# Existence and Global Uniform Asymptotic Stability of Pseudo Almost Periodic Solutions for  Cohen-Grossberg Neural Networks with Discrete and Distributed Delays

Authors: Hongying Zhu; Chunhua Feng
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

This paper studies the existence and uniform asymptotic stability of pseudo almost periodic solutions to Cohen-Grossberg neural networks (CGNNs) with discrete and distributed delays by applying Schauder fixed point theorem and constructing a suitable Lyapunov functional. An example is given to show the effectiveness of the main results.


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## Body

## 1. Introduction

Since the model of Cohen-Grossberg neural networks (CGNNs) was first proposed and studied by Cohen and Grossberg [1], it has been widely investigated because of the theoretical interest as well as the application considerations such as optimization, pattern recognition, automatic control, image processing, and associative memories. In recent years, there are many important results on dynamic behaviors of CGNNs. For instance, many sufficient conditions have been successively obtained to ensure the existence and stability of equilibrium point of CGNNs [1–10]. Some attractivity and asymptotic stability results have also been published [3, 11–14]. Many authors specially devote themselves to study the existence and global exponential stability of periodic or almost periodic solution to CGNNs [15–30]; for the other dynamic properties, see also the literatures [31, 32]. However, to the best of our knowledge, few authors have discussed the existence and the global uniform asymptotic stability of pseudo almost periodic solutions to CGNNs.In this paper, we discuss the existence and the global uniform asymptotic stability of pseudo almost periodic solutions to the following CGNNs:(1)xi′(t)=-ai(xi(t))×[∑j=1m‍pij(t)∫-∞t‍bi(xi(t))-∑j=1m‍cij(t)fj(xj(t))-∑j=1m‍dij(t)gj(xj(t-τij(t))),-∑j=1m‍pij(t)∫-∞t‍Gij(t-s)hj(xj(s))ds-Ii(t)],t≥0,xi(t)=Φi(t),t<0,
where cij(t), dij(t), pij(t), Ii(t),Φi(t)∈C(R,R), τij(t)∈C(R,R+) are pseudo almost periodic functions.The organization of this paper is as follows. In Section2, some basic definitions, marks, and lemmas are given. In Section 3, some results are given to ascertain the existence of pseudo almost periodic solution to the system (1) by applying Schauder fixed point theorem. In Section 4, the global uniform asymptotic stability of pseudo almost periodic solutions to the system (1) is obtained. In Section 5, an example is provided to demonstrate the effectiveness of the main results. In Section 6, the final conclusions are drawn.


