Chaos, Solitons and Fractals 42 (2009) 773–780

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Chaos, Solitons and Fractals

journal homepage: www.elsevier .com/locate /chaos

Existence and exponential stability of almost periodic solutionfor stochastic cellular neural networks with delay

Zaitang Huang a,b, Qi-Gui Yang a,*

a School of Mathematical Sciences, South China University of Technology, Guangzhou 510640, PR Chinab School of Mathematics and Physics, Wuzhou University, Wuzhou 543002, PR China

a r t i c l e i n f o

Article history:Accepted 9 February 2009

Communicated by Prof. Ji-Huan He

0960-0779/$ - see front matter � 2009 Elsevier Ltddoi:10.1016/j.chaos.2009.02.008

* Corresponding author.E-mail addresses: huangzaitang@126.com (Z. Hu

a b s t r a c t

The paper considers the problems of existence of quadratic mean almost periodic and glo-bal exponential stability for stochastic cellular neural networks with delays. By employingthe Holder’s inequality and fixed points principle, we present some new criteria ensuringexistence and uniqueness of a quadratic mean almost periodic and global exponential sta-bility. These criteria are important in signal processing and the design of networks. More-over, these criteria are also applied in others stochastic biological neural systems.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

In recent years, Hopfield neural networks and their various generalizations have attracted the attention of many scientists(e.g., mathematicians, physicists, computer scientists and so on), due to their potential for the tasks of classification, asso-ciative memory, parallel computational and their ability to solve difficult optimization problems [1–5]. Although discretetime delays in the delayed feedback neural networks having a small number of cells serve usually as good approximationof the prime models, a real system is usually affected by external perturbations. Therefore, it is significant and of primeimportance to consider stochastic effects to the stability property of the neural networks with delays. Considering this, manyresearchers have studied the stability analysis problem for delayed neural networks with environmental noise [6–10]. On theother hand, periodic oscillation in neural networks is an interesting phenomenon, like many biological and cognitive activ-ities. It is worth noting that coexistence of periodic orbits is necessary in practical applications such as associative memorystorage, pattern recognition, decision making, digital selection and analogy amplification [11–14]. We use the notion of‘‘quadratic mean almost periodic” to describe coexistence of periodic orbits. To the best of our knowledge, the quadraticmean almost periodic is seldom considered for stochastic cellular neural networks, the concept of almost periodicity isimportant in probability for investigating stochastic processes [15–20]. Such a notion is also of interest for applications aris-ing in mathematical physics and statistics. The concept of almost periodicity for stochastic processes, which is the centralquestion in this paper was first introduced in the literature at the end of thirties by Slutsky [17], who then obtained reason-able sufficient conditions for sample paths of a stationary process to be almost periodic in the sense of Besicovitch. In thispaper, we shall consider existence of quadratic mean almost periodic and global exponential stability for stochastic cellularneural networks with delays. By employing the Holder’s inequality and fixed points principle, we present some new criteriaensuring existence and uniqueness of a quadratic mean almost periodic and global exponential stability. These criteria arealso applied in others stochastic biological neural systems.

The rest of this paper is organized as follows: in Section 2, we introduce a class of stochastic neural networks with delays,and the relating notations, definitions and lemmas which would be used later; in Section 3, one present some new criteriaensuring existence and uniqueness of a quadratic mean almost periodic and global exponential stability.

. All rights reserved.

ang), yangqg168@126.com (Q.-G. Yang).

