed strengths of connectivity, respectively; f1j, f2j, g1i, g2i are activation functions; Ii(t), Jj(t) stands for the external inputs; τij(t), τ-ij(t), δji(t), and δ-ji(t) correspond to the delays, they are nonnegative; ci(t), dj(t)>0 represent the rate with which the ith neuron of X layer and the jth neuron of Y layer will reset its potential to the resting state in isolation when disconnected from the networks.Throughout this paper, we assume the following.(H1)
ci(t), dj(t), aij(t), pji(t), bji(t), qij(t), τij(t), τ-ij(t), δji(t), δ-ji(t), Ii(t), and Jj(t) are continuous almost periodic functions. Moreover, we let
(2.2)ci+=supt∈R{ci(t)},ci-=inft∈R{ci(t)}>0,dj+=supt∈R{dj(t)},dj-=inft∈R{dj(t)}>0,aij=supt∈R{|aij(t)|}<∞,bji=supt∈R{|bji(t)|}<∞,pji=supt∈R{|pji(t)|}<∞,qij=supt∈R{|qij(t)|}<∞,Ii=supt∈R{|Ii(t)|}<∞,Jj=supt∈R{|Jj(t)|}<∞.(H2)
f1j, f2j, g1i, and g2i are Lipschitz continuous with the Lipschitz constants F1j, F2j, G1i, G2i, and f1j(0)=f2j(0)=g1i(0)=g2i(0)=0.(H3)
Consider(2.3)α=max{max1≤i≤nmax{1ci-,1+ci+ci-}(∑j=1m‍aijF1j+∑j=1n‍bjiF2j),ccccccmax1≤j≤mmax{1dj-,1+dj+dj-}(∑i=1n‍pjiG1i+∑i=1m‍qijG2i)(∑j=1m‍aijF1j+∑j=1n‍bjiF2j)}<1.The initial conditions of system (2.1) are of the following form:
(2.4)xi(t)=φi(t),t∈[-δ,0],δ=supt∈Rmaxi,jmax{δji(t),δ-ji(t)},yj(t)=ϕj(t),t∈[-τ,0],τ=supt∈Rmaxi,jmax{τij(t),τ-ij(t)},
where i=1,2,…,n; j=1,2,…,m; φi(t), ϕj(t) are continuous almost periodic functions.LetX={ψ|ψ=(φ1,φ2,…,φn,ϕ1,ϕ2,…,ϕm)T,where φi, ϕj:R→R are continuously differentiable almost periodic functions. For any ψ∈X, ψ(t)=(φ1(t),φ2(t),…,φn(t),ϕ1(t),ϕ2(t),…,ϕm(t))T. We define ∥ψ(t)∥1=max{∥ψ(t)∥0,∥ψ˙(t)∥0}, where ∥ψ(t)∥0=max{max1≤i≤n{|φi(t)|},max1≤j≤m{|ϕi(t)|}}, and ψ˙(t) is the derivative of ψ at t. Let ∥ψ∥=supt∈R∥ψ(t)∥1, then X is a Banach space.The following definitions and lemmas will be used in this paper.Definition 2.1 (see [11]).
Letx(t):R→Rn be continuous in t. x(t) is said to be almost periodic on R, if for any ε>0, the set T(x,ε)={w|x(t+w)-x(t)<ε,forallt∈R} is relatively dense, that is, for all ε>0, it is possible to find a real number l=l(ε)>0, for any interval length l(ε), there exists a number τ=τ(ε) in this interval such that |x(t+τ)-x(t)|<ε, for all t∈R.Definition 2.2 (see [11]).
Letx∈C(R,Rn) and Q(t) be  n×n continuous matrix defined on R. The following linear system:
(2.5)x˙(t)=Q(t)x(t)
is said to admit an exponential dichotomy on R if there exist constants K, α, projection P, and the fundamental solution X(t) of (2.5) satisfying
(2.6)|X(t)PX-1(s)|≤Ke-α(t-s),t≥s,|X(t)(I-P)X-1(s)|≤Ke-α(s-t),t≤s.Definition 2.3.
