Abstract

In this paper we consider the existence and stability of solutions to stochastic neutral functional differential equations with finite delays. Under suitable conditions, the existence and exponential stability of solutions were obtained by using the semigroup approach and Banach fixed point theorem.

1. Introduction

In natural world, stochastic phenomena are everywhere. Many stochastic facts exist in biology, chemistry, physics, and economical systems. In addition, delays also appear sometimes to change the results. These facts imply the necessity to study stochastic functional differential equations (SFDEs), although there are a lot of papers on the related topics for deterministic partial functional differential equations [1–4].

A large amount of basic knowledge about stochastic differential equations (SDEs) has been given in [5, 6]. Recently, many researchers studied the existence, stability, and other properties of solutions to SFDEs. Some of these topics have been solved by the semigroup approach and others have been solved by the variational one. There are many types of equations worth attentions; say [7] is focused on retarded equations with finite delays while [8] is concerned with neutral ones. In fact the stability of SDEs is most important among all qualitative properties and also determines whether the model is significant. Aside from all kinds of stability, other properties have drawn many attentions. For example, Wu et al. [9, 10] investigated the stochastic delay population dynamics under regime switching by using generalized Itô’s formula, Gronwall inequality, and Young’s inequality. Furthermore, some stochastic systems have added the impulse factors [11].

On the other hand, although stochastic partial FDEs with finite delays also seem very important, the corresponding properties of these systems have not been studied in great detail.

Taniguchi et al. [12] discussed the stochastic partial FDEs with finite delay as follows:Under some suitable conditions, the existence and asymptotic behavior of solutions were obtained employing the semigroup approach and fixed point theorem.

In this paper, we shall discuss the existence and uniqueness of mild solutions to a class of stochastic NFDEs with finite delays, where is -measurable. is a linear and densely closed operator which generates an analytic semigroup ,  , on a real separable Hilbert space with the inner product and norm .

Notice that (1) is of retarded type; here we put a neutral type one, (2). When , our equation is the same as that in [12]. In this paper we assume ,   and define with the norm for , where is the domain of the fractional power operator . Let and ,  . is another real separable Hilbert space with the inner product and norm , is a -valued Wiener process with a finite trace nuclear covariance operator . Assume ,   are measurable mappings, satisfying that ,  ,   are locally bounded in -norm and -norm, respectively.

This paper is organized as follows. In Section 2 some preliminary results are given, which are fundamental for the subsequent developments. The existence and uniqueness of solutions are investigated in Section 3. At last, we obtain the almost sure exponential stability of the solutions.

2. Preliminaries

Let be a probability space and ,   is a continuous -adapted, -valued stochastic process. We give process ,  , by setting ,  

Let ,  , be the space of all -measurable -valued functions :Next we give three important assumptions.

Assumption A. (a) is a separable Hilbert space on which there is an analytic semigroup ,  , and is the infinitesimal generator of .
(b) There exist and such that ,  ,  .
(c) ,  , for any , where ,  .
(d) ,  ,  .
The details can be seen in Pazy [13].
Under Assumption A, we will consider the stochastic integral equation next instead of (2) by carrying out a semigroup type argument mentioned above:The function which satisfies (4) is called the mild solution of (2).

Lemma 1 (see [12]). Suppose ,   If ,  ,  , and ,   then

Assumption B. For any and , there exist three positive real constants , , and such thatUnder Assumption B, we get a real number such thatfor , where is any fixed time.

Assumption C. There exist and such thatfor every ,  , and

3. Existence of Mild Solutions

Definition 2. If function satisfies (4) on and is small, it is called local solution. If satisfied (4) on , it is called global solution.

Theorem 3. Let . Suppose that Assumptions A, B, and C hold. There exists a unique local continuous solution to (4) for any initial value with and .

In order to prove Theorem 3, we need some lemmas. Assume is a fixed time and is the subspace of all continuous processes which belong to the space with , where Introduce the following mapping on :

Lemma 4. For , is continuous on in the -sense.

Proof. Let For any fixed , From Assumptions A, B, and C we get thatbecause of the continuity of ,  , when . There exist and let ,   such thatIn the similar way, there exist and we haveAt last Then, there exist and such thatSince , it follows that ,  ,  ,  ,   tend to zero, as . Therefore, the proof of the lemma is complete.

Lemma 5. Suppose the operator mapping and the corresponding domain are defined as above; then

The proof is similar to that of Lemma 4.

Proof of Theorem 3. Let , and then for any fixed , Next, let andand then by the use of the stochastic Fubini Theorem again we haveSo we can choose a suitable to make sufficiently small; we obtain such thatfor any Thus, by Banach fixed point theorem we get a unique fixed point . Setting , we obtainThe proof is complete.