## Dynamical Aspects of Initial/Boundary Value Problems for Ordinary Differential Equations 2014

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# Existence of Square-Mean Almost Automorphic Solutions to Stochastic Functional Integrodifferential Equations in Hilbert Spaces

**Academic Editor:**Yonghuia Xia

#### Abstract

The existence and uniqueness of square-mean almost automorphic mild solution to a stochastic functional integrodifferential equation is studied. Under some appropriate assumptions, the existence and uniqueness of square-mean almost automorphic mild solution is obtained by Banachâ€™s fixed point theorem. Particularly, based on Schauderâ€™s fixed point theorem, the existence of square-mean almost automorphic mild solution is obtained by using the condition which is weaker than Lipschitz conditions. Finally, an example illustrating our main result is given.

#### 1. Introduction

The almost periodic type solutions to stochastic differential equations are among the most attractive topics in mathematical analysis due to their extensive applications in areas such as physics, economics, mathematical biology, and engineering. And the concept of almost automorphic functions, which was initially introduced in the literature by Bochner [1], is an important generalization of the almost periodic functions. From then on, the almost automorphic functions and the almost automorphic solutions for differential systems have been investigated by many mathematicians [2â€“13].

Integrodifferential equations arose naturally in mechanics, electromagnetic theory, heat flow, nuclear reactor dynamics, and population dynamics [14]. The papers [5, 14â€“20] are concerned with the existence of almost periodic type solutions to stochastic functional integrodifferential equations. Ding et al. [21] investigated the existence of pseudo almost periodic solutions for an equation arising in the study of heat conduction in materials with memory, which could be transformed into the following abstract integrodifferential equation: Diagana et al. [5] established the existence and uniqueness of asymptotically almost automorphic mild solution to an abstract partial neutral integrodifferential equation with unbounded delay where .

Furthermore, noise or stochastic perturbation is unavoidable and omnipresent in nature as well as in man-made systems. This paper is mainly focused on the existence and uniqueness of square-mean almost automorphic mild solutions to the following stochastic functional integrodifferential equations in the abstract form: where , , and are linear, closed, and densely defined operators on and is a two-sided standard one-dimensional Brownian motion defined on the filtered probability space , where . Here , , , , and are appropriate functions to be specified later.

The Lipschitz condition is a very important condition in the field of the existence and uniqueness of solutions for differential equations. In [15], Chang et al. established a new composition theorem for square-mean almost automorphic functions under conditions which were different from Lipschitz conditions in the literature. And they apply this new composition theorem to investigate the existence of square-mean almost automorphic mild solutions for a stochastic differential equation. It can be proved that the conditions in [15] (see assumption (H3)(ii)) are weaker than Lipschitz condition. In other words, a function with Lipschitz condition is satisfied (H3)(ii). However, there exists such a function which satisfies (H3)(ii) but does not satisfy Lipschitz condition (see Remark 11).

In this paper, firstly, upon using Lipschitz condition and other some appropriate assumptions, some sufficient conditions for the existence and uniqueness of square-mean almost automorphic mild solution to (3) are given. Secondly, by virtue of new composition theorem in [15] together with Schauderâ€™s fixed point theorem, we investigate the existence of square-mean almost automorphic mild solutions for a stochastic differential equation in a real separable Hilbert space, which is different from Lipschitz condition in the literature. Finally, we discuss the existence and uniqueness of an almost automorphic mild solution to a concrete integrodifferential equation, which is an illustration to demonstrate our main analyses.

#### 2. Preliminaries

Throughout this paper, we assume that and are two real separable Hilbert spaces. Let be a complete probability space. The notation stands for the space of all -valued random variables such that For , let Then it is routine to check that is a Hilbert space equipped with the norm . The notations and stand for the collection of all continuous stochastic processes from into and the space of all bounded continuous stochastic processes , respectively. It is then easy to check that is a Banach space when it is endowed with the norm . Let denote the space of all linear bounded operators from into , which are equipped with the usual operator norm ; in particular, it is simply denoted by when . In addition, is a two-sided standard one-dimensional Brownian motion defined on the filtered probability space , where .