Letz*(t)=(x*(t),y*(t))T=(x1*(t),…,xn*(t),y1*(t),…,ym*(t))T be a continuously differentiable almost periodic solution of (2.1) with initial value ψ*=(φ*,ϕ*)T=(φ1*,…,φn*,ϕ1*,…,ϕm*)T. If there exist constants λ>0, M>1 such that for every solution z(t) = (x(t),y(t))T = (x1(t),…,xn(t),y1(t),…,ym(t))T of (2.1) with any initial value ψ = (φ,ϕ)T = (φ1,…,φn,ϕ1,…,ϕm)T, if
(2.7)∥z(t)-z*(t)∥1≤Meλt∥ψ-ψ*∥,fort>0,
where φi*(t),ϕj*(t),φi(t), and ϕj(t) are almost periodic functions. Then z*(t) is said to be globally exponentially stable.Lemma 2.4 (see [11]).
If the linear system (2.5) admits an exponential dichotomy, then the almost periodic system
(2.8)x˙(t)=Q(t)x(t)+f(t)
has a unique almost periodic solution
(2.9)ψ(t)=∫-∞t‍X(t)PX-1(s)f(s)ds-∫t+∞‍X(t)(I-P)X-1f(s)ds.Lemma 2.5 (see [11]).
Letqi(t) be an almost periodic function on R and
(2.10)M[qi]=limT→+∞1T∫tt+T‍qi(t)ds>0,i=1,2,…,n,
then the linear system z˙(t)=diag{-q1(t),…,-qn(t)}z(t) admits exponential dichotomy on R.


## 3. Existence and Uniqueness of Almost Periodic Solutions

In this section, we consider the existence and uniqueness of almost periodic solutions by fixed-point theorem.Theorem 3.1.
Under the assumptions(H1)–(H3), the system (2.1) has a unique almost periodic solution in the region ∥ψ-ψ0∥≤αβ/(1-α).(H4)
If(3.1)M[ci]=limT→+∞1T∫tt+T‍ci(s)ds>0,i=1,2,…,n,M[dj]=limT→+∞1T∫tt+T‍dj(s)ds>0,j=1,2,…,mholds, where
(3.2)β=max{max1≤i≤nmax{Iici-,Ii+Iici+ci-},max1≤j≤mmax{Jjdj-,Jj+Jjdj+dj-}},ψ0(t)=(∫-∞t‍e-∫st‍c1(u)duI1(s)ds,…,∫-∞t‍e-∫st‍cn(u)duIn(s)ds,c∫-∞t‍e-∫st‍d1(u)duJ1(s)ds,…,∫-∞t‍e-∫st‍dm(u)duJm(s)ds)T.Proof.
For any(φ,ϕ)T=(φ1,…,φn,ϕ1,…,ϕm)T∈X, we consider the the following system:
(3.3)x˙i(t)=-ci(t)xi(t)+∑j=1m‍aij(t)f1j(ϕj(t-τij(t)))+∑j=1n‍bji(t)f2j(φ˙j(t-δ-ji(t)))+Ii(t),y˙j(t)=-dj(t)yj(t)+∑i=1n‍pji(t)g1i(φi(t-δji(t)))+∑i=1m‍qij(t)g2i(ϕ˙i(t-τ-ij(t)))+Jj(t).
From (H4) and Lemma 2.5, we know the following linear system:
(3.4)x˙i(t)=-ci(t)xi(t),y˙j(t)=-dj(t)yj(t)
admits an exponential dichotomy on R. By Lemma 2.4, System (3.3) has an almost periodic solution z(φ,ϕ)T(t) which can be expressed as follows:
(3.5)z(φ,ϕ)T(t)=(∫-∞t‍e-∫st‍c1(u)du(A1(s)+I1(s))ds,…,∫-∞t‍e-∫st‍cn(u)du(An(s)+In(s))ds,c∫-∞t‍e-∫st‍d1(u)du(A-1(s)+J1(s))ds,…,∫-∞t‍e-∫st‍dm(u)du(A-m(s)+Jm(s))ds)T,
where
(3.6)Ai(s)=∑j=1m‍aij(s)f1j(ϕj(s-τij(s)))+∑j=1n‍bji(s)f2j(φ˙j(s-δ-ji(s))),i=1,2,…,n,A-j(s)=∑i=1n‍pji(s)g1i(φi(s-δji(s)))+∑i=1m‍qij(s)g2i(ϕ˙i(s-τ-ij(s))),j=1,2,…,m.