## 2. Preliminaries

In this section, some basic definitions, lemmas, and assumptions are introduced.Definition 1 (see [33]).
f
(
t
)
∈
B
C
(
R
,
R
) is said to be Bohr almost periodic if, for all ϵ>0, set
(2)T(f,ϵ)={|f(t+τ)-f(t)|<ϵ,∀t∈R}
is relatively dense. Namely, for any ϵ>0 there exists a number l=l(ϵ)>0 such that every interval [a,a+l] contains at least one point of τ=τ(ϵ) such that |f(t+τ)-f(t)|<ϵ for every t∈R. The collection of those functions is denoted by AP(R,Rm). Define the class of functions PAP0(R,Rm) as follows:
(3)PAP0(R,Rm)={lim⁡T→+∞12T∫-TT‍∥f(t)∥f∈BC(R,Rm)∣lim⁡T→+∞12T∫-TT‍∥f(t)∥dt=0}.Definition 2 (see [34]).
A functionf∈BC(R,Rm) is called pseudo almost periodic if it can be expressed as
(4)f=f1+f0,
where f1∈AP(R,Rm) and f0∈PAP0(R,Rm). The collection of such functions will be denoted by PAP(R,Rm).Remark 3.
From the definitions above, we haveAP(R,Rm)⊂PAP(R,Rm).Lemma 4 (see [3]).
PAP(R,Rm) is a Banach space with the norm |ϕ|=sup⁡t∈R|ϕ(t)|.Lemma 5 (see [19]).
Iff(t,u)∈C(R×D,Rm), where D is an open set in Rm or D=Rm, C(R×D,Rm) denote continuous function class. Suppose f∈PAP(R×D) satisfies the Lipschitz condition
(5)|f(t,u)-f(t,v)|≤L|u-v|,∀t∈R,u,v∈D;
if ϕ(t)∈PAP(R), then the composite function f(t,ϕ(t))∈PAP(R). Suppose f:R×C→Rm; then the equation
(6)x′(t)=f(t,xt)
is called lagging-type almost periodic differential equation. The following system (7) is defined as the product systems of (6):
(7)x′(t)=f(t,xt(t)),y′(t)=f(t,yt(t)).Lemma 6.
Supposeϕ(t)∈PAP(R,Rm); then ϕ(t-τ)∈PAP(R,Rm) for all τ∈R.Proof.
From Definition2 of the PAP, we have ϕ=ϕ1+ϕ0, where ϕ1∈AP(R,Rm) and ϕ0∈PAP0(R,Rm). Clearly ϕ(t-τ)=ϕ1(t-τ)+ϕ0(t-τ); it is easy to know ϕ1(t-τ)∈AP(R,Rm) and
(8)0≤12T∫-TT‍|ϕ0(t-τ)|dt=12T∫-(T+τ)T-τ‍|ϕ0(t)|dt≤T+τT·12(T+τ)∫-(T+τ)T+τ‍|ϕ0(t)|dt.
This indicates that ϕ0(t-τ)∈PAP0(R,Rm). So ϕ(t-τ)∈PAP(R,Rm).Definition 7.
Assume thatx*(t) is a pseudo almost periodic solution of system (1). By a translation transformation y(t)=x(t)-x*(t), system (1) is transformed into a new system. If the zero solution of new system is globally uniformly asymptotically stable, then the pseudo almost periodic solution of system (1) is said to be globally uniformly asymptotically stable. As for the uniform asymptotical stability, see [35].Lemma 8 (see [33]).
There is a continuous functionalV(t,φ,ψ) for t≥0, φ,ψ∈CH, CH={φ:φ∈C,|φ|<H}, |φ|=sup⁡θ∈[-r,0]|φ(θ)| such that(H′2.1)u(|φ-ψ|)≤V(t,φ,ψ)≤v(|φ-ψ|);(H′2.2)|V(t,φ1,ψ1)-V(t,φ2,ψ2)|≤k(|φ1-φ2|+|ψ1-ψ2|);(H′2.3)V(7)′(t,φ,ψ)≤-aV(t,φ,ψ),where a is a positive constant and u(s) and v(s) are continuous nondecreasing functions; when s→0, u(s)→0,k is a positive constant. At this time, if (7) has a bounded solution x(t,σ,φ) such that |x(t,σ,φ)|≤H1, where t≥σ≥0, H>H1>0, then (6) in CH has a unique almost periodic solution which is uniformly asymptotically stable.Throughout this paper, we make the following assumptions.(H2.1): Functions ai(u) are continuous bounded and there are positive constants ai+,ai- such that
(9)0<ai-≤ai(u)≤ai+,∀u∈R,i=1,2,…,m.(H2.2): Functions bi(u)∈C(R,R) and there exist positive constants bi-,bi+ such that
(10)bi-≤bi(u)-bi(v)u-v≤bi+,u≠v,∀u,v∈R,bi(0)=0.(H2.3): cij(t), dij(t), pij(t), Ii(t)∈C(R,R), τij(t)∈C(R,R+) are pseudo almost periodic functions:
(11)sup⁡t∈Rcij(t)=cij+>0,sup⁡t∈Rdij(t)=dij+>0,sup⁡t∈Rpij(t)=pij+>0,sup⁡t∈RIi(t)=Ii+>0,whereR+=[0,∞), i,j=1,2,…,m.(H2.4): Delay kernel functions Gij:[0,+∞)→[0,+∞) are piecewise continuous and integrable
(12)∫0+∞‍Gij(u)du=1,∫0∞‍uGij(u)du<+∞,i,j=1,2,…,m.(H2.5): Functions fj(u),gj(u),hj(u)∈C(R,R) satisfy the Lipschitz condition; namely, there exist nonnegative constants Ljf, Ljg, and Ljh such that
(13)|fj(u)-fj(v)|≤Ljf|u-v|,∀u,v∈R,j=1,2,…,m,|gj(u)-gj(v)|≤Ljg|u-v|,∀u,v∈R,j=1,2,…,m,|hj(u)-hj(v)|≤Ljh|u-v|,∀u,v∈R,j=1,2,…,m.