774 Z. Huang, Q.-G. Yang / Chaos, Solitons and Fractals 42 (2009) 773–780

2. Preliminaries

In this paper, we consider the following the stochastic cellular neural networks with delays describe described by:

dxðtÞ ¼ �AxðtÞ þ Bf ðt; xðtÞÞ þ Cf ðt; xðt � rÞÞdt þ Gðt; xðtÞÞdWðtÞ; t 2 R; ð2:1Þ

where A ¼ diagðaiÞn�n; ai are positive constants, they denote the rate with which the cell ith will reset their potential to theresting state in isolation when isolated form the other cells and inputs; B ¼ ðbijÞn�n and C ¼ ðcijÞn�n, bji and cji are the connec-tion weights of the neural network; r is the transmission delay of the neural and is constant. The activation function f showshow neurons respond to each other. WðuÞ ¼ ðw1ðu1Þ;w2ðu2Þ; � � � ;wnðvnÞÞ> is an Brownian motion defined on a completeprobability space ðX;I;mþ nÞ with a natural filtration fItgtP0;Gðt; xÞ is locally Lipschitz continuous and satisfies the lineargrowth conditions.

Let ðB; k � kÞ be a Banach space and let ðX; F;PÞ be a probability space. Define LpðP;BÞ for p P 1 to be the space of allB-value random variable Y such that

EkYkp ¼Z

XkYkpdP <1: ð2:2Þ

It is then routine to check that LpðP;BÞ is a Banach space when it is equipped with its natural norm k � kp define by:

kYkp ¼Z

XkYkpdP

� �1p

;

for each Y 2 LpðP;BÞ.This setting requires the following preliminary definitions.

Definition 2.1. A stochastic process X : R! LpðP;BÞ is said to be continuous whenever

limt!s

EkXðtÞ � XðsÞkp � 0:

Definition 2.2. A stochastic process X : R! LpðP;BÞ is said to be stochastically bounded whenever

limN!1

supt2R

PfkXðtÞ � XðsÞk > Ng ¼ 0:

Definition 2.3. A continuous stochastic process X : R! LpðP;BÞ is said to be p-mean almost periodic if for each e > 0 thereexists lðeÞ > 0 such that any interval of length lðeÞ contains at least a number s for which

supt2R

EkXðt þ sÞ � XðtÞkp< 0:

The number s will be called an e-translation of X and the set of all e-translation of X is denoted by @ðe;XÞ.

Remark 2.4. The following properties of e-translations can be easily obtained:

(i) If e0 > e, then @ðe;XÞ � @ðe0;XÞ;(ii) if s 2 @ðe;XÞ, then so is �s;

(iii) if s1 2 @ðe1;XÞ and if s2 2 @ðe2;XÞ, then both s1 þ s2 and s1 � s2 belong to @ðe1 þ e2;XÞ.

The collection of all stochastic processes X : R! LpðP;BÞ which are p-mean almost periodic is then denote byAPðR; LpðP;BÞÞ.

The net lemma provides with some properties of p-mean almost periodic processes.

Lemma 2.5 [18]. If X belongs to APðR; LpðP;BÞÞ, then

(i) the mapping t ! EkXðtÞkp is uniformly continuous;(ii) there exist a constant M > 0 such that EkXðtÞkp

< M, for each t 2 R;(iii) X is stochastically bounded.

Lemma 2.6 [18]. APðLpðP;BÞÞ � CUPðLpðP;BÞÞ is a closed subspace.

Z. Huang, Q.-G. Yang / Chaos, Solitons and Fractals 42 (2009) 773–780 775

Definition 2.7 [19]. A function f : R� LpðP;B1Þ ! LpðP;B2Þ; ðt; yÞ ! f ðt; yÞ, which is jointly continuous, is said to be p-meanalmost periodic in t 2 R uniformly in y 2 K where K � LpðP;B1ÞÞ is compact if for any e > 0, there exists lðe;KÞ > 0 such thatany interval of length lðe;KÞ constants at least a number s for which

supt2RðEkf ðt þ s; yÞ � f ðt; yÞkpÞ

1p < e

for each stochastic process y : R! K.Here again, the number s will be called an e-translation of f and the set of all e-translation of f is denoted by @ðe; F;KÞ.