So, we can define a mapping T:X→X, by letting
(3.7)T(φ,ϕ)T(t)=z(φ,ϕ)T(t),∀(φ,ϕ)T∈X.
Set X0={ψ|ψ∈X,∥ψ-ψ0∥≤αβ/(1-α)}; clearly, X0 is a closed convex subset of X, so we have
(3.8)∥ψ0∥=max{supt∈Rmax1≤i≤n|∫-∞t‍e-∫st‍ci(u)duIi(s)ds|,supt∈Rmax1≤i≤n|(∫-∞t‍e-∫st‍ci(u)duIi(s)ds)′|,cccccsupt∈Rmax1≤j≤m|∫-∞t‍e-∫st‍dj(u)duJj(s)ds|,supt∈Rmax1≤j≤m|(∫-∞t‍e-∫st‍dj(u)duJj(s)ds)′|}≤max{max1≤i≤nmax{Iici-,Ii+Iici+ci-},max1≤j≤mmax{Jjdj-,Jj+Jjdj+dj-}}=β.
Therefore,
(3.9)∥ψ∥≤∥ψ-ψ0∥+∥ψ0∥≤αβ1-α+β=β1-α,∀ψ∈X0.
First, we prove that the mappingT is a self-mapping from X0 to X0. In fact, for any ψ=(φ-1,…,φ-n,ϕ-1,…,ϕ-m)T∈X0, let
(3.10)Bi(s)=∑j=1m‍aij(s)f1j(ϕ-j(s-τij(s)))+∑j=1n‍bji(s)f2j(φ-˙j(s-δ-ji(s))),i=1,2,…,n,B-j(s)=∑i=1n‍pji(s)g1i(φ-i(s-δji(s)))+∑i=1m‍qij(s)g2i(ϕ-˙i(s-τ-ij(s))),j=1,2,…,m.
From (H2) and (H3), we have
(3.11)∥Tψ-ψ0∥=max{supt∈Rmax1≤i≤n{|∫-∞t‍e-∫st‍ci(u)duBi(s)ds|},csupt∈Rmax1≤i≤n{|-ci(t)∫-∞t‍e-∫st‍ci(u)duBi(s)ds+Bi(t)|},ccccccccsupt∈Rmax1≤j≤m{|∫-∞t‍e-∫st‍dj(u)duB-j(s)ds|},ccccccccsupt∈Rmax1≤j≤m{|-dj(t)∫-∞t‍e-∫st‍dj(u)duB-j(s)ds+B-j(t)|}}≤max{supt∈Rmax1≤i≤n{∫-∞t‍eci-(s-t)|Bi(s)|ds},ccccccccsupt∈Rmax1≤i≤n{ci+∫-∞t‍eci-(s-t)|Bi(s)|ds+|Bi(t)|},ccccccccsupt∈Rmax1≤j≤m{∫-∞t‍edj-(s-t)|B-j(s)|ds},ccccccccsupt∈Rmax1≤j≤m{dj+∫-∞t‍edj-(s-t)|B-j(s)|ds+|B-j(t)|}}≤max{max1≤i≤n{1ci-(∑j=1m‍aijF1j+∑j=1n‍bjiF2j)},ccccccccmax1≤i≤n{(1+ci+ci-)(∑j=1m‍aijF1j+∑j=1n‍bjiF2j)},ccccccccmax1≤j≤m{1dj-(∑i=1n‍pjiG1i+∑i=1m‍qijG2i)},ccccccccmax1≤j≤m{(1+dj+dj-)(∑i=1n‍pjiG1i+∑i=1m‍qijG2i)}}∥ψ∥=max{max1≤i≤nmax{1ci-,1+ci+ci-}(∑j=1m‍aijF1j+∑j=1n‍bjiF2j),ccccccccmax1≤j≤mmax{1dj-,1+dj+dj-}(∑i=1n‍pjiG1i+∑i=1m‍qijG2i)(∑j=1m‍aijF1j+∑j=1n‍bjiF2j)}∥ψ∥=α∥ψ∥≤αβ1-α.
This implies that T(ψ)∈X0, so T is a self-mapping from X0 to X0.
