Shunting inhibitory cellular neural networks are studied. Some sufficient criteria are obtained for the existence and uniqueness of pseudo almost-periodic solution of this system. Our results improve and generalize those of the previous studies. This is the first paper considering the pseudo almost-periodic SICNNs. Furthermore, several methods are applied to establish sufficient criteria for the globally exponential stability of this system. The approaches are based on constructing suitable Lyapunov functionals and the well-known Banach contraction mapping principle.
1. Introduction
It is well known that the cellular neural networks (CNNs) are widely applied in signal processing, image processing, pattern recognition, and so on. The theoretical and applied studies of CNNs have been a new focus of studies worldwide (see [1–12]). Bouzerdoum and Pinter in [1] have introduced a new class of CNNs, namely, the shunting inhibitory CNNs (SICNNs). Shunting neural networks have been extensively applied in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing. Recently, Chen and Cao [9] have studied the existence of almost-periodic solutions of the following system of SICNNs: ẋij(t)=-aijxij(t)-∑Ckl∈Nr(i,j)cijklf(xij(t-τ))xij(t)+Lij(t),
where Cij denotes the cell at the (i,j) position of the lattice, the r-neighborhood Nr(i,j) of Cij is Nr(i,j)={Cijkl:max(|k-l|,|l-j|≤r),1≤k≤m,1≤l≤n},xij is the activity of the cell Cij, Lij(t) is the external input to Cij, the constant aij>0 represents the passive decay rate of the cell activity, Cijkl≥0 is the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell Cij, and the activation function f(xkl) is a positive continuous function representing the output or firing rate of the cell Cij. Since studies on neural dynamic systems not only involve a discussion of stability properties, but also involve many dynamic properties such as periodic oscillatory behavior, almost-periodic oscillatory properties, chaos, and bifurcation. To the best of our knowledge, few authors have studied almost-periodic solutions for SINNs with delays and variable coefficients, and most of them discuss the stability, periodic oscillation in the case of constant coefficients. In their paper, they investigated the existence and stability of periodic solutions of SINNs with delays and variable coefficients. They considered the SICNNs with delays and variable coefficients ẋij(t)=-aij(t)xij(t)-∑Bkl∈Nr(i,j)Bijkl(t)fij(xkl(t))xij(t)-∑Ckl∈Nr(i,j)Cijkl(t)gij(xkl(t-τkl))xij(t)+Lij(t),
where, for each i=1,2,…,n,j=1,2,…,m,aij(t),Bijkl(t),Cijkl(t),and Lij(t) are all continuous ω- periodic functions and aij(t)>0,Bijkl(t)≥0,Cijkl(t)≥0, and τij is a positive constant.
In this paper, we consider the following more general SICNNs: ẋij(t)=-aij(t)xij(t)-∑Bkl∈Nr(i,j)Bijkl(t)fij(xkl(t))xij(t)-∑Ckl∈Nr(i,j)Cijkl(t)gij(xkl(t-τkl(t)))xij(t)+Lij(t).
By using the Lyapunov functional and contraction mapping, a set of criteria are established for the globally exponential stability, the existence, and uniqueness of pseudo almost-periodic solution for the SICNNs. This is the first paper considering the pseudo almost-periodic solution of SICNNs. Since the nature is full of all kinds of tiny perturbations, either the periodicity assumption or the almost-periodicity assumption is just approximation of some degree of the natural perturbations. A well-known extension of almost periodicity is the asymptotically almost periodicity, which was introduced by Frechet. In 1992, Zhang [13, 14] introduced a more general extension of the concept of asymptotically almost periodicity, the so-called pseudo almost periodicity, which has been widely applied in the theory of ODEs and PDEs. However, it is rarely applied in the theory of neural networks or mathematical biology. This paper is expected to establish criteria that provide much flexibility in the designing and training of neural networks and to shed some new light on the application of pseudo almost periodicity in neural networks, population dynamics, and the theory of differential equations.
Throughout this paper, we will use the notations gM=supt∈Rg(t),gL=supt∈Rg(t), where g(t) is a bounded continuous function on R.
In this paper, we always use i=1,2,…,m;j=1,2,…,n; unless otherwise stated.
In this paper, we always consider system (1.4) together with the following assumptions.
aij(t) are almost periodic on R with aij(t)>0, and Lij(t) and τij(t) are pseudo almost periodic on R with Lij>0.
