# Existence and Global Exponential Stability of Almost Periodic Solutions for SICNNs with Nonlinear Behaved Functions and Mixed Delays

Authors: Xinsong Yang; Jinde Cao; Chuangxia Huang; Yao Long
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

By using the Leray-Schauder fixed point theorem and differential inequality techniques,
several new sufficient conditions are obtained for the existence and global exponential
stability of almost periodic solutions for shunting inhibitory cellular neural networks with
discrete and distributed delays. The model in this paper possesses two characters: nonlinear
behaved functions and all coefficients are time varying. Hence, our model is general and
applicable to many known models. Moreover, our main results are also general and can be
easily deduced to many simple cases, including some existing results. An example and its
simulation are employed to illustrate our feasible results.


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## Body

## 1. Introduction

Consider the following shunting inhibitory cellular neural networks (SICNNs) with discrete and distributed delays (mixed delays):(1.1)xij′(t)=-aij(t,xij(t))+∑Bkl∈Nr(i,j)Bijkl(t)fij(xkl(t-τkl(t)))xij(t)+∑Ckl∈Nr(i,j)Cijkl(t)∫-∞tkij(t-s)gij(xkl(s))dsxij(t)+Iij(t),
where i=1,2,…,n, j=1,2,…,m.  Cij(t) denotes the cell at the (i,j) position of the lattice at time t, the r-neighborhood Nr(i,j) of Cij(t) is(1.2)Nr(i,j)={Ckl(t):max(|k-i|,|l-j|)≤r,1≤k≤m,1≤l≤n}.xij(t) is the activity of the cell Cij(t), Iij(t) is the external input to Cij(t), aij(t,xij(t)) represents an appropriately behaved function of the cell Cij(t) at time t; Bijkl(t) and Cijkl(t) are the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell Cij(t) depending upon discrete delays and distributed delays, respectively; the activation functions fij(·) and gij(·) are continuous representing the output or firing rate of the cell Ckl(t); τij(t) represent axonal signal transmission delays; Bijkl(t),Cijkl(t),fij(·),gij(·),Iij(t),τij(t) are all continuous almost periodic functions.Since Bouzerdoum and Pinter described SICNNs as a new cellular neural networks [1–3], SICNNs have been extensively studied and found many important applications in different areas such as psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing. There have been some results on the existence of periodic and almost solutions for SICNNs with discrete or distributed delays (distributed delay was first introduced in [4]) [5–21]. We find that all the behaved functions in the models in [5–21] are linear. Actually, aij(t,xij(t)) may be nonlinear. Moreover, we find that the models in [5–20] are special cases of (1.1). For example, let aij(t,xij(t))=aij(t)xij(t), then Cijkl(t)≡0 in [5],  Bijkl(t)≡0 in [7], Cijkl(t)≡0 in [8], aij(t),Cijkl(t) are constants and Bijkl(t)≡0 in [9], aij(t),Bijkl(t),τkl(t) are constants, and Cijkl(t)≡0 in [10, 11]. To the best of our knowledge, few authors have considered the existence and global exponential stability of almost periodic solutions for SICNNs with nonlinear behaved functions, periodic coefficients and mixed delays. Obviously, (1.1) is general and is worth to continue to investigate its dynamical properties such as existence and global exponential stability of almost periodic solutions.The main purpose of this paper is to get sufficient conditions on the existence and global exponential stability of almost periodic solutions for SICNNs (1.1) by using the Leray-Schauder fixed point theorem and differential inequality techniques. Our results are general and possess infinitely adjustable real parameters and can be deduced to many simple results, including some existing results as special cases. Therefore, our results provide a wider application criteria for neural networks.The remaining part of this paper is organized as follows. We first state some useful definitions and lemmas in Section2. In Section 3, we study the existence of almost periodic solutions of system (1.1) by using the Leray-Schauder's fixed point theorem. In Section 4, by using Lemma 2.7, we will derive sufficient conditions for the global exponential stability of the almost periodic solution of system (1.1). A useful corollary is also obtained. An illustrative example and its simulation are given in Section 5.


## 2. Preliminaries

For convenience, we denote(2.1)x={xij(t)}=(x11(t),x12(t),…,x1m(t),…,xn1(t),xn2(t),…,xnm(t))T.Definition 2.1.
The continuous functionxij(t):ℝ→ℝ is called almost periodic on ℝ, if for any ɛ>0, it is possible to find a real number l=l(ɛ)>0 such that, for any interval with length l, there exists a number τ=τ(ɛ) in this interval such that |xij(t+τ)-xij(t)|<ɛ, for any t∈ℝ.Definition 2.2 (see [22, page 21]).
