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Abstract and Applied Analysis
Volume 2014, Article ID 934534, 11 pages
http://dx.doi.org/10.1155/2014/934534
Research Article

Asymptotically Almost Periodic Solutions for a Class of Stochastic Functional Differential Equations

1Educational Technology Center, Yulin Normal University, Yulin 537000, China
2School of Mathematics and Information Science, Yulin Normal University, Yulin 537000, China

Received 7 February 2014; Revised 29 March 2014; Accepted 29 March 2014; Published 6 May 2014

Academic Editor: Yonghui Xia

Copyright © 2014 Aimin Liu 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

This work is concerned with the quadratic-mean asymptotically almost periodic mild solutions for a class of stochastic functional differential equations . A new criterion ensuring the existence and uniqueness of the quadratic-mean asymptotically almost periodic mild solutions for the system is presented. The condition of being uniformly exponentially stable of the strongly continuous semigroup is essentially removed, which is generated by the linear densely defined operator , only using the exponential trichotomy of the system, which reflects a deeper analysis of the behavior of solutions of the system. In this case the asymptotic behavior is described through the splitting of the main space into stable, unstable, and central subspaces at each point from the flow’s domain. An example is also given to illustrate our results.

1. Introduction

The theory of almost periodic functions was first developed by the Danish mathematician H. Bohr in 1925-1926. Then Bohr’s work was developed substantially by S. Bochner, J. Favard, V. V. Stepanov, and others. Generalization of the classical theory of almost periodic functions has been taken in several directions. These works were recapitulated in literatures [1] and [2]. The concept of almost periodicity is important in probability for investigating stochastic processes [37]. Such a notion is also of interest for applications arising in mathematical physics and statistics. Literature [8] developed the notion of -mean almost periodicity based on the concept of quadratic mean uniformly almost periodic utilized by [7] and pointed out that each -mean almost periodic process is uniformly continuous and stochastically bounded [9]. Literature [8] also pointed that the collection of all -mean almost periodic processes is a Banach space when it is equipped with some norm obtained through the norm of , where is a Banach space.

The asymptotically almost periodic functions were first introduced by Fréchet. In the modern theory of differential equations, many authors [1, 2] applied successfully the asymptotic property to determine the existence of almost periodic solutions. Along with the development of such equations as the evolution partial differential equations, functional differential equations, and so forth, where the phase spaces are infinite, the theory of Banach valued asymptotically almost periodic functions had been developed [1012]. Some techniques in functional analysis and harmonic analysis were applied to such equations; for example, in [1, 13], the authors applied spectrum theory to get almost periodic solutions for some linear abstract evolution differential equations. More recently, [14] developed the notion of -mean asymptotical almost periodicity for stochastic processes. Among others, it showed that each -mean asymptotically almost periodic stochastic process is stochastically bounded.

Recently, [8] studied the existence and uniqueness of quadratic-mean almost periodic solutions for the class of stochastic differential equations Literature [15] investigated the existence and stability of quadratic-mean almost periodic mild solutions for stochastic functional differential equations They both assumed that the strongly continuous semigroup is uniformly exponentially stable, which is generated by the linear densely defined operator . For other works, we refer the reader to [1621] and the references therein.

One should point that the following condition (C) is very much important in the above-mentioned literatures.(C)The operator is the generator of a uniformly exponentially stable semigroup such that there exist constants with .

It is clear that the condition (C) is too strict [22] so that it cannot be satisfied even if for simple or . Literature [22] presented some new criteria ensuring the existence and uniqueness of quadratic-mean almost periodic solution for stochastic differential equation (1), and only assumes that the linear system admits exponential dichotomy. It is clear that when , the system (3) admits exponential dichotomy. More generally, in the case , a constant matrix, the system (3) admits exponential dichotomy if and only if the eigenvalues of have a nonzero real part. Literature [14, 17] has obtained the existence and uniqueness of quadratic-mean almost automorphic solutions or asymptotically almost periodic solutions for stochastic functional differential equations under a hyperbolic and analytic semigroup . At the same time, one notices that the case that the eigenvalues of have a zero real part is very common; for example, . Therefore, it is interesting to ask, what is that, when the semigroup is not exponentially stable, which is generated by the family , or when the semigroup is nonhyperbolic? This question will be considered in the paper.

In the present paper, motivated by [8, 14, 15], we discuss the existence and uniqueness of quadratic-mean asymptotically almost periodic solution to the following stochastic functional differential equation on , where is the probability measure of the probability space and is a real separable Hilbert space: We present the new criterion ensuring the existence of a unique quadratic-mean asymptotically almost periodic solution for the stochastic functional differential equation (1), by employing the properties of almost periodic function and the technique of inequality. We essentially remove the above conditions (C) and only assume that the linear system (2) admits exponential trichotomy (see Definition 11). We also point out that exponential trichotomy is the most complex asymptotic property of dynamical systems arising from the central manifold theory. Starting from the idea that the center manifold of an equilibrium point of a dynamical system consists of orbits whose behavior around the equilibrium point is not controlled by either the attraction of the stable manifold or the repulsion of the unstable manifold. The exponential trichotomy reflects a deeper analysis of the behavior of solutions of dynamical systems. In this case the asymptotic behavior is described through the splitting of the main space into stable, unstable, and central subspaces at each point from the flow’s domain.