τij(t) is bounded, continuous and differentiable with 0≤τij≤τ,1-τ>0 for t∈R, where τ is constant.
fij,gij∈C(R,R) are bounded, and continuous, and there exist positive numbers μij,νij such that |fij(x)-fij(y)|≤μij|x-y|,|gij(x)-gij(y)|≤νij|x-y|, for all x,y∈R.
2. Preliminaries and Basic Results of Pseudo Almost-Periodic Function
In this section, we explore the existence of pseudo almost-periodic solution of (1.4). First, we would like to recall some basic notations and results of almost periodicity and pseudo almost periodicity [15, 16] which will come into play later on.
Let Ω⊂Cn be close and let ℒ(R) (resp., ℒ(R×Ω)) denote the C*-algebra of bounded continuous complex-valued functions on R (respectively, R×Ω) with supremum norm |·|∞ denoting the Euclidean norm in Cn; that is, |x|∞=max1≤i≤n|xi|, x∈Cn.
Definition 2.1.
A function g∈ℒ(R) is called almost periodic if, for each ϵ>0, there exists an lϵ>0 such that every interval of length lϵ contains a number τ with the property that |g(t+τ)-g(t)|<ϵ,t∈R.
Definition 2.2.
A function g∈ℒ(R×Ω) is called almost periodic in t∈R, uniformly in Z∈Ω, if, for each ϵ>0 and any compact set M of Ω, there exists an lϵ>0 such that every interval of length lϵ contains a number τ with the property that |g(t+τ,z)-g(t,z)|<ϵ,t∈R,Z∈Ω. The number τ is called an ϵ-translation number of g. Denote by 𝒜𝒫(R)(𝒜𝒫(R×Ω)) the set of all such function.
A function f∈ℒ(R)(ℒ(R×Ω)) is called pseudo almost periodic (pseudo almost periodic in t∈R, uniformly in Z∈Ω), if f=g+φ, where g∈𝒫𝒜(R)(𝒫𝒜(R×Ω)) and φ∈𝒫𝒜𝒫0(R)(𝒫𝒜𝒫0(R×Ω)). The function g and φ are called the almost periodic component and the erigodic perturbation, respectively, of the function f. Denote by 𝒫𝒜𝒫(R)(𝒫𝒜𝒫(R×Ω)) the set of all such functions f.
Define ∥x∥∞=supt∈R|x(t)|∞,x∈𝒫𝒜𝒫. It is trivial to show that 𝒫𝒜𝒫 is a Banach space with ∥·∥∞.
Let A(t)=(aij(t)) be a complex n×n matrix-valued function with elements (entries) which are continuous on R. We consider the homogeneous linear ODE and nonhomogeneous linear ODE as follows:dxdt=A(t)x,dxdt=A(t)x+f(t),
where x denotes an n-column vector.
Definition 2.4 (see [15, 16]).
The homogeneous linear ODE (2.2) is said to admit an exponential dichotomy if there exist a linear projection p(i.e.,p2=P) on Cn and positive constants k,α,β such that
∥X(t)PX-1(s)∥≤ke-α(t-s),t≥s,∥X(t)(I-P)X-1(s)∥≤ke-β(s-t),t≤s,
where X(t) is a fundamental matrix of (2.2) with X(0)=E; E is the n×n identity matrix.
Definition 2.5 (see [15, 16]).
The matrix A(t) is said to be row dominant if there exists a number δ>0 such that |Reaij(t)|≥∑j=1,j≠in|aij(t)|+δ for all t∈(-∞,∞) and i=1,2,…,n.
Lemma 2.6 (see [15, 16]).
If A(t) is a bounded, continuous, and row-dominant n×n matrix function on R, and there exists k≤n such that Reaij<0(i=1,2,…,k). Then (2.2) has a fundamental matrix solution X(t) satisfying
∥X(t)PX-1(s)∥≤ke-δ(t-s),t≥s,∥X(t)(I-P)X-1(s)∥≤ke-δ(s-t),t≤s,
where K is a positive constant, and P=diag(Ek,0) with Ek being a k×k identity matrix.
For H=(h1,h2,…,hn)∈ℒ(R)n, suppose that H(t)∈Ω for all t∈R. Define H×l:R→Ω×R by H×l(t)=(h1(t),h2(t),…,hn(t),t)(t∈R).
Lemma 2.7 (see [15, 16]).