*Definition 1 (see [18]). *A family of bounded linear operators from into is a resolvent operator family for the problem
if the following conditions are satisfied:(a) (the identity operator on ) and the function is continuous on for every ;(b) for all and for , is continuous on , and is continuously differentiable on ;(c)for , the following resolvent equations hold:

For more details on semigroup theory and resolvent operators, we refer the reader to [16, 18, 22].

*Definition 2 (see [17]). *A stochastic process is said to be stochastically continuous if

*Definition 3 (see [17]). *A stochastically continuous stochastic process is said to be square-mean almost automorphic if, for every sequence of real numbers, there exists a subsequence and a stochastic process such that
for each . The collection of all square-mean almost automorphic stochastic processes is denoted by .

*Definition 4. *A function , , which is jointly continuous, is said to be square-mean almost automorphic if is square-mean almost automorphic in uniformly for all , where is any bounded subset of . That is to say, for every sequence of real numbers , there exist a subsequence and a function such that
for each and each .

Lemma 5 (see [17]). * is a Banach space when it is equipped with the norm
**
for .*

Lemma 6 (see [17]). *If , , and are all square-mean almost automorphic stochastic processes, then the following hold:*(i)* is square-mean almost automorphic;*(ii)* is square-mean almost automorphic for every scalar ;*(iii)*there exists a constant such that . That is, is bounded in .*

Theorem 7 (see [17]). *Let , be square-mean almost automorphic in for each , and assume that satisfies a Lipschitz condition in the following sense:
**
for all and for each , where is independent of . Then for any square-mean almost automorphic process , the stochastic process given by is square-mean almost automorphic.*

Theorem 8 (see [15]). *Let , be square-mean almost automorphic, and assume that is uniformly continuous on each bounded subset uniformly for ; that is, for all , there exists such that and imply that for all . Then for any square-mean almost automorphic process , the stochastic process given by is square-mean almost automorphic.*

#### 3. Some Lemmas

In this section, we apply Theorems 7 and 8 to investigate the existence of square-mean almost automorphic mild solutions to problem (3).

*Definition 9. *A stochastically continuous stochastic process is called a mild solution of the system (3) if and satisfies
for all .

*Remark 10. *It is easy to see that if is exponentially stable, then we obtain that the stochastic process is a mild solution to problem (3) if and only if satisfies the stochastic integral equation
for all .

Let us list the following assumptions.(H1)There exists a resolvent operator of (3), and is exponentially stable; that is, â€‰for all and some constants . Moreover, is compact for .(H2)The functions and there exist positive numbers , , such that â€‰for all and each .(H3),â€‰â€‰.(H4)The functions satisfy the following conditions.(i), , are square-mean almost automorphic and , , are uniformly continuous in every uniformly bounded subset for .(ii)There exist integrable functions and continuous nondecreasing functions such that for all .(iii)Let be uniformly bounded in and uniformly convergent in each compact subset of . Then , , and are relatively compact in .

*Remark 11. *Obviously, when the continuous function satisfies Lipschitz condition like (H2) and , then function satisfies (H4)(ii). However, the function satisfies (H4)(ii) but does not satisfy (H2).

In the proof of the existence theorem, we need the following technical lemmas.

Lemma 12. *Assume that conditions (H1)â€“(H3) hold. Let be the operator defined by, for each ,
**
Then maps into itself.*

*Proof. *Firstly, let ; then is in as . Hence, by (H2) and Theorem 7, one can easily see that belongs to .

Similarly, and belong to whenever .

Secondly, we show that is square-mean almost automorphic.

Let be an arbitrary sequence of real numbers. Since , there exists a subsequence of such that, for a certain stochastic process ,
hold for each . Moreover, if we let , by using Cauchy-Schwarz inequality, we have

Thus, by (19), we immediately obtain that
for each . And we can show in a similar way that
for each . Thus we conclude that .

Thirdly, we show that is square-mean almost automorphic.

Since , there exists a subsequence of such that, for a certain stochastic process ,
hold for each .

This is more complicated than the previous case because of the involvement of the Brownian motion . To overcome such a difficulty, we make extensive use of the properties of defined by
for each . Note that is a Brownian motion and has the same distribution as . Moreover, if we let , then, by making a change of variable , we get
Then, using an estimate on the Ito integral established, we obtain
Thus, by (23), we immediately obtain that
for each . And we can show in a similar way that
for each . Thus we conclude that .