Finally, we prove thatT is a contraction mapping. In fact, for any ψ1=(α1,…,αn,β1,…,βm)T, ψ2=(α-1,…,α-n,β-1,…,β-m)T∈X0. Let
(3.12)Hi(s)=∑j=1m‍aij(s)[f1j(βj(s-τij(s)))-f1j(β-j(s-τij(s)))]+∑j=1n‍bji(s)[f2j(α˙j(s-δ-ji(s)))-f2j(α-˙j(s-δ-ji(s)))],i=1,2,…,n,H-j(s)=∑i=1n‍pji(s)[g1i(αi(s-δji(s)))-g1i(α-i(s-δji(s)))]+∑i=1m‍qij(s)[g2i(β˙i(s-τ-ij(s)))-g2i(β-˙i(s-τ-ij(s)))],j=1,2,…,m.
We have
(3.13)∥Tψ1-Tψ2∥=max{supt∈Rmax1≤i≤n{|∫-∞t‍e-∫st‍ci(u)duHi(s)ds|},ccccccsupt∈Rmax1≤i≤n{|-ci(t)∫-∞t‍e-∫st‍ci(u)duHi(s)ds+Hi(t)|},ccccccsupt∈Rmax1≤j≤m{|∫-∞t‍e-∫st‍dj(u)duH-j(s)ds|},ccccccsupt∈Rmax1≤j≤m{|-dj(t)∫-∞t‍e-∫st‍dj(u)duH-j(s)ds+H-j(t)|}}≤max{max1≤i≤nmax{1ci-,1+ci+ci-}(∑j=1m‍aijF1j+∑j=1n‍bjiF2j),cccccccccmax1≤j≤mmax{1dj-,1+dj+dj-}(∑i=1n‍pjiG1i+∑i=1m‍qijG2i)(∑j=1m‍aijF1j+∑j=1n‍bjiF2j)}∥ψ1-ψ2∥=α∥ψ1-ψ2∥.
Notice that α<1, it means that the mapping T is a contraction mapping. By Banach fixed-point theorem, there exists a unique fixed-point ψ*∈X0 such that Tψ*=ψ*, which implies system (2.1) has a unique almost periodic solution.


## 4. Global Exponential Stability of the Almost Periodic Solution

In this section, we consider the exponential stability of almost periodic solution, and we give two corollaries.Theorem 4.1.
Under the assumptions(H1)–(H4), then system (2.1) has a unique almost periodic solution which is global exponentially stable.Proof.
It follows from Theorem3.1 that system (2.1) has a unique almost periodic solution z*(t)=(x*(t),y*(t))T=(x1*(t),…,xn*(t),y1*(t),…,ym*(t))T with the initial value ψ* = (φ*,ϕ*)T = (φ1*,…,φn*,ϕ1*,…,ϕm*)T. Set z(t) = (x(t),y(t))T = (x1(t),…,xn(t),y1(t),…,ym(t))T is an arbitrary solution of system (2.1) with initial value ψ = (φ,ϕ)T = (φ1,…,φn,ϕ1,…,ϕm)T. Let ui(t)=xi(t)-xi*(t), vj(t)=yj(t)-yj*(t), Ψi=φi-φi*, Φj=ϕj-ϕj*. Then z(t)-z*(t) = (u1(t),…,un(t),v1(t),…,vm(t))T, where i=1,2,…,n; j=1,2,…,m. Then system (2.1) is equivalent to the following system:
(4.1)u˙i(s)+ci(s)ui(s)=Fi(s),s>0,v˙j(s)+dj(s)vj(s)=F-j(s),s>0,
with the initial value
(4.2)Ψi(s)=φi(s)-φi*(s),s∈[-δ,0],Φj(s)=ϕj(s)-ϕj*(s),s∈[-τ,0],
where
(4.3)Fi(s)=∑j=1m‍aij(s)[f1j(yj*(s-τij(s))+vj(s-τij(s)))-f1j(yj*(s-τij(s)))]+∑j=1n‍bji(s)[f2j(x˙j*(s-δ-ji(s))+u˙j(s-δ-ji(s)))-f2j(x˙j*(s-δ-ji(s)))],F-j(s)=∑i=1n‍pji(s)[g1i(xi*(s-δji(s))+ui(s-δji(s)))-g1i(xi*(s-δji(s)))]+∑i=1m‍qij(s)[g2i(y˙i*(s-τ-ij(s))+v˙i(s-τ-ij(s)))-g2i(y˙i*(s-τ-ij(s)))].