Assume that the function f(t,z)∈𝒫𝒜𝒫(R×Ω) is continuous in z∈M uniformly in t∈R for all compact subsets M⊂Ω and F∈𝒫𝒜𝒫(R)n such that F(R)⊂Ω. Then f∘(F×l)∈𝒫𝒜𝒫(R).
It is obvious that if f satisfies a Lipschitz condition; that is, there is an L>0 such that |f(z′,t)-f(z,t)|≤L|z′-z|(z′,z∈M,t∈R),
then f is continuous in z∈M uniformly in t∈R. Obviously, if f(t)∈𝒫𝒜𝒫(R) is uniformly continuous in t∈R and φ∈𝒫𝒜𝒫(R) such that φ(R)⊂Imf, then f∘φ∈𝒫𝒜𝒫(R).
Lemma 2.8 (see [15, 16]).
Assume that A(t) is an almost-periodic matrix function and f(t)∈𝒫𝒜𝒫(Rn). If (2.2) satisfies an exponential dichotomy, then (2.3) has unique pseudo almost periodic solution x(t) reading
x(t)=∫-∞tX(t)PX-1(s)f(s)ds-∫t∞X(t)(E-P)X-1(s)f(s)ds
and satisfying ∥x∥≤(K/α+K/β)∥f∥, where X(t) is a fundamental matrix solution of (2.2).
Definition 2.9.
System (1.4) is said to be globally exponentially stable (GES), if for any two solutions x(t) and y(t) of (1.4), there exist positive numbers M and ɛ such that
|x(t)-y(t)|p≤Me-ɛ(t-t0)∥φ-ψ∥p,t>t0,
where x(t)=x(t,φ) and y(t)=y(t,ψ) denoting the solution of (1.4) through (t0,φ) and (t0,ψ) respectively. Here ɛ is called the Lyapunov exponent of (1.4).
3. Existence and Stability of Pseudo Almost-Periodic SolutionTheorem 3.1.
Then (1.4) has a unique pseudo almost-periodic solution, say x*(t), satisfying ∥x*∥∞≤L/(1-r).
Proof.
For any φ∈𝒫𝒜𝒫(R), consider
ẋij=-aij(t)xij(t)-∑Bhl∈Nr(ij)Bijhl(t)fij(φhl(t))φij(t)-∑Chl∈Nr(ij)Cijhl(t)gij(φhl(t-τhl(t)))φij(t)+Lij(t).
Since -aij(t)<0, from Lemmas 2.6, 2.7, and 2.8, it follows that (3.2) has a unique pseudo almost-periodic solution, which is given by
xφ(t)=(∫-∞te-∫staij(η)dη[-∑Bhl∈Nr(i,j)Bijhl(s)fij(φhl(s))φij(s)-∑Chl∈Nr(i,j)Cijhl(s)gij(φhl(s-τhl(s)))φij(s)+Lij(s)]ds)mn×1.
Define the mapping 𝒯:𝒫𝒜𝒫(R)→𝒫𝒜𝒫(R) by 𝒯(φ)(t)=xφ(t),φ∈𝒫𝒜𝒫(R).
Let B*={φ∣φ∈𝒫𝒜𝒫(R),∥φ-φ0∥∞≤(γ/(1-γ))L}, where φ0(t)=(∫-∞te-∫sta11(η)dηL11(s)ds,…,∫-∞te-∫staij(η)dηLij(s)ds,…,∫-∞te-∫stamn(η)dηLmn(s)ds)T.
Clearly, B* is closed and convex in B. Note that∥φ0∥∞=supt∈Rmax(i,j){|∫-∞te-∫staij(η)dηLij(s)ds|}≤supt∈Rmax(i,j){|∫-∞te-aij-(s-t)Lij+ds|}≤supt∈Rmax(i,j){Lij+aij-}=max(i,j){Lij+aij-}=L.
Therefore, for any φ∈B*, we have
∥φ∥≤∥φ-φ0∥+∥φ0∥≤γ1-γL+L=L1-γ.