Theorem 2.8 [19]. Let f : R� LpðP;B1ÞÞ ! LpðP;B2ÞÞ; ðt; xÞ ! f ðt; xÞ, be a p-mean almost periodic process in t 2 R uniformly inx 2 K, where K � LpðP;B1ÞÞ is compact. Suppose that f is Lipschitzian in the following sense:

Ekf ðt; xÞ � f ðt; yÞkp2 6 MEkx� ykp

1

for all x; y 2 LpðP;B1ÞÞ and for each t 2 R, where M > 0. Then for any p-mean almost periodic / : R! LpðP;B1ÞÞ, then stochasticprocesses t ! f ðt;/ðtÞÞ is p-mean almost periodic.

Throughout the rest of this section, we require the following assumptions:

� ðH0Þ The operator �A : Dð�AÞ � L2ðP; HÞ ! L2ðP; HÞ is the infinitesimal generator of a uniformly exponentially stable semi-group ðTðtÞÞt>0 defined on L2ðP;HÞ such that there exist constant M > 0; d > 0 with

kTðtÞk 6 Me�dt ; t P 0:

� ðH1Þ the activation function f : R� L2ðP; HÞ ! L2ðP; HÞðt; xÞ#f ðt; xÞ be a square-mean almost periodic in t 2 R uniformly inx 2 OðO � L2ðP; HÞ being a compact subspace). Moreover, f is Lipschitz in the following sense: there exists K > 0 for which

Ekf ðt; xÞ � f ðt; yÞk26 KEkx� yk2

for all stochastic processes x; y 2 L2ðP; HÞðt; xÞ and t 2 R.� ðH2Þ The function G : R� L2 ðP; HÞ ! L2ðP; HÞðt; xÞ#Gðt; xÞ be a square-mean almost periodic in t 2 R uniformly in

x 2 O0ðO0 � L2ðP; HÞ being a compact subspace). Moreover, G is Lipschitz in the following sense: there exists K 0 > 0 fort which

EkGðt; xÞ � Gðt; yÞk26 K 0Ekx� yk2

for all stochastic processes x; y 2 L2ðP; HÞ and t 2 R.

3. Almost periodic and exponential stability

In this section, it will be shown that, under certain conditions, system (2.1) has a unique of a square-mean almost periodicsolution which is globally exponentially stable.

Theorem 3.1. Assume that conditions ðH0Þ; ðH1Þ and ðH2Þ are satisfied. If there exist positive constant K; d;K 0;M and the followingconditions hold:

H ¼ 3M2KkBk2

d2 þ 3M2KkCk2

d2 þ 3M2K 0

2d< 1:

Then (2.1) has a unique square-mean almost periodic solution, which is an globally exponentially stable.

Proof. By (2.1), we can obtain

xðtÞ ¼ Tðt � t0Þxðt0Þ þZ t

0Tðt � sÞBf ðs; xðsÞÞ þ Tðt � sÞCf ðs; xðs� rÞÞdsþ

Z t

0Tðt � sÞGðt; xðtÞÞdWðsÞ; ð3:1Þ

for all t P t0 for each t0 2 R, and hence xðtÞ given by (3.1) is solution to (2.1)

Define

/xðtÞ :¼Z t

0Tðt � sÞBf ðs; xðsÞÞ þ Tðt � sÞCf ðs; xðs� rÞÞds

and

wxðtÞ :¼Z t

0Tðt � sÞGðt; xðtÞÞdWðsÞ:

Let us show that /xð�Þ is square-mean almost periodic whenever x dose. Indeed, assuming that x is square-mean almostperiodic and using ðH1Þ and Theorem 2.8, one can easily see that s#f ðs; xðsÞÞ is square-mean almost periodic. Therefore, foreach e > 0 there exists lðeÞ > 0 such that any interval of length lðeÞ contains at least s for which

776 Z. Huang, Q.-G. Yang / Chaos, Solitons and Fractals 42 (2009) 773–780

Ekf ðsþ s; xðsþ sÞÞ � f ðs; xðsÞÞk2<

d2e4kBk2M

;