## 3. The Existence of Pseudo Almost Periodic Solution

In this section, we study the existence of pseudo almost periodic solution to system (1).It follows from(H2.1) that the antiderivative of 1/ai(xi) exists. Then we choose an antiderivative Fi(xi) of 1/ai(xi) that satisfies Fi(0)=0. Clearly, Fi′(xi)=1/ai(xi). Because ai(xi)>0, Fi(xi) is increasing about xi and the inverse function Fi-1(xi) of Fi(xi) is existential, continuous, and differential. Then (Fi-1(xi))′=ai(xi). Denote Fi′(xi)xi′(t)=xi′(t)/ai(xi(t))≐ui′(t); we get xi(t)=Fi-1(ui(t)). Substituting these equations into system (1), we get the following equivalent equation:
(14)ui′(t)=-bi(Fi-1(ui(t)))+∑j=1m‍cij(t)fj(Fj-1(uj(t)))+∑j=1m‍dij(t)gj(Fj-1(uj(t-τij(t))))+∑j=1m‍pij(t)∫-∞t‍Gij(t-s)hj(Fj-1(uj(s)))ds+Ii(t),t≥0,ui(t)=Fi-1(Φi(t))=φi(t),t<0.
From (14), we get bi(Fi-1(ui(t)))=[bi(Fi-1(θiui(t)))]′ui(t)≐bi~(ui(t))ui(t), where 0≤θi≤1. Putting it into (14), we obtain
(15)ui′(t)=-bi~(ui(t))ui(t)+∑j=1m‍cij(t)fj(Fj-1(uj(t)))+∑j=1m‍dij(t)gj(Fj-1(uj(t-τij(t))))+∑j=1m‍pij(t)∫-∞t‍Gij(t-s)hj(Fj-1(uj(s)))ds+Ii(t),t≥0,ui(t)=Fi-1(Φi(t))=φi(t),t<0.
Thus, system (1) has at least one pseudo almost periodic solution if and only if the system (15) has at least one pseudo almost periodic solution. So we only consider the pseudo almost periodic solution of system (15). By Lagrange theorem, we have
(16)|Fi-1(u)-Fi-1(v)|=|[Fi-1(v+θi(u-v))]′(u-v)|=|ai(v+θi(u-v))||u-v|.
Again by (H2.1), we get
(17)ai-|u-v|≤|Fi-1(u)-Fi-1(v)|≤ai+|u-v|.
Combined with (H2.2), we have(H3.6): bi-ai-≤bi′(Fi-1(·))≤bi+ai+.In order to prove the main results, we give the following lemma.Lemma 9.
Suppose that assumptions(H2.2)–(H2.5) hold and if ϕ(t)∈PAP(R,Rm), then
(18)Cij=∫-∞t‍Gij(t-s)ϕj(s)ds∈PAP(R,R),i=1,2,…,m.Proof.
From Definition2, we have ϕj=ϕj1+ϕj0; then
(19)Cij=∫-∞t‍Gij(t-s)ϕj1(s)ds+∫-∞t‍Gij(t-s)ϕj0(s)ds=Cij1+Cij0.
Firstly, we proveCij1∈AP(R,R). For any ϵ>0, there exists a number l=l(ϵ)>0 such that every interval [a,a+l] contains at least one point of τ=τ(ϵ) such that |ϕj1(t+τ)-ϕj1(t)|≤ϵ for every t∈R. Therefore, from (H2.2)–(H2.4), we obtain