Now, we will show that 𝒯 maps B* into itself. In fact, for any φ∈B*, by using L/(1-γ)≤1, we have
∥𝒯φ-φ0∥∞=supt∈Rmax(i,j){|∫-∞te-∫staij(η)dη[-∑Bhl∈Nr(i,j)Bijhl(s)fij(φhl(s))φij(s)-∑Chl∈Nr(i,j)Cijhl(s)gij(φhl(s-τhl(s)))φij(s)]ds|)≤supt∈Rmax(i,j){∫-∞te-∫staij(η)dη[μij∑Bhl∈Nr(i,j)Bijhl(s)|(φhl(s))||φij(s)|+νij∑Chl∈Nr(i,j)Cijhl(s)|φhl(s-τhl(s))||φij(s)|]ds)≤max(i,j){∫-∞te-∫staij(η)dη(μij∑Bhl∈Nr(i,j)Bijhl(s)+νij∑Chl∈Nr(i,j)Cijhl(s))ds}∥φ∥2.≤γ∥φ∥≤γ∥φ∥≤γL1-γ.
For any φ,ϕ∈B*, it follows from L/(1-γ)≤1 that
∥𝒯φ-𝒯ψ∥∞=supt∈Rmax(i,j){∑Chl∈Nr(i,j)Cijhl(s)|∫-∞te-∫staij(η)dη×[∑Bhl∈Nr(i,j)Bijhl(s)|fij(φhl(s))φij(s)-fij(ψhl(s))ψij(s)|+∑Chl∈Nr(i,j)Cijhl(s)|gij(φhl(s-τhl(s)))φij(s)-gij(ψhl(s-τhl(s)))ψij(s)|]ds|)≤supt∈Rmax(i,j){∑Bhl∈Nr(i,j)∫-∞te-∫staij(η)dη×[∑Bhl∈Nr(i,j)Bijhl(s)(|fij(φhl(s))|·|φij(s)-ψij(s)|+|fij(φhl(s))-fij(ψhl(s))|·|ψij(s)|)+∑Chl∈Nr(i,j)Cijhl(s)(|gij(φhl(s-τhl(s)))|×|φij(s)-ψij(s)|+|gij(φhl(s-τhl(s)))-gij(ψhl(s-τhl(s)))|·|ψij(s)|)]ds∑Bhl∈Nr(i,j)}≤max(i,j){∫-∞te-∫staij(η)dη[μij∑Bhl∈Nr(i,j)Bijhl(s)+νij∑Chl∈Nr(i,j)Cijhl(s)](∥φ∥∞+∥ψ∥∞)∥φ-ψ∥ds}=γ·2L1-γ∥φ-ψ∥=2Lγ1-γ∥φ-ψ∥=δ∥φ-ψ∥.
Since δ<1, 𝒯 is a contraction mapping. Therefore, there exists a unique fixed point x*∈B* such that 𝒯x*=x*. That is, system (1.4) has a unique pseudo almost-periodic solution x*∈B* with ∥x*-φ0∥≤(γ/(1-γ))L.
Now we go ahead with the GES of (1.4). The approaches involve constructing suitable Lyapunov functions and application of a generalized Halanay's delay differential inequality. We will stop here to see our first criteria for the globally exponential stability of (1.4), which is delay dependent.
Theorem 3.2.
In addition to (A1)–(A4), if one further assumes that
c=min(i,j)inf(i,j){βij(2aij(t)-3βijL1-γ[μij∑Bhl∈Nr(i,j)Bijhl(t)+νij∑Chl∈Nr(i,j)Cijhl(t)])+βhl(μhlL1-γ∑Bij∈Nr(h,l)Bhlij(t)+L1-γνhl∑Chl∈Nr(i,j)Chlij(ξij-1(t))1-τij(ξij-1(t)))}>0
or
c=inf(i,j){βij(aij(t)-βijL1-γ[μij∑Bhl∈Nr(i,j)Bijhl(t)+νij∑Chl∈Nr(i,j)Cijhl(t)])-βhl(μhlL1-γ∑Bij∈Nr(h,l)Bhlij(t)+L1-γνhl∑Chl∈Nr(i,j)Chlij(ξij-1(t))1-τij(ξij-1(t)))}>0.
Then there exists a unique pseudo-almost periodic solution of system (1.4) and all other solutions converge exponentially to the (pseudo) almost-periodic attractor.
Proof.