Let𝔼 be a Banach space, 𝔻 an open subset in 𝔼 and f(t,x)∈C(ℝ×𝔻,𝔼). For ∀x∈𝔻, f(t,x) is called uniformly almost periodic about t, if for any ɛ>0 and any compact subset 𝕊⊂𝔻, there exists a real number l=l(ɛ,𝕊)>0 such that, for any interval with length l, there exists a number τ=τ(ɛ,𝕊) in this interval such that ∥f(t+τ,x)-f(t,x)∥<ɛ, for any (t,x)∈ℝ×𝕊.The initial conditionφ={φij(s)} of (1.1) is of the form(2.2)xij(s)=φij(s),s∈(-∞,0],
where φij(s),i=1,2,…,n,j=1,…,m, are continuous almost periodic solutions.Definition 2.3.
Letx*(t) be an almost periodic solution of (1.1) with initial value φ*. If there exist constants α>0 and P>1 such that for every solution x(t) of (1.1) with initial value φ(2.3)|xij(t)-xij*(t)|≤P∥φ-φ*∥e-αt,∀t>0,i=1,2,…,n,j=1,2,…,m,
where ∥φ-φ*∥=max(i,j)sups≤0{|φij(s)-φij*(s)|}. Then x*(t) is said to be globally exponentially stable.Lemma 2.4 (see [22, page 136]).
Suppose thatf(t,x) is uniformly continuous on ℝ×𝕊, where 𝕊 is any compact set on ℝn, and that there exists a nonsingular matrix P(t)∈C1, such that  (a)
there exists a constantρ satisfies ∥P(t)∥<ρ, for all t∈ℝ,(b)
the eigenvaluesλi(t),i=1,2,…,n, of P(t) satisfy |λi(t)|≥ϑ>0, where ϑ is a constant, and there are k negative eigenvalues, n-k nonnegative eigenvalues,(c)
all the eigenvaluesλi(t,x),i=1,2,…,n, of the following symmetric matrix:
(2.4)M(t,x)=P(t)f(t,x)+fT(t,x)P(t)+Ṗ(t)
satisfy
(2.5)λi(t,x)≤-δ<0,∀∥x∥≤R0,
where δ,R0 are constants.Then, for any fixed ϕ(t)∈C(ℝ,ℝn) and any t∈ℝ satisfying ∥ϕ(t)∥<R0, the linear differential equation
(2.6)ẋ=f(t,ϕ(t))x
admits an exponential dichotomy on ℝ:
(2.7)|Xϕ(t)QϕXϕ-1(s)|≤Mρϑe-(δ/2ρ)(t-s),t≥s,|Xϕ(t)(I-Qϕ)Xϕ-1(s)|≤Mρϑe-(δ/2ρ)(t-s),s≥t,
where Xϕ(t) is fundamental solution matrix of (2.6) satisfying Xϕ(0)=In×n, In×n is the identity matrix, Qϕ is a constant projection, the constant M has no relationship with ϕ(t).Lemma 2.5 (see [22, page 139]).
Suppose that then×n matrix function f(t,x) and the n-dimensional vector function g(t,x) are uniformly almost periodic on ℝ×ℝn, and that there exists real  symmetric nonsingular matrix P(t) satisfying the conditions (a)–(c) in Lemma 2.4, then,
(2.8)ẋ=f(t,ϕ)x+g(t,ϕ)
has a unique almost periodic solution x(t), where ϕ∈C(ℝ,ℝn) is almost periodic function and
(2.9)x(t)=∫-∞tXϕ(s)QϕXϕ-1(s)g(s,ϕ(s))ds-∫t+∞Xϕ(s)(I-Qϕ)Xϕ-1(s)g(s,ϕ(s))ds.Lemma 2.6 (Leray-Schauder).
Let𝔼 be a Banach space, and let the operator Φ:𝔼→𝔼 be completely continuous. If the set {∥x∥∣x∈𝔼,x=λΦx,0<λ<1} is bounded, then Φ has a fixed point in 𝕋, where
(2.10)𝕋={x∣x∈E,∥x∥≤R},R=sup{∥x∥∣x=λΦx,0<λ<1}.Lemma 2.7 (see [23]).
Leta≥0, bk≥0(k=1,2,…,m), the following inequality holds
(2.11)a∏k=1mbkqk≤1r∑k=1mqkbkr+1rar,
where qk>0,(k=1,2,…,m) is some constants, ∑k=1mqk=r-1, and r>1.Obviously, inequality (2.11) also holds for a≥0, bk≥0, r=1, and qk=0,k=1,2,…,m. Hence, we always assume that a≥0, bk≥0, r≥1, and qk≥0,k=1,2,…,m in (2.11)  in the later sections of this paper.Furthermore, throughout this paper, we assume that(H1)
aij(t,u)∈C(ℝ2,ℝ) is continuous almost periodic about the first argument and, there exists a positive continuous almost periodic function μij(t) such that ∂aij(t,u)/∂u≥μij(t), u∈ℝ, and aij(t,0)=0, i=1,2,…,n,j=1,2,…,m,(H2)
there exist nonnegative constantsMij and Nij,  i=1,2,…,n,j=1,2,…,m, such that
(2.12)|fij(u)|≤Mij,|gij(u)|≤Nij,∀u∈ℝ;(H3)
the delay kernelskij:[0,+∞)→ℝ are continuous, integrable and there are positive constants k¯ij such that
(2.13)∫0+∞|kij(s)|ds≤k¯ij,i=1,2,…,n,j=1,2,…,m;(H4)
there exists a constantα0>0 such that
(2.14)∫0+∞|kij(s)|eα0sds<+∞,i=1,2,…,n,j=1,2,…,m.(H5)
the following inequality holds:(2.15)max(i,j)supt∈ℝ{∑Bkl∈Nr(i,j)|Bijkl(t)|Mij+∑Ckl∈Nr(i,j)|Cijkl(t)|k¯ijNijμij(t)}=η<1;(H6)
there are nonnegative constantsαij,βij such that
(2.16)αij=supu≠v|fij(u)-fij(v)u-v|,βij=supu≠v|gij(u)-gij(v)u-v|
for all u,v∈ℝ,u≠v,i=1,2,…,n,j=1,2,…,m.