Let
(4.4)Γi(ξi)=ci--ξi-∑j=1m‍aijF1jeτξi-∑j=1n‍bjiF2jeδξi,Γ-i(ξ-i)=ci--ξ-i-(ci++ci-)(∑j=1m‍aijF1jeτξ-i+∑j=1n‍bjiF2jeδξ-i),
where ξi, ξ-i≥0, i=1,2,…,n. From (H3), we know Γi(0)>0, Γ-i(0)>0. Since Γi(·) and Γ-i(·) are continuous on [0,∞] and Γi(ξi), Γ-i(ξ-i)→-∞ as ξi, ξ-i→+∞, so there exist ξi*, ξ-i*>0 such that Γi(ξi*)=Γ-i(ξ-i*)=0 and Γi(ξi)>0 for ξi∈(0,ξi*), Γ-i(ξ-i)>0 for ξ-i∈(0,ξ-i*). By choosing ξ=min{ξ1*,…,ξn*,ξ-1*,…,ξ-n*}, we obtain Γi(ξ), Γ-i(ξ)≥0. So we can choose a positive constant λ1, 0<λ1<min{ξ,ci-,…,cn-} such that Γi(λ1), Γ-i(λ1)>0. For the same reason, we define
(4.5)Gj(ηj)=dj--ηj-∑i=1n‍pjiG1ieδηj-∑i=1m‍qijG2ieτηj,G-j(η-j)=dj--η-j-(dj-+dj+)(∑i=1n‍pjiG1ieδη-j+∑i=1m‍qijG2ieτη-j).
There exists λ2, 0<λ2<dj-, j=1,2,…,m, such that Gj(λ2), G-j(λ2)>0. Taking λ=min{λ1,λ2}, since Γi(·), Γ-i(·), Gj(·), and G-j(·) are strictly monotonous decrease functions, therefore, Γi(λ), Γ-i(λ), Gj(λ), G-j(λ)>0, which implies
(4.6)ri:=1ci--λ(∑j=1m‍aijF1jeτλ+∑j=1n‍bjiF2jeδλ)<1,r-i:=(1+ci+ci--λ)(∑j=1m‍aijF1jeτλ+∑j=1n‍bjiF2jeδλ)<1,i=1,2,…,n;1dj--λ(∑i=1n‍pjiG1ieδλ+∑i=1m‍qijG2ieτλ)<1,(1+dj+dj--λ)(∑i=1n‍pjiG1ieδλ+∑i=1m‍qijG2ieτλ)<1,j=1,2,…,m.
Multiplying the two equations of system (4.1) by e∫0s‍ci(u)du and e∫0s‍dj(u)du, respectively, and integrating on [0,t], we get
(4.7)ui(t)=ui(0)e-∫0t‍ci(u)du+∫0t‍e-∫st‍ci(u)duFi(s)ds,vj(t)=vj(0)e-∫0t‍dj(u)du+∫0t‍e-∫st‍dj(u)duF-j(s)ds.
Taking
(4.8)M=max{max1≤i≤nci-∑j=1maijF1j+∑j=1nbjiF2j,max1≤j≤mdj-∑i=1npjiG1i+∑i=1mqijG2i},
then M>1, thus
(4.9)∥z(t)-z*(t)∥1=∥ψ(t)-ψ*(t)∥1≤∥ψ-ψ*∥≤M∥ψ-ψ*∥eλt,t≤0,
where λ>0 as in (4.6). We claim that
(4.10)∥z(t)-z*(t)∥1≤M∥ψ-ψ*∥eλt,t>0.
To prove (4.10), we first show for any p>1, the following inequality holds:
(4.11)∥z(t)-z*(t)∥1<pM∥ψ-ψ*∥eλt,t>0.