(20)|Cij1(t+τ)-Cij1(t)|=|∫-ϕt+τ‍Gij(t+τ-s)ϕj1(s)ds-∫-∞t‍Gij(t-s)ϕj1(s)ds|≤∫-∞t‍|Gij(t-s)||φj1(s+τ)-φj1(s)|ds≤ϵ∫-∞t‍|Gij(t-s)|ds≤ϵ
so that Cij1∈AP(R,R).
And then we show thatCij0∈PAP0(R,R) because
(21)lim⁡T→+∞12T∫-TT‍|Cij0|dt=sup⁡t∈Rlim⁡T→+∞12T∫-TT‍|∫-∞t‍Gij(t-s)ϕj0(s)ds|dt≤sup⁡t∈Rlim⁡T→+∞12T∫0+∞‍|Gij(u)|∫-(T+u)T+u‍|ϕj0(v)dv|du=0.
Thus Cij0∈PAP0(R,R). So Cij∈PAP(R,R).Theorem 10.
Suppose that(H2.1)–(H2.5) and (H3.6) hold; if
(22)δ=max⁡1≤i≤m{ai+bi-ai-∑j=1m‍(Ljfcij++Ljgdij++Ljhpij+)}<1,
then the system (1) has at least one pseudo almost periodic solution.Proof.
For allz(t)=ϕ(t)T=(ϕ1(t),…,ϕm(t))T∈PAP(R,Rm), we define the nonlinear operator T:z(t)→T(z)(t)=z(ϕ)T(t)=(xϕ(t))T, where
(23)xϕi(t)=∫-∞t‍e-∫st‍bi~(ϕi(τ))dτ×[∑j=1m‍cij(s)fj(Fj-1(ϕj(s)))+∑j=1m‍dij(s)gj(Fj-1(ϕj(s-τij(s))))+∑j=1m‍pij(s)∫-∞t‍Gij(s-v)hj(Fj-1(ϕj(v)))dv+Ii(s)∑j=1m‍cij(s)fj(Fj-1(ϕj(s)))]ds.
Now, we prove that
(24)T:PAP(R,Rm)⟶PAP(R,Rm).
Let
(25)Eij=∑j=1m‍cij(s)fj(Fj-1(ϕj(s)))+∑j=1m‍dij(s)gj(Fj-1(ϕj(s-τij(s))))+∑j=1m‍pij(s)∫-∞t‍Gij(s-v)hj(Fj-1(ϕj(v)))dv+Ii(s).
Forz(t)∈PAP(R,Rm), conditions (H2.2)–(H2.4), Lemmas 5, 6, and 9, and the composition theorem in [16], we will get Eij∈PAP(R,R), ∀i,j=1,2,…,m.
From Definition2, we have Eij=Eij1+Eij0, ∀i,j=1,2,…,m. Where Eij1∈AP(R,R) and Eij0∈PAP0(R,R). Then
(26)xϕi(t)=∫-∞t‍e-∫st‍bi~(ϕi(τ))dτEij1(s)ds+∫-∞t‍e-∫st‍bi~(ϕi(τ))dτEij0(s)ds=Tij1+Tij0,
where Tij1=∫-∞t‍e-∫st‍bi~(ϕi(τ))dτEij1(s)ds and Tij0=∫-∞t‍e-∫st‍bi~(ϕi(τ))dτEij0(s)ds.
BecauseEij1∈AP(R,R), for any ϵ>0, there exists a number l=l(ϵ)>0 such that every interval [a,a+l] contains at least one point of δ=δ(ϵ) such that sup⁡t∈R|Eij1(t+δ)-Eij1(t)|≤ϵ for every t∈R and ∀i,j=1,2,…,m. Hence, we obtain
(27)|Tij1(t+δ)-Tij1(t)|=|∫-∞t+δ‍e-∫st+δ‍bi~(ϕi(τ))dτEij1(s)ds-∫-∞t‍e-∫st‍bi~(ϕi(τ))dτEij1(s)ds|≤∫-∞t+τ‍|e-bi-ai-(t-s)∣Eij1(s+δ)-Eij1(s)|ds≤ϵbi-ai-
so that Tij1∈AP(R,R).
And because(28)lim⁡T→+∞12T∫-TT‍|∫-∞t‍e-∫st‍bi~(ϕi(τ))dτEij0(s)ds|dt≤lim⁡T→+∞12T∫-TT‍|∫-Tt‍e-∫st‍bi~(ϕi(τ))dτEij0(s)ds|dt+lim⁡T→+∞12T∫-TT‍|∫-∞-T‍e-∫st‍bi~(ϕi(τ))dτEij0(s)ds|dt≤lim⁡T→+∞12T∫-TT‍∥Eij0(t)∥dt∫-Tt‍e-bi-ai-(t-s)ds+lim⁡T→+∞sup⁡t∈R|Eij0(t)|2T∫-TT‍dt(∫-∞-T‍|e-bi-ai-(t-s)|ds)≤lim⁡T→+∞12Tbi-ai-∫-TT‍∥Eij0(t)∥dt+lim⁡T→+∞sup⁡t∈R|Eij0(t)|2T(bi-ai-)2(1-e-bi-ai-(2T))=0+lim⁡T→+∞sup⁡t∈R|Eij0(t)|2T(bi-ai-)2(1-e-bi-ai-(2T))=0,