## 3. Existence of Almost Periodic Solutions

Letξij, i=1,2,…,n, j=1,2,…,m be constants. Make the following transformation:(3.1)xij=ξijyij(t),i=1,2,…,n,j=1,2,…,m,
then (1.1) can be reformulated as(3.2)yij′(t)=-ξij-1aij(t,ξijyij(t))+∑Bkl∈Nr(i,j)Bijkl(t)fij(ξklykl(t-τkl(t)))yij(t)+∑Ckl∈Nr(i,j)Cijkl(t)∫-∞tkij(t-s)gij(ξklykl(s))dsyij(t)+ξij-1Iij(t).System (3.2) can be rewritten as(3.3)yij′(t)=-dij(t,yij(t))yij(t)+∑Bkl∈Nr(i,j)Bijkl(t)fij(ξklykl(t-τkl(t)))yij(t)+∑Ckl∈Nr(i,j)Cijkl(t)∫-∞tkij(t-s)gij(ξklykl(s))dsyij(t)+ξij-1Iij(t),
where dij(t,yij(t))≜(∂aij(t,z)/∂z)|z=eij, eij is between 0 and ξijyij(t), eij∈ℝ. By (H1), we know that aij(t,ξijyij) is strictly monotone increasing about yij. Hence, dij(t,yij(t)) is unique for any yij(t). Obviously, dij(t,yij(t)) is continuous almost periodic about the first argument and dij(t,yij(t))≥μij(t).Take𝕏={ϕ={ϕij(t)}∣ϕij:ℝ→ℝ is an almost periodic function, i=1,…,n, j=1,…,m}. Then 𝕏 is a Banach space with the norm(3.4)∥ϕ∥=max(i,j){|ϕij|0},|ϕij|0=supt∈ℝ|ϕij(t)|,i=1,…,n,j=1,…,m.For for allϕ∈𝕏, we consider the following auxiliary equation:(3.5)yij′(t)=-dij(t,ϕij(t))yij(t)+∑Bkl∈Nr(i,j)Bijkl(t)fij(ξklϕkl(t-τkl(t)))ϕij(t)+∑Ckl∈Nr(i,j)Cijkl(t)∫-∞tkij(t-s)gij(ξklϕkl(s))dsϕij(t)+ξij-1Iij(t).
From (H1), we know that(3.6)dij(t,yij),i=1,2,…,n,j=1,2,…,m,
are uniformly almost periodic functions on ℝ×ℝ. Since μij(t),i=1,2,…,n,j=1,2,…,m, are positive continuous almost periodic functions, there exists a positive constant δ such that(3.7)dij(t,ϕij(t))≥μij(t)≥δ>0.
Hence, the conditions in Lemma 2.5 are satisfied (take P(t)=Inm×nm).According to the fact thatBijkl(t),ϕij(t),Cijkl(t),Iij(t) are almost periodic functions, in view of Lemma 2.5, we know that system (3.5) has a unique almost periodic solution(3.8)yϕ(t)={∫-∞te-∫stdij(u,ϕij(u))du×[∑Bkl∈Nr(i,j)Bijkl(s)fij(ξklϕkl(s-τkl(s)))ϕij(s)+∑Ckl∈Nr(i,j)Cijkl(s)∫-∞skij(s-u)gij(ξklϕkl(u))duϕij(s)+ξij-1Iij(s)]ds}.Set a mappingΦ:𝕏→𝕏 by setting(3.9)(Φϕ)(t)=yϕ(t),∀ϕ∈𝕏.Before using Lemma2.6 to obtain conditions of the existence of almost periodic solution for (1.1), we have to prove the following lemma.Lemma 3.1.
Suppose that (H1)–( H5) hold. Then Φ:𝕏→𝕏 is completely continuous.Proof.
Under our assumptions, it is clear that the operatorΦ is continuous. Next, we show that Φ is compact.