Ekf ðsþ s� r; xðsþ s� rÞÞ � f ðs� r; xðs� rÞÞk2<

d2e4kCk2M

;

for each s 2 R.Now

k/xðt þ sÞ � /xðtÞk ¼Z t

0Tðt � sÞB½f ðsþ s; xðsþ sÞÞ � f ðs; xðsÞÞ�

���� þ Tðt � sÞC½f ðsþ s; xðsþ s� rÞÞ � f ðs; xðs

� rÞÞ�dsk6Z t

0kTðt � sÞkkBkkf ðsþ s; xðsþ sÞÞ � f ðs; xðsÞÞ

����dsþZ t

0kTðt � sÞkkCkkf ðs

þ s; xðsþ s� rÞÞ � f ðs; xðs� rÞÞkds

6 MkBZ t

0e�dðt�sÞkf ðsþ s; xðsþ sÞÞ � f ðs; xðsÞÞ

���� ����dsþMkCZ t

0e�dðt�sÞ

���� ����f ðsþ s; xðsþ s� rÞÞ

� f ðs; xðs� rÞÞkds ð3:2Þ

and hence, using Cauchy-Schwarz inequality we can write:

Ek/xðt þ sÞ � /xðtÞk26 2MkBk2

E

Z t

0e�dðt�sÞkf ðsþ s; xðsþ sÞÞ � f ðs; xðsÞÞkds

� �2

þ 2MkCk2E

Z t

0e�dðt�sÞkf ðsþ s; xðsþ s� rÞÞ � f ðs; xðs� rÞÞkds

� �2

6 2MkBk2E

Z t

0e�

dðt�sÞ2 e�

dðt�sÞ2 kf ðsþ s; xðsþ sÞÞ � f ðs; xðsÞÞkds

� �2

þ 2MkCk2E

Z t

0e�

dðt�sÞ2 e�

dðt�sÞ2 kf ðsþ s; xðsþ s� rÞÞ � f ðs; xðs� rÞÞkds

� �2

6 2MkBk2E

Z t

0e�dðt�sÞds

� � Z t

0e�dðt�sÞkf ðsþ s; xðsþ sÞÞ � f ðs; xðsÞÞk2ds

� �� �þ 2MkCk2

E

Z t

0e�dðt�sÞds

� � Z t

t0

e�dðt�sÞkf ðsþ s; xðsþ s� rÞÞ � f ðs; xðs� rÞÞk2ds� �� �

6 2MkBk2Z t

0e�dðt�sÞds

� � Z t

0e�dðt�sÞEkf ðsþ s; xðsþ sÞÞ � f ðs; xðsÞÞk2ds

� �þ 2MkCk2

Z t

0e�dðt�sÞds

� � Z t

t0

e�dðt�sÞEkf ðsþ s; xðsþ s� rÞÞ � f ðs; xðs� rÞÞk2ds� �

6 2MkBk2Z t

0e�dðt�sÞds

� �2

supt2R

Ekf ðsþ s; xðsþ sÞÞ � f ðs; xðsÞÞk2 þ 2MkCk2Z t

0e�dðt�sÞds

� �2

� supt2R

Ekf ðsþ s; xðsþ s� rÞÞ � f ðs; xðs� rÞÞk2

<d2

2eZ t

0e�dðt�sÞds

� �2

þ d2

2eZ t

0e�dðt�sÞds

� �2

¼ e: ð3:3Þ

In view of the above, Ek/xðt þ sÞ � /xðtÞk2< e for each t 2 R, that is, t periodic.

Similarly, assuming that x is square-mean almost periodic and using, ðH2Þ and Theorem 2.8, one can easily see thats#Gðs; xðsÞÞ is square-mean almost periodic. Therefore, for each e > 0 there exists lðeÞ > 0 such that any interval of lengthlðeÞ > 0 contains at least s for which

EkGðsþ s; xðsþ sÞÞ � Gðs; xðsÞÞk2<

2d

M2 e

for each s 2 R. The next step consists of proving the square-mean almost periodicity of wxð�Þ. Of course, this is more compli-cated than the previous case because of the involvement of the Brownian motion W. To overcome such a difficulty, we makeextensive use of the Itôs isometry identity and the properties of fW defined by fW :¼Wðsþ sÞ �WðsÞ for each s. Note that fWis also a Brownian motion and has the same distribution as W.