thus Tij0∈PAP0(R,R). So ∀i,j=1,2,…,m,xϕi(t)∈PAP(R,R). Therefore z(ϕ)T(t)∈PAP(R,Rm).
From Lemma9, X=PAP(R,Rm) is a Banach space. If
(29)δ<1,
then there exists a sufficiently large β≥1 such that
(30)δ≤1-β-1I,
where
(31)I=max⁡1≤i≤m{Ii+bi-ai-}.
We choose a closed subset
(32)B={z(t)=ϕ(t)T=(ϕ1(t),…,ϕm(t))T∈X:∥z∥≤β}.
Firstly, we prove thatT:B→B; that is, TB⊂B.
From (29)–(32) and for ∀z∈B, we get
(33)|xϕi(t)|≤max⁡1≤i≤m{ai+bi-ai-∑j=1n‍(Ljfcij++Ljgdij++Ljhpij+)}×∥z∥+max⁡1≤i≤m{Ii+bi-ai-}≤βδ+I≤β.
Secondly, we prove that the mappingT is completely continuous.
By the continuity of the functionfj, gj, hj, for any ɛ>0, there is γ=γ(ɛ,β) such that
(34)|fj(u)-fj(v)|≤Ljfɛδ,|u-v|≤γ,∀u,v∈B,j=1,2,…,m,|gj(u)-gj(v)|≤Ljgɛδ,|u-v|≤γ,∀u,v∈B,j=1,2,…,m,|hj(u)-hj(v)|≤Ljhɛδ,|u-v|≤γ,∀u,v∈B,j=1,2,…,m.
Let w(t)=(ψ1(t),…,ψm(t)), z,w∈B, and ∥z-w∥≤γ; then ∥z∥≤γ,∥w∥≤γ and ϕj(t),ψj(t)∈C(R,B); then, for any s∈R, we get |ϕj(s)-ψ1(s)|≤γ. So, we have
(35)|fj(Fj-1(ϕj(s)))-fj(Fj-1(ψj(s)))|≤Ljfɛδ,|u-v|≤γ,|gj(Fj-1(ϕj(s)))-fj(Fj-1(ψj(s)))|≤Ljgɛδ,|u-v|≤γ,|hj(Fj-1(ϕj(s)))-fj(Fj-1(ψj(s)))|≤Ljhɛδ.|u-v|≤γ.
Thus
(36)∥T(z)(t)-T(w)(t)∥≤max⁡1≤i≤m{1bi-ai-∑j=1m‍ai+(Ljfcij++Ljgdij++Ljhpij+)}ɛδ≤δɛδ≤ɛ.
Therefore, T is continuous.
Thirdly, we show thatT is compact.
LetS={z(t)∈X:∥z∥≤K}, where K>0 to be any constant. We denote ρ=max⁡1≤i≤m{(ai+/bi-ai-)∑j=1m‍K(Ljfcij++Ljgdij++Ljhpij+)+Ii+}. Then we have
(37)∥T(z)(t)∥=sup⁡t∈Rmax⁡1≤i≤m|xϕi(t)|≤ρ,∀z∈S.
Hence, T is uniformly bounded. Then, from (23), we get
(38)[|xϕi(t)|]′=-bi~(xϕi(t))xϕi(t)+∑j=1m‍cij(t)fj(Fj-1(ϕj(t)))+∑j=1m‍dij(t)gj(Fj-1(ϕj(t-τij(t))))+∑j=1m‍pij(t)∫-∞t‍Gij(t-s)hj(Fj-1(ϕj(s)))ds+Ii(t)≤bi+ρ+∑j=1m‍aj+K(Ljfcij++Ljgdij++Ljhpij+)+Ii+≤L,
where
(39)L=max⁡1≤i≤m{bi+ρ+∑j=1m‍aj+K(Ljfcij++Ljgdij++Ljhpij+)+Ii+}.
Therefore, T is equicontinuous. By the Ascoli-Arzela theorem, the operator T is compact; then it is completely continuous. By the Schauder fixed point theorem, the system (1) has at least one pseudo almost periodic solution.