For any constantD>0, let Ω={ϕ∣ϕ∈𝕏,∥ϕ∥<D}. Then, for any ϕ∈Ω, we have
(3.10)∥(Φϕ)∥=max(i,j)supt∈ℝ{|∫-∞te-∫stdij(u,ϕij(u))du×[∑Bkl∈Nr(i,j)Bijkl(s)fij(ξklϕkl(s-τkl(s)))ϕij(s)+∑Ckl∈Nr(i,j)Cijkl(s)∫-∞skij(s-u)gij(ξklϕkl(u))duϕij(s)+ξij-1Iij(s)∑Ckl∈Nr(i,j)Cijkl(s)∫-∞skij]ds|}≤max(i,j)supt∈ℝ{∫-∞te-∫stμij(u)du[(∑Bkl∈Nr(i,j)|Bijkl(s)|Mij+∑Ckl∈Nr(i,j)|Cijkl(s)|k¯ijNij)×|ϕij(s)|+ξij-1|Iij(s)|∑Ckl∈Nr(i,j)]ds∑Ckl∈Nr(i,j)}≤max(i,j)supt∈ℝ{∫-∞te-∫stμij(u)duμij(s)ds}η∥ϕ∥+max(i,j){I¯ijξijμ̲ij}<ηD+max(i,j){I¯ijξijμ̲ij},
where I¯ij=maxt∈R|Iij(t)|, μ̲ij=inft∈ℝμij(t). Hence, Φ(Ω) is uniformly bounded.
By the definition ofΦ, we get
(3.11)(Φϕ)ij′(t)=ddt(∫-∞te-∫stdij(u,ϕij(u))du×[∑Bkl∈Nr(i,j)Bijkl(s)fij(ξklϕkl(s-τkl(s)))ϕij(s)+∑Ckl∈Nr(i,j)Cijkl(s)∫-∞skij(s-u)gij(ξklϕkl(u))duϕij(s)+ξij-1Iij(s)]ds)=-dij(t,ϕij(t))(Φϕij)(t)+∑Bkl∈Nr(i,j)Bijkl(t)fij(ξklϕkl(t-τkl(t)))ϕij(t)+∑Ckl∈Nr(i,j)Cijkl(t)∫-∞tkij(t-s)gij(ξklϕkl(s))dsϕij(t)+ξij-1Iij(t).
Since dij(t,ϕij(t)),i=1,2,…,n,j=1,2,…,m, are uniformly almost periodic functions on ℝ×Ω, there exists a positive constant Θ such that
(3.12)|dij(t,ϕij(t))|≤Θ,fort∈ℝ,ϕij(t)∈Ω,i=1,2,…,n,j=1,2,…,m.
Hence,
(3.13)∥(Φϕ)ij′(t)∥≤Θ(ηD+max(i,j){I¯ijξijμ̲ij})+max(i,j){∑Bkl∈Nr(i,j)B¯ijklMijD+∑Ckl∈Nr(i,j)C¯ijklk¯ijNijD+ξij-1I¯ij},
where B¯ijkl=supt∈R|Bijkl(t)|, C¯ijkl=supt∈ℝ|Cijkl(t)|. So, Φ(Ω)⊆𝕏 is a family of uniformly bounded and equicontinuous subsets. By using the Arzela-Ascoli theorem, Φ:𝕏→𝕏 is compact. Therefore, Φ:𝕏→𝕏 is completely continuous. This completes the proof.Theorem 3.2.
Suppose that(H1)–(H5) hold. Let ξij,i=1,2,…,n,j=1,2,…,m, be constants. Then system (1.1) has an almost periodic solution x*(t) with ∥x*∥≤max(i,j){ξij}R̃≜R0, where
(3.14)R̃=max(i,j){I¯ij/ξijμ̲ij}1-η.Proof.
Letϕ∈𝕏. From Lemma 3.1, we get that Φ:𝕏→𝕏 is completely continuous. Consider the following operator equation:
(3.15)ϕ=λΦϕ,λ∈(0,1).
If ϕ is a solution of (3.15), we obtain
(3.16)∥ϕ∥≤∥Φϕ∥≤θ∥ϕ∥+max(i,j){I¯ijξijμ̲ij}.
This and (H5) imply that
(3.17)∥ϕ∥≤R̃.
In view of Lemma 2.6, we obtain that Φ has a fixed point ϕ*(t) with ∥ϕ*∥≤R̃. From (3.5) and (3.8), we know that ϕ* satisfies (3.3) and (3.2). Hence, system (3.2) has an almost periodic solution ϕ*(t)={ϕij*(t)} with ∥ϕ*∥≤R̃. It follows from (3.1) that x*(t)={xij*(t)}={ξijϕij*(t)} is one almost periodic solution of (1.1) with
(3.18)∥x*∥≤max(i,j){ξij}R̃≜R0.
This completes the proof.


## 4. Stability of Almost Periodic solution

In this section, we prove that, under suitable conditions, the almost periodic solution obtained in Theorem3.2 is globally exponentially stable.Theorem 4.1.