Now

wðxðt þ sÞÞ � wðxðtÞÞ ¼Z t

0Tðt � sÞ½Gðsþ s; xðsþ sÞÞ � Gðs; xðsÞ�dWðsÞ:

Z. Huang, Q.-G. Yang / Chaos, Solitons and Fractals 42 (2009) 773–780 777

Next, make a change of variables s ¼ l� s to get

Ekwðxðt þ sÞÞ � wðxðtÞÞk2 ¼ E

Z t

t0

Tðt � sÞ½Gðsþ s; xðsþ sÞÞ � Gðs; xðsÞ�dfW ðsÞ���� ����2

:

Thus using Itôs isometry identity we obtain:

Ekwðxðt þ sÞÞ � wðxðtÞÞk2 ¼Z t

t0

EkTðt � sÞ½Gðsþ s; xðsþ sÞÞ � Gðs; xðsÞ�k2ds

6 M2Z t

t0

e�2dðt�sÞEk½Gðsþ s; xðsþ sÞÞ � Gðs; xðsÞ�k2ds

6 M2Z t

t0

e�2dðt�sÞds� �

sups2R

Ek½Gðsþ s; xðsþ sÞÞ � Gðs; xðsÞ�k2< e

and therefore wðxð�ÞÞ is square-mean almost periodic.Define

ðLxÞðtÞ :¼ Tðt � t0Þxð0Þ þZ t

0Tðt � sÞBf ðs; xðsÞÞ þ Tðt � sÞCf ðs; xðs� rÞÞdsþ

Z t

0Tðt � sÞGðt; xðtÞÞdWðsÞ:

In view of the above, it is clear that L maps APðR; L2ðP;HÞÞ into itself. To complete the proof, it suffices to prove that L hasa unique fixed-point.

Clearly,

kðLxÞðtÞ � ðLyÞðtÞk ¼Z t

0Tðt � sÞB½f ðs; xðsÞÞ � f ðs; yðsÞÞ�ds

���� þZ t

0Tðt � sÞC½f ðs; xðs� rÞÞ � f ðs; yðs� rÞÞ�ds

þZ t

0Tðt � sÞ½Gðs; xðsÞÞ � Gðs; yðsÞÞ�dWðsÞ

����6 MkBk

Z t

0e�dðt�sÞkf ðs; xðsÞÞ � f ðs; yðsÞÞkdsþMkCk

Z t

0e�dðt�sÞkf ðs; xðs� rÞÞ � f ðs; yðs� rÞÞkds

þZ t

0Tðt � sÞ½Gðs; xðsÞÞ � Gðs; yðsÞÞ�dWðsÞ

���� ����:

Since ðaþ bþ cÞ2 < 3a2 þ 3b2 þ 3c2, we can write

EkðLxÞðtÞ � ðLyÞðtÞk26 3M2kBk2

E

Z t

0e�dðt�sÞkf ðs; xðsÞÞ � f ðs; yðsÞÞkds

� �2

þ 3M2kCk2E

Z t

0e�dðt�sÞkf ðs; xðs� rÞÞ � f ðs; yðs� rÞÞkds

� �2

þ 3E

Z t

0Tðt � sÞ½Gðs; xðsÞÞ � Gðs; yðsÞÞ�dWðsÞ

���� ����� �2

:

We first evaluate the first term of the right-hand side as follows:

3M2kBk2E

Z t

0e�dðt�sÞkf ðs; xðsÞÞ � f ðs; yðsÞÞkds

� �2

6 3M2kBk2E

Z t

0e�dðt�sÞds

� � Z t

0e�dðt�sÞkf ðs; xðsÞÞ � f ðs; yðsÞÞk2ds

� �� �6 3M2kBk2

Z t

0e�dðt�sÞds

� � Z t

0e�dðt�sÞEkf ðs; xðsÞÞ � f ðs; yðsÞÞk2ds

� �6 3M2kBk2K

Z t

0e�dðt�sÞds

� �2

supt2R

EkxðsÞ � yðsÞk2

¼ 3M2kBk2KZ t

0e�dðt�sÞds

� �2

kx� yk1 63M2kBk2K

d2 kx� yk1:

As to the second term, we use isometry identity and obtain:

778 Z. Huang, Q.-G. Yang / Chaos, Solitons and Fractals 42 (2009) 773–780

3M2kCk2E

Z t

0e�dðt�sÞkf ðs� r; xðs� rÞÞ � f ðs� r; yðs� rÞÞkds

� �2

6 3M2kCk2E

Z t

0e�dðt�sÞds

� � Z t

0e�dðt�sÞkf ðs� r; xðs� rÞÞ � f ðs� r; yðs� rÞÞk2ds

� �� �6 3M2kCk2

Z t

0e�dðt�sÞds

� � Z t

0Ee�dðt�sÞkf ðs� r; xðs� rÞÞ � f ðs; yðsÞÞk2ds

� �6 3M2kCk2K

Z t

0e�dðt�sÞds

� �2

supt2R

EðkxðsÞ � yðsÞk2 ¼ 3M2kCk2KZ t

0e�dðt�sÞds

� �2

kx� yk1 63M2kCk2K

d2 kx� yk1:

As to the third term, we use isometry identity and obtain:

E

Z t

0Tðt � sÞ½Gðs; xðsÞÞ � Gðs; yðsÞÞ�dWðsÞ

���� ����� �2

¼ E

Z t

0kTðt � sÞGðs; xðsÞÞ � Gðs; yðsÞÞk2ds

� �6 E

Z t

0kTðt � sÞk2kGðs; xðsÞÞ � Gðs; yðsÞÞk2ds

� �6 M2

Z t

0e�2dðt�sÞEkGðs; xðsÞÞ � Gðs; yðsÞÞk2ds

6 M2K 0Z t

0e�2dðt�sÞds

� �supt2R

EkxðsÞ � yðsÞk26

M2K 0

2dkx� yk1:

Thus, by combining, it follows that

EkðLxÞðtÞ � ðLyÞðtÞk26

3M2KkBk2

d2 þ 3M2KkCk2

d2 þ 3M2K 0

2d

!kx� yk1

and therefore,

kðLxÞðtÞ � ðLyÞðtÞk21 6

3M2KkBk2

d2 þ 3M2KkCk2

d2 þ 3M2K 0

2d

!kx� yk1 ¼ Hkx� yk1: ð3:4Þ

Consequently, if H < 1 thus by (3.4), we know that L is a contraction mapping. Hence by the contraction mapping prin-ciple, L has a unique fixed point x(t) in R, then (2.1) has a unique fixed-point, which obviously is the unique square-meanalmost periodic solution (2.1).

Remark. From the conditions of Theorem 3.1, one can know that the external input, system parameters and activationfunctions have key effect on the unique of quadratic mean almost periodic.

Now we prove that system (2.1) is global exponential stability. One choose a positive constant a such that 0 < a < d. Itfollows from (3.1) that

eatEkxðtÞk2k 6 3eatEkTðt � t0Þxðt0Þk2 þ 6eatEkZ t

t0

Tðt � sÞBf ðs; xðsÞÞdsk2 þ 6eatE

Z t

t0

Tðt � sÞCf ðs; xðs� rÞÞds���� ����2

þ 3E

Z t

t0

Tðt � sÞGðt; xðtÞÞdWðsÞ���� ����2

: ð3:5Þ

Now we estimate the term on the right-hand side of first of (3.5), by the condition ðH0Þ, we obtain