## 4. The Global Uniform Asymptotic Stability of Pseudo Almost Periodic Solution

In order to discuss the global uniform asymptotic stability of pseudo almost periodic solution to system (1), we give the following assumptions:(H4.1): delay functions τij(t)∈C1(R,R+) satisfy that τ˙ij(t)≤τij*<1, i,j=1,2,…,m;(H4.2): N=min⁡1≤i≤m{Ni}>0, where Ni=bi--∑j=1m‍cij+Ljf-∑j=1m‍(dij+Ljg/(1-τij*))-∑j=1m‍pij+Ljh.Theorem 11.
Assume that (H2.1)–(H2.5) and (H4.1)-(H4.2) hold; then the pseudo almost periodic solution of system (1) is globally uniformly asymptotically stable.Proof.
The product system of the system (1) is
(40)xi′(t)=-ai(xi(t))×[bi(xi(t))-∑j=1m‍cij(t)fj(xj(t))-∑j=1m‍dij(t)gj(xj(t-τij(t)))-∑j=1m‍pij(t)∫-∞t‍Gij(t-s)hj(xj(s))ds-Ii(t)],yi′(t)=-ai(yi(t))×[bi(yi(t))-∑j=1m‍cij(t)fj(yj(t))-∑j=1m‍dij(t)gj(yj(t-τij(t)))-∑j=1m‍pij(t)∫-∞t‍Gij(t-s)hj(yj(s))ds-Ii(t)].
In order to apply the conclusion of Lemma 8, we construct a Lyapunov functional about product system (40)
(41)V(t)=V1(t)+V2(t)+V3(t),
where
(42)V1(t)=∑i=1m‍|∫yi(t)xi(t)‍1ai(s)ds|;V2(t)=∑i=1m‍∑j=1m‍dij+Ljg1-τij*∫t-τij(t)t‍|xi(s)-yi(s)|ds;V3(t)=∑i=1m‍∑j=1m‍pij+Ljh∫0+∞‍Gij(s)∫t-st‍|xi(u)-yi(u)|duds.
LetX(t)=x(t)T=(x1(t),…,xm(t))T and Y(t)=y(t)T=(y1(t),…,ym(t))T. For product system (X,Y), we receive
(43)|X-Y|≤V(t,X(t),Y(t))≤∑i=1m‍{1ai-+∑j=1m‍dij+Ljgτij+1-τij*+∑j=1n‍pij+Lih∫0+∞‍Gij(s)sds}×|xi(t)-yi(t)|≤Mi∑i=1m‍|xi(t)-yi(t)|≤M|X-Y|,
where Mi=(1/ai-)+∑j=1m‍(dij+Ljgτij+/(1-τij*))+∑j=1n‍pij+Lih∫0+∞‍Gij(s)sds and M=max⁡1≤i≤m{Mi}. Let u(s)=s and v(s)=Ms; we easily know it satisfies condition (H′2.1) of Lemma 8. Then we obtain
(44)|V(t,X,Y)-V(t,X*,Y*)|=|∑i=1m‍∑j=1m‍pij+Ljh∫0+∞‍Gij(s)∫t-st‍|xi*(u)-yi*(u)|d∑i=1m‍|∫yi(t)xi(t)‍1ai(s)ds|-∑i=1m‍|∫yi*(t)xi*(t)‍1ai(s)ds|+∑i=1m‍∑j=1m‍dij+Ljg1-τij*∫t-τij(t)t‍|xi(s)-yi(s)|ds-∑i=1m‍∑j=1m‍dij+Ljg1-τij*∫t-τij(t)t‍|xi*(s)-yi*(s)|ds-∑i=1m‍∑j=1m‍pij+Ljh∫0+∞‍Gij(s)∫t-st‍|xi(u)-yi(u)|duds-∑i=1m‍∑j=1m‍pij+Ljh∫0+∞‍Gij(s)∫t-st‍|xi*(u)-yi*(u)|duds|≤∑i=1m‍1ai-|xi(t)-xi*(t)|+∑i=1m‍1ai-|yi(t)-yi*(t)|+∑i=1m‍∑j=1m‍dij+Ljg1-τij*×∫t-τij(t)t‍(|xi(s)-xi*(s)|+|yi(s)-yi*(s)|)ds+∑i=1m‍∑j=1m‍pij+Ljh∫0+∞‍Gij(s)×∫t-st‍ds(|xi(t)-xi*(t)|+|yi(t)-yi*(t)|)≤M(|X-X*|+|Y-Y*|).
We also know that it satisfies condition (H′2.2) of Lemma 8.
Calculating the upright derivative ofV1(t), V2(t), and V3(t) along the system (40), respectively, and noting that ((1-τ˙ij(t))/(1-τij*))>1, we have