Assume that(H1)–(H4) and (H6) hold and  (H7)
there are constantsos,ɛs,ls,γs,δs,ds∈ℝ,q≥1, ps≥0, σ>0, ξij>0,s=1,2,…,m′+1, i=1,2,…,n,j=1,2,…,m, such that
(4.1)Λ=max(i,j)supt∈R{(-qμij(t)+q∑Bkl∈Nr(i,j)|Bijkl(t)|Mij+q∑Ckl∈Nr(i,j)|Cijkl(t)|k¯ijNij+∑Bkl∈Nr(i,j)∑s=1m′ps|Bijkl(t)|qosαijqɛsR0qls+∑Ckl∈Nr(i,j)∑s=1m′ps|Cijkl(t)|qγsβijqδsR0qds)ξij+∑Bkl∈Nr(i,j)|Bijkl(t)|qom′+1αijqɛm′+1R0qlm′+1ξkl+∑Ckl∈Nr(i,j)|Cijkl(t)|qγm′+1βijqδm′+1R0qdm′+1k¯ijqξkl}≤-σ<0,
where, om′+1+∑s=1m′osps=ɛm′+1+∑s=1m′ɛsps=lm′+1+∑s=1m′lsps=γm′+1+∑s=1m′γsps=δm′+1+∑s=1m′δsps=dm′+1+∑s=1m′dsps=1, ∑s=1m′ps=q-1.Then (1.1) has unique almost periodic solution, which is globally exponentially stable.Proof.
Obviously, that(H7) holds implies that (H5) holds. By Theorem 3.2, there exists an almost periodic solution x*(t) of (1.1) with initial value φ*(t) and ∥x*∥≤R0. Suppose that x(t) is an arbitrary solution of (1.1) with initial value φ(t). Set z(t)=x(t)-x*(t). Then, from (1.1), we have
(4.2)zij′(t)=-[aij(t,xij(t))-aij(t,xij*(t))]+∑Bkl∈Nr(i,j)Bijkl(t)(fij(xkl(t-τkl(t)))xij(t)-fij(xkl*(t-τkl(t)))xij*(t))+∑Ckl∈Nr(i,j)Cijkl(t)(∫-∞tkij(t-s)gij(xkl(s))dsxij(t)-∫-∞tkij(t-s)gij(xkl*(s))dsxij*(t)),i=1,2,…,n,j=1,2,…,m.
Set
(4.3)Γij(λ,t)=(λ-qμij(t)+q∑Bkl∈Nr(i,j)|Bijkl(t)|Mij+q∑Ckl∈Nr(i,j)|Cijkl(t)|k¯ijNij+∑Bkl∈Nr(i,j)‍∑s=1m′ps|Bijkl(t)|qosαijqɛsR0qls+∑Ckl∈Nr(i,j)‍∑s=1m′ps|Cijkl(t)|qγsβijqδsR0qds)ξij+∑Bkl∈Nr(i,j)|Bijkl(t)|qom′+1αijqɛm′+1R0qlm′+1eλτkl(t)ξkl+∑Ckl∈Nr(i,j)|Cijkl(t)|qγm′+1βijqδm′+1R0qdm′+1(∫0+∞|kij(s)|e(λ/q)sds)qξkl,i=1,2,…,n,j=1,2,…,m.
Clearly, Γij(λ,t), i=1,2,…,n, j=1,2,…,m, are continuous functions about λ and almost periodic about t on ℝ.
From(H7), we get
(4.4)Γij(0,t)≤Λ≤-σ<0,i=1,2,…,n,j=1,2,…,m.
In addition to
(4.5)∂Γij(λ,t)∂λ=ξij+∑Bkl∈Nr(i,j)|Bijkl(t)|qom′+1αijqɛm′+1R0qlm′+1τkl(t)eλτkl(t)ξkl+∑Ckl∈Nr(i,j)|Cijkl(t)|qγm′+1βijqδm′+1R0qdm′+1(∫0+∞|kij(s)|se(λ/q)sds)qξkl>0
and Γij(+∞,t)=+∞ for ∀t∈R, we obtain that Γij(λ,t),i=1,2,…,n, j=1,2,…,m, are strictly monotone increasing functions about λ. Therefore, for any i∈{1,2,…,n},  j∈{1,2,…,m}, there is unique real number λ0ij such that
(4.6)supt∈RΓij(λ0ij,t)=0.
In view of (4.4), we have λ0ij>0. Let θ=min{λ011,λ012,…,λ0nm}, we obtain
(4.7)supt∈RΓij(θ,t)≤0,i=1,2,…,n,j=1,2,…,m.
Now, we choose a positive constant d such that qdξij≥1,i=1,2,…,n,j=1,2,…,m. It is obvious that
(4.8)|zi(t)|q≤∥φ-φ*∥q≤qdξi∥φ-φ*∥qe-θt,for∈(-∞,0],i=1,2,…,n,
where ∥φ-φ*∥ is defined as that in Definition 2.3.