3eatEkTðt � t0Þxð0Þk26 3eða�2dÞtM2kxðt0Þk2 ! 0 as t !1: ð3:6Þ

Secondly, Holder’s inequality and condition ðH0Þ yield

6eatEkZ t

0Tðt � sÞBf ðs; xðsÞÞdsk2

6 6eatM2K2kBk2E

Z t

0eð�dðt�sÞkxðsÞÞkds

� �2

¼ 6eatM2K2kBk2E

Z t

0eð�dðt�sÞ

2 eð�dðt�sÞ

2 kxðsÞÞkds� �2

6eatM2K2kBk2Z t

0e�dðt�sÞds

�Z t

0e�dðt�sÞEkxðsÞÞk2ds

6 6M2K2kBk2 1d

eatZ t

0e�dðt�sÞEkxðsÞÞk2ds

6 6M2K2kBk2 1d

e�ðd�aÞtZ t

0eðd�aÞseasEkxðsÞÞk2ds: ð3:7Þ

Z. Huang, Q.-G. Yang / Chaos, Solitons and Fractals 42 (2009) 773–780 779

For any xðtÞ 2 R and any e > 0, there exists a t1 > 0 such that easEkxðsÞÞk2< e for t P t1. Thus

6eatEkZ t

0Tðt � sÞBf ðs; xðsÞÞdsk2

6 6M2K2kBk2 1d

e�ðd�aÞtZ t

t1

eðd�aÞseasEkxðsÞÞk2dsþ 6M2K2kBk2 1d

e�ðd�aÞt

�Z t1

0eðd�aÞseasEkxðsÞÞk2ds:

6 6M2K2kBk2 1d

e�ðd�aÞtZ t1

0eðd�aÞseasEkxðsÞÞk2dsþ 6M2K2kBk2 1

dðd� aÞ e: ð3:8Þ

As e�ðd�aÞt ! 0 as t !1, there exists t2 P t1 such that for any t P t2, we have

6M2K2kBk2 1d

e�ðd�aÞtZ t1

0eðd�aÞseasEkxðsÞÞk2ds 6 e� 6M2K2kBk2 1

dðd� aÞ e: ð3:9Þ

So from (3.8) and (3.9), we obtain for any t P t2

6eatEkZ t

0Tðt � sÞBf ðs; xðsÞÞdsk2

< e:

That is to say,

6eatEkZ t

0Tðt � sÞBf ðs; xðsÞÞdsk2 ! 0 as t !1: ð3:10Þ

Similar to the discussion the third term on the right-hand side of (3.5) for any xðtÞ 2 R; t 2 ½�r;1Þ, one have

6eatEkZ t

0Tðt � sÞCf ðs; xðs� rÞÞdsk2 ! 0 as t !1: ð3:11Þ

As for the fourth term on the right-hand side of (3.5) for any xðtÞ 2 R; t 2 ½�r;1Þ, one have

3eatE

Z t

t0

Tðt � sÞGðt; xðtÞÞdWðsÞ���� ����2

6 3eatM2Z t

0e�2dðt�sÞEkGðt; xðtÞÞk2ds 6 3eatM2K 02

Z t

0e�2dðt�sÞEkxðtÞk2ds: ð3:12Þ

Similar to the proof of the second term on the right-hand side of (3.5) for any xðtÞ 2 R; t 2 ½�r;1Þ, we obtain

3eatE

Z t

t0

Tðt � sÞGðt; xðtÞÞdWðsÞ���� ����2

! 0 as t !1: ð3:13Þ

Thus, from (3.5), (3.6), (3.10), (3.11) and (3.13), we know that eatEkxðtÞk2 ! 0 as t !1. So we conclude that (2.1) hasa unique square-mean almost periodic solution, which is an exponentially stable.

4. Conclusions

In this paper, we study stochastic neural networks model with delays. By employing the Holder’s inequality and fixedpoints principle, we present some new criteria ensuring existence almost periodic and global exponential stability. Fromthe conditions of Theorem 3.1, one can know that the external input, system parameters and activation functions havekey effect on the unique of quadratic mean almost periodic.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 10871074), Foundation of EducationDepartment of GuangXi Province (No. 200807MS121) and Main Foundation of Wuzhou University (No. 2008B011).

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