(45)D+V1(t)|(40)=∑i=1m‍Sgn(xi(t)-yi(t))[xi′(t)ai(xi(t))-yi′(t)ai(yi(t))]≤∑i=1m‍{∑j=1n‍pij+Ljh∫0+∞‍-bi(xi(t))-bi(yi(t))xi(t)-yi(t)|xi(t)-yi(t)|+∑j=1m‍cij+Ljf|xj(t)-yj(t)|+∑j=1m‍dij+Ljg|xj(t-τij(t))-yj(t-τij(t))|+∑j=1n‍pij+Ljh∫0+∞‍Gij(s)|xj(t-s)-yj(t-s)|ds}≤∑i=1m‍{-bi-|xi(t)-yi(t)|+∑j=1m‍cij+Ljf|xj(t)-yj(t)|+∑j=1m‍dij+Ljg|xj(t-τij(t))-yj(t-τij(t))|+∑j=1n‍pij+Ljh∫0+∞‍Gij(s)|xj(t-s)-yj(t-s)|ds};D+V2(t)|(40)=∑i=1m‍∑j=1m‍dij+Ljg1-τij*|xi(t)-yi(t)|-∑i=1m‍∑j=1m‍dij+Ljg1-τij*|xi(t-τij(t))-yi(t-τij(t))|;D+V3(t)|(40)=∑i=1m‍∑j=1m‍pij+Ljh∫0+∞‍Gij(s)|xi(t)-yi(t)|ds-∑i=1m‍∑j=1m‍pij+Ljf∫0+∞‍Gij(s)|xi(t-s)-yi(t-s)|ds.
Combining (45) and assumptions (H4.1) and (H4.2), we get
(46)D+V(t)|(40)≤∑i=1m‍{-bi-+∑j=1m‍cij+Ljf+∑j=1m‍dij+Ljg1-τij*+∑j=1m‍pij+Ljh}×|xi(t)-yi(t)|≤-N∑j=1m‍|xi(t)-yi(t)|≤-aV(t).
From assumption (H4.2), we have a=N/M>0.
By Lemma8, the pseudo almost periodic solutions of system (1) are globally uniformly asymptotically stable. This completes the proof.Corollary 12.
Consider the following periodic CGNNs systems:(47)xi′(t)=-ai(xi(t))×[∑j=1m‍pij(t)∫-∞t‍bi(xi(t))-∑j=1m‍cij(t)fj(xj(t))-∑j=1m‍dij(t)fj(xj(t-τij(t)))-∑j=1m‍pij(t)∫-∞t‍Gij(t-s)fj(xj(s))ds-Ii(t)],t≥0,xi(t)=Φi(t),t<0,
where i=1,2,…,m, and the following assumptions hold.(G4.1): Functions ai(u) are continuous bounded and there are positive constants ai+,ai- such that
(48)0<ai-≤ai(u)≤ai+,∀u∈R,i=1,2,…,m.(G4.2): Functions bi(u)∈C(R,R) and there exist positive constants bi-,bi+ such that
(49)bi-≤bi(u)-bi(v)u-v≤bi+,u≠v,∀u,v∈R,bi(0)=0.(G4.3): cij(t), dij(t) and pij(t), Ii(t)∈C(R,R), τij(t)∈C(R,R+) are all periodic functions, and
(50)sup⁡t∈Rcij(t)=cij+>0,sup⁡t∈Rdij(t)=dij+>0,sup⁡t∈Rpij(t)=pij+>0,sup⁡t∈RIi(t)=Ii+>0,whereR+=[0,∞), i,j=1,2,…,m.(G4.4): Delay kernel functions Gij:[0,+∞)→[0,+∞) are piecewise continuous and integrable
(51)∫0+∞‍Gij(u)du=1,∫0∞‍uGij(u)du<+∞,i,j=1,2,…,m.(G4.5): Functions fj(u)∈C(R,R) satisfy the Lipschitz condition; namely, there exist nonnegative constants Lj such that
(52)|fj(u)-fj(v)|≤Lj|u-v|,∀u,v∈R,j=1,2,…,m.(G4.6): If δ=max⁡1≤i≤m⁡{(ai+/bi-ai-)∑j=1mLj(cij++dij++pij+)}<1,then the system (47) has at least one periodic solution.Corollary 13.
Assume that (G4.1)–(G4.6) hold and suppose further that (G4.7): delay functions τij(t)∈C1(R,R+) satisfy that τ˙ij(t)≤τij*<1, i,j=1,2,…,m.(G4.8): N=min⁡1≤i≤m{Ni}>0, where Ni=bi--∑j=1m‍cij+Lj-∑j=1m‍(dij+Lj/(1-τij*))-∑j=1m‍pij+Lj; then the periodic solution of system (47) is globally uniformly asymptotically stable.Remark 14.
Recently, the global exponential stability of periodic or almost periodic solution to CGNNs is studied by many scholars (see [15–30]). However, few authors pay attention to the global uniform asymptotic stability. Corollaries 12 and 13 provide some new results.


