Journal of Applied MathematicsVolume 2012 (2012), Article ID 414320, 14 pagesdoi:10.1155/2012/414320

Research Article

Mean Square Almost Periodic Solutions for Impulsive Stochastic Differential Equations with Delays

Ruojun Zhang,¹ Nan Ding,2,3 and Linshan Wang¹

¹School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China²College of Information Engineering, Ocean University of China, Qingdao 266100, China³Department of Mathematics, Chongqing Three Gorges University, Chongqing 404100, China

Received 28 December 2011; Revised 15 March 2012; Accepted 15 March 2012

Academic Editor: Zhiwei Gao

Copyright © 2012 Ruojun Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We establish a result on existence and uniqueness on mean square almost periodic solutions for a class of impulsive stochastic differential equations with delays, which extends some earlier works reported in the literature.

1. Introduction

Impulsive effects widely exist in many evolution processes of real-life phenomena in which states are changed abruptly at certain moments of time, involving such areas as population dynamics and automatic control [1–3]. Because delay is ubiquitous in the dynamical system, impulsive differential equations with delays have received much interesting in recent years, intensively researched, some important results are obtained [4–9]. And almost periodic solutions for abstract impulsive differential equations and for impulsive neural networks with delay have been discussed by G. T. Stamov and I. M. Stamova [10], and Stamov and Alzabut [11].

However, besides delay and impulsive effects, stochastic effects likewise exist in real system. A lot of dynamic systems have variable structures subject to stochastic abrupt changes, which may result from abrupt phenomena such as stochastic failures and repairs of components, changes in the interconnections of subsystems, sudden environment changes, and so on [12–14]. Moreover, differential descriptor systems also have abrupt changes [15, 16]. Recently, a large number of stability criteria of stochastic system with delays have been reported [17–19]. Almost periodic solutions to some functional integro-differential stochastic evolution equations and to some stochastic differential equations have been studied by Bezandry and Diagana [20], and Bezandry [21]. Huang and Yang investigated almost periodic solution for stochastic cellular neural networks with delays [22]. Because it is not easy to deal with the case of coexistence of impulsive, delay and stochastic effects in a dynamical system, there are few results about this problems [23–25]. To the best of our knowledge, there exists no result on the existence and uniqueness of mean square almost periodic solutions for impulsive stochastic differential equations with delays.

Motivated by the above discussions, the main aim of this paper is to study the mean square almost periodic solutions for impulsive stochastic differential equations with delays. By employing stochastic analysis, delay differential inequality technique and fixed points theorem, we obtain some criteria to ensure the existence and uniqueness of mean square almost periodic solutions.

The rest of this paper is organized as follows: in Section 2, we introduce a class of impulsive stochastic differential equations with delays, and the relating notations, definitions and lemmas which would be used later; in Section 3, a new sufficient condition is proposed to ensure the existence and uniqueness of mean square almost periodic solutions; in Section 4, an example is constructed to show the effectiveness of our results. Finally, a conclusion is given in Section 5.

2. Preliminaries

Let , , and be the set of all sequence unbounded and strictly increasing. For and , let be any vector norm, and denote the induced matrix norm and the matrix measure, respectively, by The norm and measure of vector and matrix are , , .

Consider the following a class of Itô impulsive stochastic differential equations with delay where is the solution process, are constant matrices, , , , is the diffusion coefficient matrix, is impulsive function, is delay; is impulsive time, is a constant vector, is an -dimensional Brown motion defined on a complete probability space with a natural filtration generated by , and denote by the associated -algebra generated by with the probability measure . Moreover, the initial conditions . Denote by the family of all bounded -measurable, -valued random variable , satisfying , where is continuous}. denotes the expectation of stochastic process.

Let be a Hilbert space and be a complete probability space. Define to be the space of all -value random variable such that

It is then routine to check that is a Hilbert space when it is equipped with its natural norm defined by for each .

Definition 2.1 (see [25]). For any , a function is said to be solution of system (2.2) on satisfying initial value condition, if the following conditions hold:(i) is absolutely continuous on each interval ;(ii) for any , and exist and ;(iii) satisfies (2.2) for almost everywhere in and at impulsive points situated in , may have discontinuity points of the first kind.Obviously, the solution defined by definition 1 is piecewise continuous.

Definition 2.2 (see [26]). The set of sequences , is said to be uniformly almost periodic if for any , there exists relatively dense set of -almost periods common for any sequences.

Definition 2.3. A piecewise continuous function with discontinuity points of first kind at is said to be mean square almost periodic, if(i) the set of sequence is uniformly almost periodic;(ii) for any , there exists , such that if the points and belong to one and the same interval of continuity of and satisfy the inequality , then ;(iii) for any , there exists a relatively dense set such that if , then for all satisfying the condition .