LetVij(t)=(1/q)eθt|zij(t)|q,i=1,2,…,n,j=1,2,…,m. In view of (4.2) and (2.11), for t≥0, we obtain
(4.9)d+Vij(t)dt=eθt|zij(t)|q-1×sgnzij{-[∫-∞tkijaij(t,xij(t))-aij(t,xij*(t))]+∑Bkl∈Nr(i,j)Bijkl(t)(fij(xkl(t-τkl(t)))xij(t)-fij(xkl*(t-τkl(t)))xij*(t))+∑Ckl∈Nr(i,j)Cijkl(t)(∫-∞tkij(t-s)gij(xkl(s))dsxij(t)-∫-∞tkij(t-s)gij(xkl*(s))dsxij*(t))}+1qθeθt|zij(t)|q≤eθt|zij(t)|q-1×{-μij(t)|zij(t)|+∑Bkl∈Nr(i,j)|Bijkl(t)|×(|fij(xkl(t-τkl(t)))||zij(t)|+|fij(xkl(t-τkl(t)))-fij(xkl*(t-τkl(t)))||xij*(t)|)+∑Ckl∈Nr(i,j)|Cijkl(t)|(∫-∞t|kij(t-s)||gij(xkl(s))|ds|zij(t)|+∫-∞t|kij(t-s)||gij(xkl(s))-gij(xkl*(s))|ds|xij*(t)|)}+1qθeθt|zij(t)|q≤eθt|zij(t)|q-1×{-μij(t)|zij(t)|+∑Bkl∈Nr(i,j)|Bijkl(t)|(Mij|zij(t)|+αijR0|zkl(t-τkl(t))|)+∑Ckl∈Nr(i,j)|Cijkl(t)|(k¯ijNij|zij(t)|+βijR0∫-∞t|kij(t-s)||zkl(s)|ds)}+1qθeθt|zij(t)|q≤(θ-qμij(t)+q∑Bkl∈Nr(i,j)|Bijkl(t)|Mij+q∑Ckl∈Nr(i,j)|Cijkl(t)|k¯ijNij+∑Bkl∈Nr(i,j)∑s=1m'ps|Bijkl(t)|qosαijqεsR0qls+∑Ckl∈Nr(i,j)‍∑s=1m'ps|Cijkl(t)|qγsβijqδsR0qds)Vij(t)+∑Bkl∈Nr(i,j)|Bijkl(t)|qom′+1αijqεm′+1R0qlm′+1eθτkl(t)Vkl(t-τkl(t))+∑Ckl∈Nr(i,j)|Cijkl(t)|qγm′+1βijqδm′+1R0qdm′+1(∫-∞t|kij(t-s)|e(θ/q)(t-s)Vkl(s)1/qds)q.
We now prove that the following statement is true:
(4.10)Vij(t)=1qeθt|zij(t)|q≤dξij∥ϕ-ϕ*∥q,i=1,2,…,n,j=1,2,…,m,∀t>0.
Contrarily, there must exist some i∈{1,2,…,n}, j∈{1,2,…,m}, and t̃>0, such that
(4.11)Vij(t̃)=dξij∥ϕ-ϕ*∥q,d+Vij(t̃)dt>0,Vkl(t)≤dξkl∥ϕ-ϕ*∥q,
for all  t∈(-∞,t̃],k=1,2,…,n,l=1,2,…,m. Together with (4.9), (4.11), and (4.7), we obtain
(4.12)0<d+Vij(t̃)dt≤d∥ϕ-ϕ*∥q{(θ-qμij(t̃)+q∑Bkl∈Nr(i,j)|Bijkl(t̃)|Mij+q∑Ckl∈Nr(i,j)|Cijkl(t̃)|k¯ijNij+∑Bkl∈Nr(i,j)∑s=1m′ps|Bijkl(t̃)|qosαijqɛsR0qls+∑Ckl∈Nr(i,j)∑s=1m′ps|Cijkl(t̃)|qγsβijqδsR0qds)ξij+∑Bkl∈Nr(i,j)|Bijkl(t̃)|qom′+1αijqɛm′+1R0qlm′+1eθτkl(t̃)ξkl+∑Ckl∈Nr(i,j)|Cijkl(t̃)|qγm′+1βijqδm′+1R0qdm′+1(∫0+∞|kij(s)|e(θ/q)sds)qξkl}≤0,
which is a contradiction. Hence, (4.10) holds. It follows that
(4.13)|xij(t)-xij*(t)|=|zij(t)|≤(qdξij)1/q∥φ-φ*∥e-(θ/q)t,∀t>0,i=1,2,…,n,j=1,2,…,m.
Let P=max(i,j){(qdξij)1/q+1}, then
(4.14)|xij(t)-xij*(t)|≤P∥φ-φ*∥e-(θ/q)t,∀t>0,i=1,2,…,n,j=1,2,…,m.
By means of Definition 2.3, the almost periodic solution x*(t) of (1.1) is globally exponentially stable. This completes the proof.Since Theorem3.2 is general, the expression of (H7) is complex. In order to check the applicability of the results easily, we give the following corollary.Corollary 4.2.