This paper is organized as follows. In Section 2, the relating notations, definitions, and the basic results are introduced, which would be used later. In Section 3, a new criterion ensuring the existence and uniqueness of a quadratic-mean asymptotically almost periodic mild solution for stochastic functional differential equations is presented. In Section 4, an example is given to illustrate our results. Finally, conclusions are drawn in Section 5.

2. Preliminaries

Let be a probability space, for a Banach space and , denoted by , the Banana space of all -value random variable , such that

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

This setting requires the following preliminary definitions.

Definition 1. A stochastic process is said to be continuous whenever

Definition 2. A stochastic process is said to be stochastically bounded whenever

Let denote the collection of all stochastic processes , which are continuous and uniformly bounded. It is easily to check that is a Banach space when it is equipped with the norm

Definition 3. A continuous stochastic process is said to be -mean almost periodic if for each there exists such that any interval of length contains at least a number for which

The collection of all stochastic processes which are -mean almost periodic is denoted by is a closed subspace of . Therefore, is a Banach space when it is equipped with the norm (see, for example, [3]).

Let , be two Banach spaces and , their corresponding -spaces, respectively.

Definition 4 (see [8]). A function , , which is jointly continuous, is said to be -mean almost periodic in uniformly in where is compact if, for any , there exists such that any interval of length contains at least a number for which for each stochastic process .

Theorem 5. Let , , be a -mean almost periodic process in uniformly in , where is compact. Then for every real sequence , there exists and a continuous function such that exists uniformly on . Further, is also -mean almost periodic process in uniformly in .

The space of the restrictions of all -mean almost periodic stochastic processes on is denoted by , and -mean almost periodic stochastic processes in , uniformly for in compact subset of by .

Denote by the space of all continuous stochastic processes such that , and denote by the space of all continuous functions such that uniformly for in any compact subset of .

Definition 6. A stochastic process is called -mean asymptotically almost periodic if there exist two stochastic processes and such that .

By one denotes the collection of all -mean asymptotically almost periodic stochastic processes.

Definition 7. A stochastic process is called -mean asymptotically periodic if there exist two stochastic processes and such that .

By we denote the collection of such function.

Lemma 8 (see [14]). If belongs to , one has the following:(1)there exists a constant such that for each ;(2) is stochastically bounded.
It is easy to see that . Then the space of -mean asymptotically almost periodic stochastic processes is a Banach space when it is equipped with the norm .

Lemma 9 (see [14]). Let , , where is regarded as a valued stochastic process. Moreover Then .

Lemma 10 (see [14]). Let , , where is regarded as a valued stochastic process. Moreover Then .

Next, one introduces a crucial concept [23].

Suppose that is the fundamental matrix solution of the linear differential system where is a linear continuous operator, with .

Definition 11 (see [23]). System (14) is said to admit exponential trichotomy if there are linear projections , such that and constants and such that

If in the above definition we put , (16) becomes

It is clear that when , the system (14) admits exponential trichotomy. More generally, in the case , a constant matrix, the system (14) admits exponential trichotomy.

Theorem 12. Let be a continuous linear operator on and let (14) have an exponential trichotomy (16) on . If for some sequence , , uniformly on compact subintervals of , , , and the equation (which is called the hull equation (14)) has an exponential trichotomy on with projections , and the same constants , .

Proof. The translated equation has the fundamental matrix where , . Since , by restricting attention to a subsequence we can assume that , , where , are projections. Since for every , where is the fundament matrix of (21) such that , it follows that
Since the projection corresponding to an exponential trichotomy on is uniquely determined it follows that , without restriction to a subsequence.

3. Existence of Asymptotically Almost Periodic Solutions

For convenience, throughout this section, let be a real separable Hilbert space, and let be a complete probability space equipped with a normal filtration , that is, a right-continuous, increasing family of sub--algebras of . Let be the space of all bounded linear operators from to and the space of all continuous functions from to with the sup norm

For any continuous -adapted -valued stochastic process , , we associate it with a continuous -adapted -valued stochastic process , by setting .

In this section, we study the existence and uniqueness of quadratic-mean asymptotically almost periodic mild solution to stochastic functional differential equations of the form where is a linear operator and generates a strongly continuous semigroup , which is nonhyperbolic. That is to say, the linear operator may exhibit central flow. is a certain -Wiener process with covariance operator and takes its values on . and are two continuous mappings.

Throughout this section, we require the following assumptions.(H1) Suppose that . Furthermore, there exists a constant such that for all stochastic processes and for each .(H2) Suppose that . Furthermore, there exists a constant such that for all stochastic processes and for each .(H3) Suppose that the linear system of (26) admits exponential trichotomy (see Definition 11); that is, there exist constants , such that (16) holds.