According to Lemma 6, we can easily obtain that maps into itself.

Lemma 13. *Assume that conditions (H1), (H3), and (H4)(i) are satisfied. Then is continuous and maps into itself, where is defined by Lemma 12.*

*Proof. **Step **1*. We prove that maps into itself.

Firstly, let ; then is in as . Hence, by (H4)(i) and Theorem 8, one can easily see that belongs to .

Similarly, and belong to whenever .

Secondly, from Lemma 12, we can prove that and are square-mean almost automorphic.

According to Lemma 6, we can easily obtain that is continuous and maps into itself.*Step **2*. We prove that is continuous on .

Let be a sequence which converges to some with respect to ; that is, as . There exists a bounded subset such that , for , . By (H3), (H4)(i), and Theorem 8, for any , there exist and such that , which imply that
for all and , where are given in (H1). Then
for all and all . This implies that is continuous. The proof is completed.

#### 4. Existence of Square-Mean Almost Automorphic Solutions

Theorem 14. *If the assumptions (H1)â€“(H3) are satisfied, then the system (3) has a unique square-mean almost periodic automorphic mild solution, whenever , where
*

*Proof. *Let be the operator defined by

From the proof of Lemma 12, we see that . Therefore, ; thus maps into itself.

To complete the proof, it suffices to prove that is a contraction. Since , for , we obtain

Evaluating the three terms of the right-hand side, we have

Thus, we obtain that
for each .

Consequently, if , then (3) has a unique fixed point, which is the unique square-mean almost periodic automorphic mild solution to (3), such that .

Theorem 15. *Assume that conditions (H1), (H3), and (H4) are satisfied; then problem (3) admits at least one square-mean almost automorphic mild solution on provided that
*

*Proof. *For the sake of convenience, we break the proof into several steps.*Step **1*. is continuous.

From the proof of the previous lemmas it is clear that the nonlinear operator is well defined and continuous. Moreover, by Lemma 13, we infer that whenever ; that is, maps into itself.*Step **2*. maps bounded sets into bounded sets.

Let for each . We prove that there exists a number such that .

Clearly, for each positive number , is a bounded closed convex set in . We claim that there exists a positive number such that . If it is not true, then for every there exist and such that . However, on the other hand, we have
Dividing both sides by and taking the lower limit as , we obtain
which contradicts condition (37). Thus, for some positive number , .*Step **3*. maps bounded sets into equicontinuous sets.

Next we prove that the operator is completely continuous on . It suffices to prove that the following statements are true.(i) is relatively compact in for each .(ii) is a family of equicontinuous functions.

Firstly, we show that (i) holds. Let be given. For each and , we define

Since is compact and (H4)(iii), then the set is relatively compact in for each . Moreover, for every , we have

Observe that is continuous; therefore, letting , there are relatively compact sets arbitrarily close to and hence is also relatively compact in for each and is completely continuous on .

We now show that (ii) holds. Let . Then
The right-hand side tends to independently of as , which implies that the set is right equicontinuous at . By a similar procedure we can show that is left equicontinuous at . Thus, the set is equicontinuous; that is, maps into a family of equicontinuous functions.

As a consequence of Steps 1â€“3 together with Schauderâ€™s fixed point theorem, we deduce that has a fixed point in which is a square-mean almost periodic mild solution to (3). The proof is completed.

#### 5. Example

Consider the neutral stochastic partial functional integrodifferential equations of the form for , where is a two-sided standard one-dimensional Brownian motion defined on the filtered probability space , . Here and , the constants , .

For that, let be an open subset whose boundary is sufficiently regular. Let and consider the linear operator whose domain is given by and where , are real-valued functions of class on such that , and .

From Chen [23], we see that is the infinitesimal generator of a uniformly exponentially stable -semigroup on . In what follows, we will assume that are positive constants such that for all .

Let , where is the operator family defined by and assume that From the results in Grimmer [22], we see that the abstract integrodifferential system, has an associated uniformly exponentially stable resolvent of operators with for .

Let Thus, one has It follows that . Therefore, according to Theorem 14, (43) has a unique square-mean almost periodic mild solution provided that . Let Thus, one has It follows that and .

Obviously,