Assume that(H1)–(H4) and (H6) hold. Furthermore, if one of the following conditions holds  (A1)
there are constantsσ>0, q≥1, ξij>0, i=1,2,…,n, j=1,2,…,m, such that
(4.15)max(i,j)supt∈R{(-qμij(t)+q∑Bkl∈Nr(i,j)|Bijkl(t)|Mij+q∑Ckl∈Nr(i,j)|Cijkl(t)|k¯ijNij+(q-1)∑Bkl∈Nr(i,j)|Bijkl(t)|+(q-1)∑Ckl∈Nr(i,j)|Cijkl(t)|)ξij+∑Bkl∈Nr(i,j)|Bijkl(t)|αijqR0qξkl+∑Ckl∈Nr(i,j)|Cijkl(t)|βijqR0qk¯ijqξkl}≤-σ<0,(A2)
there are constantsσ>0, ξij>0, i=1,2,…,n, j=1,2,…,m, such that
(4.16)max(i,j)supt∈R{(-2μij(t)+2∑Bkl∈Nr(i,j)|Bijkl(t)|Mij+2∑Ckl∈Nr(i,j)|Cijkl(t)|k¯ijNij+∑Bkl∈Nr(i,j)|Bijkl(t)|+∑Ckl∈Nr(i,j)|Cijkl(t)|)ξij+∑Bkl∈Nr(i,j)|Bijkl(t)|αij2R02ξkl+∑Ckl∈Nr(i,j)|Cijkl(t)|βij2R02k¯ij2ξkl}≤-σ<0,(A3)
there are constantsσ>0, ξij>0, i=1,2,…,n, j=1,2,…,m, such that
(4.17)max(i,j)supt∈R{(-μij(t)+∑Bkl∈Nr(i,j)|Bijkl(t)|Mij+∑Ckl∈Nr(i,j)|Cijkl(t)|k¯ijNij)ξij+∑Bkl∈Nr(i,j)|Bijkl(t)|αijR0ξkl+∑Ckl∈Nr(i,j)|Cijkl(t)|βijR0k¯ijξkl}≤-σ<0.Then (1.1) has exactly one almost periodic solution, which is globally exponentially stable.Proof.
Letm′=1, o1=o2=γ1=γ2=1/q, ɛ1=l1=δ1=d1=0, and ɛ2=l2=δ2=d2=1, i=1,2,…,n, j=1,2,…,m, in (H7), respectively, then (H7) turns to (A1).
Letq=2 and q=1 in (A1), respectively, then we get (A2) and (A3). This completes the proof.Remark 4.3.
Corollary4.2 can still be simplified into many simple results, which include the results in [7–11]. For example, authors of [9] considered a special case of our model (1.1) and obtained sufficient conditions of existence and exponential stability of almost periodic solutions for their model. Although the results of [9] and ours are completely different, we claim that the results in [9] are also special cases of the results in this paper. The model considered in [9] is
(4.18)xij′(t)=-aijxij(t)+∑Ckl∈Nr(i,j)Cijkl∫0∞kij(s)g(xkl(t-s))dsxij(t)+Iij(t),
where aij,Cijkl≥0,|g(x)|≤M, and |g(u)-g(v)|≤μ|u-v| for u,v∈R. The other arguments are same as this paper. Obviously, system (4.18) is a special case of system (1.1). The main theorem in [9] is the following.Corollary 4.4.
Assume that there exist nonnegative constantsI,q, and η such thatmax(i,j){∑Ckl∈Nr(i,j)Cijklk¯ijM/aij} = η<1, I=max(i,j){Iij/aij}, q=η(1+μI/(1-η)M)<1. Then system (4.18) has exactly one almost periodic solution, which is globally exponentially stable.Now we prove that Corollary4.4 is a special case of Corollary 4.2. Comparing (1.1) with (4.18), we have μij(t)=aij, |Cijkl(t)|=Cijkl, |Bijkl(t)|=0, Nij=M. Let ξij=1,i=1,2…,n,j=1,2…,m. According to the proof of Theorem 3.2, we have R0=I/(1-η). So, q=η(1+μR0/M)<1⇔η((M+μR0)/M)<1. In view of the condition ∑Ckl∈Nr(i,j)Cijklk¯ijM/aij≤η, we have(4.19)∑Ckl∈Nr(i,j)Cijklk¯ijaij(M+μR0)<1,
which is equivalent to(4.20)-aij+M∑Ckl∈Nr(i,j)Cijklk¯ij+μR0∑Ckl∈Nr(i,j)Cijklk¯ij<0.
The expression is a special case of (A3) in Corollary 4.2.Similarly, from Corollary4.2, we can get the main results of [5, 11] by the condition (A3) (restrict in the case of periodic system in [5]) and the main results of [10] by condition (A2). Hence, the results of this paper are useful and important.Remark 4.5.
In (1.1), if fij(u),gij(u),u∈ℝ are replaced by fij(t,u),gij(t,u),t,u∈ℝ, where fij(t,u),gij(t,u) are bounded and almost periodic functions about the first argument, then, by the same methods used in this paper, we can also get some general results similar to Theorem 4.1 and Corollary 4.2 and extend and improve the main results in [12].Remark 4.6.