By Theorem (3.2), there exists a unique pseudo almost-periodic solution, namely, x(t)=x(t,φ). Let y=y(t,φ) be any other solution of (1.4) through (t0,φ). Assume that (A5) is satisfied and consider the auxiliary functions Fij(ɛ) defined on [0,+∞] as follows:
Fij(ɛ)=inft∈R{βij(2aij(t)-ɛ)-3βijL1-γ[μij∑Bhl∈Nr(i,j)Bijhl(t)+νij∑Chl∈Nr(i,j)Cijhl(t)]+βhl(μhlL1-γ∑Bij∈Nr(h,l)Bhlij(t)+L1-γνhl∑Chl∈Nr(i,j)Chlij(ξij-1(t))1-τij(ξij-1(t)))}.
From(A1)–(A3), one can easily show that Fij(ɛ) is well defined and is continuous. From (A5), it follows that Fij(0)>0,Fij(ɛ)→-∞ as ɛ→∞ it follows that there exists an ɛij>0 such that Fij(ɛij)>0. Let ɛ=mini,jɛij. Then we have Fij(ɛ)>0,1≤i≤n,1≤j≤m.
Consider the Lyapunov functional defined byV(t)=12∑(i,j)βij[∑Chl∈Nr(i,j)(xij(t)-yij(t))2eɛt+∑Chl∈Nr(i,j)VijL1-γ∫t-τhl(t)tChl(ξhl-1(t))1-τhl(ξhl-1)(t)(xhl(s)-yhl(s))2eɛ(s+τhlM)ds].
Calculating the upper-right derivative of V(t) and using the inequality 2ab≤a2+b2, one has
V'(t)≤∑(i,j)βij{∑Bhl∈Nr(i,j)12(xij(t)-yij(t))2ɛeɛt-eɛtaij(t)(xij-yij(t))2+eɛt∑Bhl∈Nr(i,j)Bijhl(t)|xij(t)-yij(t)|·|fij(xhl(t))xij(t)-fij(yhl(t))yij(t)|+eɛt∑Chl∈Nr(i,j)Cijhl(t)|xij(t)-yij(t)|·|gij(xhl(t-τhl(t)))xij(t)-gij(yhl(t-τhl(t)))yij(t)|+12∑Chl∈Nr(i,j)νijL1-γCijhl(ξhl-1(t))1-τhl(ξhl-1(t))(xhl(t)-yhl(t))2eɛ(t+τhlM)-12∑Chl∈Nr(i,j)νijL1-γCijhl(t)(xhl(t-τhl(t))-yhl(t-τhl(t)))2eɛ(t-τhl(t)+τhlM)}≤∑(i,j)βij{∑Bhl∈Nr(i,j)12(xij(t)-yij(t))2ɛeɛt-eɛtaij(t)(xij-yij(t))2+eɛt∑Bhl∈Nr(i,j)Bijhl(t)|xij(t)-yij(t)|·|fij(xhl(t))xij(t)-fij(yhl(t))yij(t)|+eɛt∑Chl∈Nr(i,j)Cijhl(t)|xij(t)-yij(t)|·|gij(xhl(t-τhl(t)))xij(t)-gij(yhl(t-τhl(t)))yij(t)|+12∑Chl∈Nr(i,j)νijL1-γCijhl(ξhl-1(t))1-τhl(ξhl-1(t))(xhl(t)-yhl(t))2eɛ(t+τhlM)-12∑Chl∈Nr(i,j)νijL1-γCijhl(t)(xhl(t-τhl(t))-yhl(t-τhl(t)))2eɛ(t)}≤eɛ(t)∑(i,j)βij{∑Bhl∈Nr(i,j)(ɛ2-aij(t))(xij(t)-yij(t))2+∑Bhl∈Nr(i,j)Bijhl(t)|xij(t)-yij(t)|[|fij(xhl(t))||xij(t)-yij(t)|+|fij(xhl(t))-fij(yhl(t))||yij(t)|]+∑Chl∈Nr(i,j)Cijhl(t)|xij(t)-yij(t)|×[|gij(xhl(t-τhl(t)))||xij(t)-yij(t)|+|gij(xhl(t-τhl(t)))-gij(yhl(t-τhl(t)))||yij(t)|]+12∑Chl∈Nr(i,j)νijL1-γCijhl(ξhl-1(t))1-τhl(ξhl-1(t))(xhl(t)-yhl(t))2eɛτhlM-12∑Chl∈Nr(i,j)νijL1-γCijhl(t)(xhl(t-τhl(t))-yhl(t-τhl(t)))2}≤eɛ(t)∑(i,j)βij{∑Chl∈Nr(i,j)(ɛ2-aij(t))(xij(t)-yij(t))2+∑Bhl∈Nr(i,j)Bijhl(t)μijL1-γ×[(xij(t)-yij(t))2+|xij(t)-yij(t)||xhl(t)-yhl(t)|]+∑Chl∈Nr(i,j)Cijhl(t)νijL1-γ×[(xij(t)-yij(t))2+|xij(t)-yij(t)|×|xhl(t-τhl(t))-yhl(t-τhl(t))|]+12∑Chl∈Nr(i,j)νijL1-γCijhl(ξhl-1(t))1-τhl(ξhl-1(t))(xij(t)-yij(t))2eɛτhlM-12∑Chl∈Nr(i,j)νijL1-γCijhl(t)(xhl(t-τhl(t))-yhl(t-τhl(t)))2}≤eɛ(t)∑(i,j)βij{∑Chl∈Nr(i,j)(ɛ2-aij(t))(xij(t)-yij(t))2+∑Bhl∈Nr(i,j)Bijhl(t)μijL1-γ×[(xij(t)-yij(t))2+12(xij(t)-yij(t))2(xhl(t)-yhl(t))2]+∑Chl∈Nr(i,j)Cijhl(t)νijL1-γ×[(xij(t)-yij(t))2+12(xij(t)-yij(t))2×(xhl(t-τhl(t))-yhl(t-τhl(t)))2]+12∑Chl∈Nr(i,j)νijL1-γCijhl(ξhl-1(t))1-τhl(ξhl-1(t))(xij(t)-yij(t))2eɛτhlM-12∑Chl∈Nr(i,j)νijL1-γCijhl(t)(xhl(t-τhl(t))-yhl(t-τhl(t)))2}≤eɛ(t)∑(i,j){βij(ɛ2-aij(t))(xij(t)-yij(t))2+32∑Bhl∈Nr(i,j)Bijhl(t)μijL1-γ(xij(t)-yij(t))2+32βij∑Chl∈Nr(i,j)Cijhl(t)νijL1-γ(xij(t)-yij(t))2+12∑Bij∈Nr(h,l)βhlBhlij(t)μhlL1-γ(xij(t)-yij(t))2+12∑Cij∈Nr(h,l)νhlL1-γChlij(ξij-1(t))1-τij(ξij-1(t))(xij(t)-yij(t))2eɛτhlM}≤-c02eɛt(xij(t)-yij(t))2≤0,
where c0>0 is defined by
c0=min(i,j)inft∈R{βij(2aij(t)-ɛ)-3βijL1-γ[μij∑Bhl∈Nr(i,j)Bijhl(t)+νij∑Chl∈Nr(i,j)Cijhl(t)]+βhl(μhlL1-γ∑Bij∈Nr(h,l)Bhlij(t)+L1-γνhl∑Chl∈Nr(i,j)Chlij(ξij-1(t))1-τij(ξij-1(t)))}>0.
From the above, we have V(t)≤V(t0),t≥t0, and
12eɛtminh,lβhl∑i=1n(xij(t)-yij(t))2≤V(t),t≥t0,V(t0)≤12eɛt0[∑(i,j)βij+∑(i,j)∑(h,l)βijνijL1-γτhlμeɛτhlμsupt∈[t0-τ,t0]Chl(ξhl-1(t))1-τhl(ξhl-1(t))]∥φ-ψ∥22.
Thus, it follows that there exists a positive constant M>1 such that
|x(t)-y(t)|2≤Me-(ɛ/2)(t-t0)∥φ-ψ∥2,t≥t0,
which implies that (1.4) is GES.
Now we assume that (A6) is satisfied. By carrying out similar arguments as above, one can easily show that there exists an ɛ>0 such thatinf(i,j){βij(aij(t)-ɛ)-βijL1-γ[μij∑Bhl∈Nr(i,j)Bijhl(t)+νij∑Chl∈Nr(i,j)Cijhl(t)]-βhl(μhlL1-γ∑Bij∈Nr(h,l)Bhlij(t)+L1-γνhl∑Chl∈Nr(i,j)Chlij(ξij-1(t))1-τij(ξij-1(t)))}>0.
Consider the Lyapunov function
V(t)=∑(i,j)βij{|xij(t)-yij(t)|eɛt+∑Chl∈Nr(i,j)νijL1-γ×∫t-τhl(t)tCijhl(ξhl-1(s))1-τhl(ξhl-1(s))|xhl(s)-yhl(s)|eɛ(t+τhlμ)ds∑Bhl∈Nr(i,j)}.