If (4.11) is false, then there must be some t1>0 and some i,l∈{1,2,…,n}, j,k∈{1,2,…,m}, such that
(4.12)∥z(t1)-z*(t1)∥1=max{|ui(t1)|,|u˙l(t1)|,|vj(t1)|,|v˙k(t1)|}=pM∥ψ-ψ*∥eλt1,(4.13)∥z(t)-z*(t)∥1<pM∥ψ-ψ*∥eλt,0<t<t1.
By (4.3)–(4.8), (4.12),  and (4.13), we have
(4.14)|ui(t1)|=|ui(0)e-∫0t1‍ci(u)du+∫0t1‍e-∫st1‍ci(u)duFi(s)ds|≤e-ci-t1∥ψ-ψ*∥+∫0t1‍e-ci-(t1-s)|Fi(s)|ds≤e-ci-t1∥ψ-ψ*∥+∫0t1‍e-ci-(t1-s)(∑j=1m‍aijF1jpM∥ψ-ψ*∥e-λ(s-τij(s))ccccccccccccccccccccccccccccccccc+∑j=1n‍bjiF2jpM∥ψ-ψ*∥e-λ(s-δ-ji(s)))ds<pM∥ψ-ψ*∥e-λt1[et1(λ-ci-)M+1-et1(λ-ci-)ci--λ(∑j=1m‍aijF1jeλτ+∑j=1n‍bjiF2jeλδ)]=pM∥ψ-ψ*∥e-λt1[(1M-ri)et1(λ-ci-)+ri]<pM∥ψ-ψ*∥e-λt1;|u˙l(t1)|=|-cl(t1)ul(0)e-∫0t1‍cl(u)du-cl(t1)∫0t1‍e-∫st1‍cl(u)duFl(s)ds+Fl(t1)|≤cl+e-cl-t1∥ψ-ψ*∥+cl+∫0t1‍e-cl-(t1-s)|Fl(s)|ds+|Fl(t1)|≤cl+e-cl-t1∥ψ-ψ*∥+cl+∫0t1‍e-cl-(t1-s)(∑j=1m‍aljF1jpM∥ψ-ψ*∥e-λ(s-τlj(s))ccccccccccccccccccccccccccccccccccccc+∑j=1n‍bjlF2jpM∥ψ-ψ*∥e-λ(s-δ-jl(s)))dscc+∑j=1m‍aljF1jpM∥ψ-ψ*∥e-λ(t1-τlj(t1))+∑j=1n‍bjlF2jpM∥ψ-ψ*∥e-λ(t1-δ-jl(t1))<pM∥ψ-ψ*∥e-λt1[(1M-rl)et1(λ-cl-)+r-l]<pM∥ψ-ψ*∥e-λt1.
We also can get
(4.15)|vj(t1)|<pM∥ψ-ψ*∥e-λt1,|v˙k(t1)|<pM∥ψ-ψ*∥e-λt1.
From (4.14)–(4.15), we have
(4.16)∥z(t1)-z*(t1)∥1=max{|ui(t1)|,|u˙l(t1)|,|vj(t1)|,|v˙k(t1)|}<pM∥ψ-ψ*∥e-λt1,
which contradicts the equality (4.12), so (4.11) holds. Letting p→1, then (4.10) holds. The almost periodic solution of system (2.1) is globally exponentially stable.Corollary 4.2.
Letbji(t)=qij(t)=0. Under assumptions (H1), (H2), and (H4), if, (H5)(4.17)α1=max{max1≤i≤n{1ci-∑j=1m‍aijF1j},max1≤j≤m{1dj-∑i=1n‍pjiG1i}}<1
holds, then system
(4.18)x˙i(t)=-ci(t)xi(t)+∑j=1m‍aij(t)f1j(yj(t-τij(t)))+Ii(t),y˙j(t)=-dj(t)yj(t)+∑i=1n‍pji(t)g1i(xi(t-δji(t)))+Jj(t)
has a unique almost periodic solution in the region ∥ψ-ψ0∥≤α1β/(1-α1), which is global exponentially stable.In fact, Zhang and Si [11, 16] and Chen et al. [17] studied system (4.18). This Corollary 4.2 is the Theorem  3.1 in [11], Theorem  1.1 in [16], and Theorem  1 in [17]. Especially, in [17], authors let(H5′)
(4.19)α-1=max1≤i≤n{1ci-∑j=1m‍aijF1j}+max1≤j≤m{1dj-∑i=1n‍pjiG1i}<1.Therefore, we extend and improve previously known results.Remark 4.3.