Define the function as the form where is the fundamental matrix solution of the linear differential system (29) with .

Definition 13. A -progressive process is called a mild solution of (26) on if for all , for each , where is defined by (30).

Theorem 14. Assume that conditions (H1)–(H3) are satisfied. And the positive constants , , , and satisfy the following condition: Then the system (26) has a unique quadratic-mean asymptotically almost periodic mild solution, which can be explicitly expressed as follows: where

We show that (26) exists as a mild solution. Note that (33) and (34) are well defined for each and satisfy (31) for all , for each . Hence given by (33) and (34) is a mild solution of (26).

Define a mapping on by where the are defined in (34).

In order to prove Theorem 14, we first prove Lemmas 15 and 16.

Lemma 15. Assume that conditions (H1)–(H3) are satisfied. Then the operator maps into itself.

Proof. First, let us show that is quadratic-mean asymptotically almost periodic whenever dose.
Indeed, assuming that is quadratic-mean asymptotically almost periodic and using condition (H1) and Lemma 10, one can easily see that is quadratic-mean asymptotically almost periodic. Therefore, there exist an almost periodic stochastic process and a stochastic process such that . Furthermore, one observes that We claim that . In fact, is quadratic-mean almost periodic. Therefore, one can find a common subsequence . One still denotes it as , such that uniformly for , and in the sense of norm uniformly for . Then From (37), (38), (39), and Lebesgue’s control convergence theorem, one sees that converges to uniformly for . Hence .
Next, let us show that . In fact Since , we deduce that .
Therefore, is quadratic-mean asymptotically almost periodic.
Similar to the proof given for , one can prove that , , and are quadratic-mean asymptotically almost periodic.
Thus, is quadratic-mean asymptotically almost periodic whenever .
Secondly, we show that is also quadratic-mean asymptotically almost periodic whenever .
Of course, this is more complicated than the previous case because of the involvement of the Brownian motion . To overcome such a difficulty, one makes extensive use of the Itô isometry identity and the properties of defined by for each . Note that is also a Brownian motion and has the same distribution as . Assuming that is quadratic-mean asymptotically almost periodic, using (H2) and Lemma 10, one can easily see that is quadratic-mean asymptotically almost periodic. Therefore, there exist an almost periodic stochastic process and a stochastic process such that . Furthermore, one observes that
We claim that . In fact, is quadratic-mean almost periodic. Therefore, one can find a common subsequence . One still denotes it as , such that uniformly for and (38) hold.
Now
Using Itô’s isometry identity, one obtains From (38), (43), (45), and Lebesgue’s control convergence theorem, one sees that uniformly for . Hence .
Next, let us show that . In fact where we make extensive use of the Itô isometry identity and the properties of defined by for each . Note that is also a Brownian motion and has the same distribution as .
Since , we deduce that .
Therefore, is quadratic-mean asymptotically almost periodic.
Similar to the proof given for , one can prove that , , and are quadratic-mean asymptotically almost periodic.
Thus, is quadratic-mean asymptotically almost periodic whenever .
In view of the above, it is clear that maps into itself. The proof of Lemma 15 is complete.

Lemma 16. Assume that conditions (H1)–(H3) are satisfied. Then the operator is a contraction providing .

Proof. Consider
Since , one can write Combining (30), one can write Thus,
We first evaluate as follows: Similar to the discussion given for , for , , and , one has Then,
Next, we evaluate as follows:
As for the first term , using Itô’s isometry identity, one obtains
Similarly, one can evaluate the second term , third term , and the fourth term of the right-hand side, respectively:
Therefore,
Thus, by combing (54) and (58), it follows that which implies that Since , by (60), we know that is a contraction mapping. The proof of Lemma 16 is complete.

Hence, combining Lemmas 15 and 16 and the contraction mapping principle, has a unique fixed point , which is obviously the unique quadratic-mean asymptotically almost periodic mild solution of (26).

This completes the proof of Theorem 14 due to Lemmas 15 and 16.

Remark 17. If the conditions of the main result of [14] and (H1) and (H2) hold, (26) admits exponential trichotomy with projections and ; hence system (26) has a unique quadratic-mean asymptotically almost periodic mild solution. So our main result improves the main result of [14].

4. Example

The deterministic nonlinear Duffing-van der Pol equation has become a paradigm for mathematicians, physicists, and engineers. There are numerous physical and engineering problems whose dynamics are described by (61) for some parameter values.

In particular, for one obtains the van der Pol equation and for one obtains the Duffing equation.

As an example, we consider the following functional differential equation for small and : It is easy to see that (61) is the general form of the Duffing-van der Pol equation (61). By perturbing the remaining parameters and the right-hand side of (62) by real or white noise, one arrives at noise, more specifically the random (for real noise) or stochastic (for white noise) functional differential equation, respectively: where is intensity parameter, functions , , and is a stationary process or white noise. As a first order system for , (63) takes the form where

Let . Suppose that vector functions and satisfy the inequality conditions of (H1) and (H2) in Theorem 14. We see that when (i)