The collection of all functions with discontinuity points of the first kind at which are mean square almost periodic is denoted by , one can check that is a Banach space when it is equipped with the norm:

Let and be Banach space and and be their corresponding -space, respectively.

Lemma 2.4 (see [20]). Let be mean square almost periodic in uniformly in , where is compact. Suppose that there exists such that for all and for each . Then for any mean square almost periodic function , is mean square almost periodic.

In this paper, we always assume that:(A1) det and the sequence , is almost periodic, where is the identity matrix;(A2) the set of is uniformly almost periodic and .

Recall [2], consider the following linear system of system(2.2) that if is the Cauchy matrix for the system then the Cauchy matrix for the system (2.7) is in the form

As the special case of Lemma  1 in [10], we have the following lemma.

Lemma 2.5. Assume that (A1), (A2) and the following condition hold. For the Cauchy matrix of system (2.7), there exist positive constants and such that Then for any , there must be exist a relatively dense set T of -almost periodic of the matrix and a positive constant such that for , it follows:

Lemma 2.6 (see [6]). Let be the Cauchy matrix of the linear system (2.7). Given a constant for all , if and , then

Introduce the following conditions:(A3) The functions are mean square almost periodic in uniformly in , where is compact, and . Moreover, there exist such that

for all stochastic processes and .(A4) The function is mean square almost periodic in uniformly in , where is compact, and . Moreover, there exists such that

for all stochastic processes and .(A5) The function is almost periodic in the sense of Bohr, is almost periodic sequence and there exists a constant , such that(A6) The sequence of functions is mean square almost periodic uniformly with respect to , where is compact. Moreover, there exists such that for all stochastic processes .

Lemma 2.7 (see [26]). If conditions (A1)–(A6) are satisfied, then for each , there exists and relatively dense sets of real numbers and of integral numbers, such that (i), , , ;(ii);(iii);(iv);(v);(vi).

Lemma 2.8 (see [26]). Let condition (A2) holds. Then for each , there exists a positive integer such that on each interval of length , there are no more than elements of the sequence , that is, where is the number of points in the interval .

3. Main Results

Theorem 3.1. Assume that (A1)–(A6) hold, then there exists a unique mean square almost periodic solution of system (2.2) if the following conditions are satisfied: There exists a constant , such that and Furthermore,

Proof. Let satisfying the equality , where .Set where , it is easy to see that given by (3.3) is the solution of system (2.2) according to [2] and Lemma  2.2 in [27].By Lemma 2.6 and the conditions of Theorem, we have For , we define the operator in the following way Define subset , , and We have Then for , from the definition of and (3.6), since , we have For , we have Since , it follows For first term of the right-hand side, using (3.7), (A3) and Cauchy-Schwarz inequality, we have As to the second term, using (3.7), (A6) and Cauchy-Schwarz inequality, we can write As far as last term is concerned, using (3.7), (A4), and the Itô isometry theorem, we obtain Thus, by combining (3.9)–(3.12), it follows that By Lemmas 2.5 and 2.6, one can obtain Let , where the sets and are determined in Lemma 2.7, and we assume that , then Therefore, we have We first evaluate the first term of the right hand side where . For the second term, we can estimate that where .For the last term, using (A4) and Itô isometry identity, we have where .Combining (3.17), (3.18) and (3.19), it follows that where . So, , that is is self-mapping from to by (3.13) and (3.20).Secondly, we will show is contracting operator in .For , By a minor modification of the proof of (3.13), we can obtain and therefore, , it follows that is contracting operator in , so there exists a unique mean square almost periodic solution of (2.2) by the fixed points theorem.

4. Example

Consider the following impulsive stochastic differential equation with delay where , for convenience, we can choose Then . Choose . By simple calculation, we have .

Let , so, by Theorem 3.1, system (4.1) has a unique mean square almost periodic solution in .

Remark 4.1. Since there exist no results for almost periodic solutions for impulsive stochastic differential equations with delays, one can easily see that all the results in [10, 11, 20–22, 28] and the references therein cannot be applicable to system (4.1). This implies that the results of this paper are essentially new.

5. Conclusion

In this paper, a class of Itô impulsive stochastic differential equations with delays has been investigated. We conquer the difficulty of coexistence of impulsive, delay and stochastic factors in a dynamic system, and give a result for the existence and uniqueness of mean square almost periodic solutions. The results in this paper extend some earlier works reported in the literature. Moreover, our results have important applications in almost periodic oscillatory stochastic delayed neural networks with impulsive control.

Acknowledgment

This work is supported by the National Science Foundation of China (no. 10771199).

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