## 5. An Example

An example is given to illustrate the feasibility of main results in this paper. Consider the following simple neural networks:(53)xi′(t)=-ai(xi(t))×[bi(xi(t))-∑j=12‍cij(t)fj(xj(t))-∑j=12‍dij(t)gj(xj(t-τij(t)))-∑j=12‍pij(t)∫-∞t‍Gij(t-s)hj(xj(s))ds-Ii(t)],t≥0,i=1,2,
where the initial functions x1(t)=4+cos⁡(πt)1, t<0, x2(t)=5+sin(2t), t<0. ai(xi(t))=4+cos⁡πt-e-|xi(t)|, bi(xi(t))=5+sin2t-e-|xi(t)|. Let
(54)(c11(t)c12(t)c21(t)c22(t))=114(cos⁡t+e-t4cos⁡4tcos⁡(2t)+e-t4sin4tcos⁡(5t)+e-t4cos⁡4tsin(2t)+e-t2cos⁡4t),(d11(t)d12(t)d21(t)d22(t))=114(sint+e-t2cos⁡2tcos⁡(4t)+e-t2cos⁡4tsin(3t)+e-t4cos⁡4tsin(2t)+e-t4cos⁡4t),(p11(t)p12(t)p21(t)p22(t))=114(cos⁡(5t)+e-t4cos⁡2tsin(3t)+e-t4cos⁡4tcos⁡(3t)+e-t2cos⁡2tsin(2t)+e-t2cos⁡4t).I
1
(
t
)
=
I
2
(
t
)
=
2
(
sin
(
3
t
)
+
e
-
t
2
cos
⁡
4
t
), fj(xj)=gj(xj)=hj(xj)=((|x+1|-|x-1|)/2), Gij(u)=e-u, τij*=4/5. Then, we have ai+=5, ai-=2, bi-=3, cij+=dij+=pij+=1/7, Ljf=Ljg=Ljh=1, where i,j=1,2. Moreover
(55)δ=max⁡1≤i≤2{ai+bi-ai-∑j=12‍(Ljfcij++Ljgdij++Ljhpij+)}=57<1,Ni=bi--∑j=12‍cij+Ljf-∑j=12‍dij+Ljg1-τij*-∑j=12‍pij+Ljh=1>0,i=1,2.
Thus, by Theorem 10, we know that system (53) has at least one pseudo almost periodic solution. It follows from Theorem 11 that the unique pseudo almost periodic solution of system (53) is globally uniformly asymptotically stable (Figure 1).Figure 1
Simulated results of the solutions.


## 6. Conclusions

In this paper, the existence and uniform asymptotic stability of pseudo almost periodic solutions of system (1) is discussed. By applying Schauder fixed point theorem and constructing a suitable Lyapunov functional, some sufficient conditions are obtained to ensure the existence and uniform asymptotic stability of pseudo almost periodic solutions of system (1). The results have important leading significance in the design and applications of CGNNs. In addition, an example is given to demonstrate the effectiveness of main results.



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*Source: 968404-2014-08-18.xml*