Similar to the above arguments, calculating the upper-right derivative D+V(t) produces
D+V(t)≤-c2eɛt∑(i,j)|xij(t)-yij(t)|≤0,
where
c2=min(i,j)inft∈R{βij(aij(t)-ɛ)-βijL1-γ[μij∑Bhl∈Nr(i,j)Bijhl(t)+νij∑Chl∈Nr(i,j)Cijhl(t)]-βhl(μhlL1-γ∑Bij∈Nr(h,l)Bhlij(t)+L1-γνhl∑Chl∈Nr(i,j)Chlij(ξij-1(t))1-τij(ξij-1(t)))}>0.
Then we have
eɛt(min(h,l)βhl)∑(i,j)|xij(t)-yij(t)|≤V(t)≤V(t0),t≥t0.
Note that
V(t0)≤eɛt0[∑(i,j)βij+∑(i,j)βij∑Cij∈Nr(h,l)τhlμeɛτhlμsupt∈[t0-τ,t0]Cijhl(ξhl-1(t))1-τhl(ξhl-1(t))]∥φ-ψ∥1.
Then there exists a positive constant M>1 such that|x(t)-y(t)|1=∑i=1n|xi(t)-yi(t)|≤M∥φ-ψ∥1eɛ(t-t0),t≥t0.
The proof is complete.
BouzerdoumA.PinterR. B.Shunting inhibitory cellular neural networks: derivation and stability analysis199340321522110.1109/81.222804MR1232563ZBL0825.93681LiuY.YouZ.CaoL.On the almost periodic solution of generalized shunting inhibitory cellular neural networks with continuously distributed delays200636011221302-s2.0-3375033260710.1016/j.physleta.2006.08.013ZhouT.LiuY.ChenA.Almost periodic solution for shunting inhibitory cellular neural networks with time-varying delays and variable coefficients20062332432552-s2.0-3374569995610.1007/s11063-006-9000-2ChenL.ZhaoH.Global stability of almost periodic solution of shunting inhibitory cellular neural networks with variable coefficients200835235135710.1016/j.chaos.2006.05.057MR2357008ZBL1140.34425ChenA.CaoJ.Almost periodic solution of shunting inhibitory CNNs with delays20022982-316117010.1016/S0375-9601(02)00469-3MR1917000ZBL0995.92003LiY.LiuC.ZhuL.Global exponential stability of periodic solution for shunting inhibitory CNNs with delays20053371-246542-s2.0-1464441478010.1016/j.physleta.2005.01.008HuangX.CaoJ.Almost periodic solution of shunting inhibitory cellular neural networks with time-varying delay2003314322223110.1016/S0375-9601(03)00918-6MR1995884ZBL1052.82022LiuB.HuangL.Existence and stability of almost periodic solutions for shunting inhibitory cellular neural networks with time-varying delays200731121121710.1016/j.chaos.2005.09.052MR2263280ZBL1161.34352ChenA.CaoJ.Almost periodic solution of shunting inhibitory CNNs with delays20022982-316117010.1016/S0375-9601(02)00469-3MR1917000ZBL0995.92003XiaY.CaoJ.HuangZ.Existence and exponential stability of almost periodic solution for shunting inhibitory cellular neural networks with impulses20073451599160710.1016/j.chaos.2006.05.003MR2335406ZBL1152.34343ZhouQ.XiaoB.YuY.PengL.Existence and exponential stability of almost periodic solutions for shunting inhibitory cellular neural networks with continuously distributed delays200734386086610.1016/j.chaos.2006.03.092MR2327611ZBL1155.34347LiuB.Almost periodic solutions for shunting inhibitory cellular neural networks without global Lipschitz activation functions2007203115916810.1016/j.cam.2006.03.016MR2313827ZBL1122.34053ZhangC. Y.1992Ontario, CanadaUniversity of Western OntarioZhangC. Y.Pseudo-almost-periodic solutions of some differential equations19941811627610.1006/jmaa.1994.1005MR1257954ZBL0796.34029HeC. Y.1992Beijing, ChinaHigher Education Publishing HouseFinkA. M.1974377Berlin, GermanySpringerviii+336Lecture Notes in MathematicsMR0460799