Note that condition(H7) in Theorem 4.1 possesses infinitely adjustable real parameters. This enables neural networks workers to obtain many special conditions on the existence and global exponential stability of almost periodic solutions and periodic solutions in the designs and applications of neural networks. We can offer many different corollaries by changing the parameters in (H7), not only Corollary 4.2. Hence, our results provide many criteria for neural network's application.


## 5. A Numerical Example

Consider the following SICNNs with mixed delays:(5.1)xij′(t)=-aij(t,xij(t))+∑Bkl∈Nr(i,j)Bijkl(t)fij(xkl(t-τkl(t)))xij(t)+∑Ckl∈Nr(i,j)Cijkl(t)∫-∞tkij(t-s)gij(xkl(s))dsxij(t)+Iij(t),
where r=1, τij(t)=(cost)2,i,j=1,2,3. Take a11(t)=6x-cosx+xsint,a12(t)=6x+0.5sinx+xcost,a13(t)=5x+cosx+xsint,a21(t)=7x+sinx+xsint,a22(t)=7x-cosx-xsint,a23(t)=5x-cosx+xcost,a31(t)=6x+sinx+xcost,a32(t)=6x+cosx+xsint,a33(t)=6x-sinx+xcost,fij(x)=gij(x)=0.1sinx, kij(t)=e-tsint, and(5.2)[B11(t)B12(t)B13(t)B21(t)B22(t)B23(t)B31(t)B32(t)B33(t)]=[0.1cost0.2sint00.3cost00.3sint00.4cost0.2sint],[C11(t)C12(t)C13(t)C21(t)C22(t)C23(t)C31(t)C32(t)C33(t)]=[0.2cost0.4sint0.3cost0.6cost00.5sint0.5sint0.6cost0.5sint],[I11(t)I12(t)I13(t)I21(t)I22(t)I23(t)I31(t)I32(t)I33(t)]=[2cos2tsin3t1+2cos2t1+cos2t2cos2t1-sin3t2sin2t1-cos3t3sin2t].
Then μ11(t)=5+sint,μ12(t)=5.5+cost,μ13(t)=4+sint,μ21(t)=6+sint,μ22(t)=6-sint,μ23(t)=4+cost,μ31(t)=5+cost,μ32(t)=5+sint,μ33(t)=5+cost, αij(t)=βij(t)=Mij=Nij=0.1.(5.3)∑Bkl∈Nr(1,1)|B11kl(t)|=0.4|cost|+0.2|sint|,∑Bkl∈Nr(1,2)|B12kl(t)|=0.4|cost|+0.5|sint|,∑Bkl∈Nr(1,3)|B13kl(t)|=0.5|sint|,∑Bkl∈Nr(2,1)|B21kl(t)|=0.4|cost|+0.6|sint|,∑Bkl∈Nr(2,2)|B22kl(t)|=0.8|cost|+0.7|sint|,∑Bkl∈Nr(2,3)|B23kl(t)|=0.7|cost|+0.4|sint|,∑Bkl∈Nr(3,1)|B31kl(t)|=0.3|cost|+0.4|sint|,∑Bkl∈Nr(3,2)|B32kl(t)|=0.7|cost|+0.5|sint|,∑Bkl∈Nr(3,3)|B33kl(t)|=0.4|cost|+0.5|sint|,∑Ckl∈Nr(1,1)|C11kl(t)|=0.8|cost|+0.4|sint|,∑Ckl∈Nr(1,2)|C12kl(t)|=1.1|cost|+0.9|sint|,∑Ckl∈Nr(1,3)|C13kl(t)|=0.3|cost|+0.9|sint|,∑Ckl∈Nr(2,1)|C21kl(t)|=1.4|cost|+0.9|sint|,∑Ckl∈Nr(2,2)|C22kl(t)|=1.7|cost|+1.9|sint|,∑Ckl∈Nr(2,3)|C23kl(t)|=0.9|cost|+1.4|sint|,∑Ckl∈Nr(3,1)|C31kl(t)|=1.2|cost|+0.5|sint|,∑Ckl∈Nr(3,2)|C32kl(t)|=1.2|cost|+1.5|sint|,∑Ckl∈Nr(3,3)|C33kl(t)|=0.6|cost|+|sint|.Takeξij=1, i,j=1,2,3. Computing by Matlab toolbox, we have η≈0.0721<1, R0=I¯13/a̲13(1-η)≈1.0777, and Λ≈-2.6849<0. It is easy to check that all the conditions needed in Corollary 4.2 with (A3) are satisfied. Therefore, system (5.1) has a unique almost periodic solution x*(t) with ∥x*∥≤1.0777, which is globally exponentially stable. See Figures 1 and 2.Figure 1
States of networks (5.1) with t∈[0,4].Figure 2
States of networks (5.1) with t∈[0,40].Remark 5.1.
In (5.1), the behaved functions are all nonlinear. So, all the results in [5–20] and the references therein are inapplicable for (5.1). This means that the results of this paper are new and useful.



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*Source: 915451-2010-03-16.xml*