Letci(t)=dj(t), aij(t)=pji(t), bji=qij(t), Ii(t)=Jj(t), τij(t)=δji(t), τ-ij(t)=δ-ji(t), n=m. Then system (2.1) is reduced to be system (1.1), hence we have the following.Corollary 4.4.
Under assumptions(H1), (H2), and (H4), if (H6)(4.20)α2=max1≤i≤nmax{1+ci+ci-}∑j=1m‍(aijF1j+bjiF2j)<1,
holds, then system (1.1) has a unique almost periodic solution in the region ∥ψ-ψ0∥≤α2β/(1-α2), which is global exponentially stable.This Corollary4.4 is the result of [19].


## 5. An Example

In this section, we give an example to illustrate the effectiveness of our results.Letn=m=2, f1(y1)=y1/10, f2(y2)=siny2/10, g1(x1)=x1/12, g2(x2)=|x2|/8, τij(t)=τ-ij(t)=cos2t, δji(t)=δ-ji(t)=0.5, I1(t)=1+sin2(t), I2(t)=1+cos2t, J1(t)=1+|sint|, and J2(t)=sin2t+0.5, then we consider the following almost periodic system:
(5.1)x˙1(t)=-c1(t)x1(t)+∑j=12‍a1j(t)fj(yj(t-cos2t))+∑j=12‍bj1(t)x˙j(t-0.5)+I1(t),x˙2(t)=-c2(t)x2(t)+∑j=12‍a2j(t)fj(yj(t-cos2t))+∑j=12‍bj2(t)x˙j(t-0.5)+I2(t),y˙1(t)=-d1(t)y1(t)+∑i=12‍p1i(t)gi(xi(t-0.5))+∑i=12‍qi1(t)y˙i(t-cos2t)+J1(t),y˙2(t)=-d2(t)y2(t)+∑i=12‍p2i(t)gi(xi(t-0.5))+∑i=12‍qi2(t)y˙i(t-cos2t)+J2(t),
where c1(t)=1+cos2t, c2(t)=1+sin2t, d1(t)=1+|cost|, d2(t)=1+|sint|, a11(t)=|sint|/4, a12(t)=cos2t/8, a21(t)=cos2t/6, a22(t)=|sint|/4, b11(t)=cos2t/8, b12(t)=0, b21(t)=0, b22(t)=sin2t/10, p11(t)=cos2t/4, p12(t)=sin2t/9, p21(t)=sin2t/8, p22(t)=|cost|/6, q11(t)=cost/8, q12(t)=0, q21(t)=0, and q22(t)=cos2t/10. By simple calculation, we obtain α=max{39/80,51/120,69/144,63/160}<1, hence this system has a unique almost periodic solution, which is global exponentially stable by Theorem 4.1. Figure 1 depicts the time responses of state variables of x1(t), x2(t), y1(t), and y2(t) with step h=0.005 and initial states [-0.2,0.2,-0.3,0.4]T for t∈[-1,0], and Figures 2, 3, and 4 depict the phase orbits of x1(t) and y1(t), x1(t), and x2(t), y1(t) and y2(t). It confirms that our results are effective for (5.1).Figure 1
Transient response of state variablex1(t), x2(t), y1(t) and y2(t).Figure 2
Phase response of state variablex1(t) and x2(t).Figure 3
Phase response of state variablex1(t) and y1(t).Figure 4
Phase response of state variablex2(t) and y2(t).


## 6. Conclusions

In this paper, a class of BAM neural networks with variable coefficients and neutral time-varying delays are investigated. By employing Banach fixed-point theorem, the exponential dichotomy and differential inequality techniques, some sufficient conditions are obtained to ensure the existence, uniqueness, and stability of the almost periodic solution. As is known to all, neural networks with neutral delays are studied rarely, and most authors solve these problems by linear matrix inequality techniques. In addition, BAM neural networks are much more complicated than the one-layer neural network. In a word, this paper is original, and novel. It also extends and improves other previously known results (see [11, 16, 17, 19]).



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*Source: 482584-2012-12-25.xml*