Abstract

We focus on a class of neutral stochastic delay partial differential equations perturbed by a standard Brownian motion and a fractional Brownian motion. Under some suitable assumptions, the existence, uniqueness, and controllability results for these equations are investigated by means of the Banach fixed point method. Moreover, an example is presented to illustrate our main results.

1. Introduction

Fractional Brownian motion (fBm for short) with Hurst parameter is a centered Gaussian process which is often used to model many complex phenomena in applications when the systems contain rough external forcing. When , the fBm is the standard Brownian motion which is a Markov process and martingale, and we can use the classical Itô theory to construct a stochastic integration with respect to the standard Brownian motion. However, when , the fBm is neither a Markov process nor a semimartingale, and the fBm behaviors become quite different. The classical techniques based on PDE cannot apply to in this context. For more details on fBm, we refer the readers to the articles [1–5].

Neutral differential equations (NDEs for short) are a family of differential equations depending on the past as well as the present state which involve derivatives with delays as well as the function itself. This family of equations has great applications in mathematical, electrical engineering, ecology, and other fields of science. In recent years, NDEs have received more and more attention; see, for example, [6–16] and the reference therein.

The observations of stock prices processes suggest that they are not self-similar. As a result, the mixed model describes the stock prices behavior in a better way. Since the pioneering paper by Cheridito [17], mixed stochastic models containing both a standard Brownian motion and fBm gained a lot of attention. The main reason for this is that they allow us to model systems driven by white noise and fractional Gaussian noise. For more details on this topic see [18–21].

The controllability problem is one of the most attractive and important problems of differential equations because of their great significance and applications in physics, population dynamics, engineering, mathematical biology, and other areas of science. Controllability for stochastic models only containing standard Brownian motion has been investigated very well (see [7, 12, 16, 22]). Recently, controllability for stochastic systems only driven by fBm has gained a lot of attention; we refer to [6, 9, 11]. Up to now, there is no paper which considers the controllability for stochastic systems driven by standard Brownian motion and fBm.

In this paper, we study the existence, uniqueness, and controllability results for a class of neutral stochastic delay partial differential equations (NSDPDEs for short) driven by a standard Brownian motion and an fBm in the abstract formwhere is the infinitesimal generator of an analytic semigroup of bounded linear operators in a real separable Hilbert space . The delay functions are continuous, and , , and are Borel measurable satisfying appropriate conditions. The control mapping takes values in , and the Hilbert space of admissible control mappings for a real separable Hilbert , is a bounded linear operator from to . denotes a standard Brownian motion in a real separable Hilbert space . denotes a fractional Brownian motion in a real separable Hilbert space with Hurst parameter . Let be the space of all -Hilbert-Schmidt operators from into .

Recently, Boufoussi and Hajji in [8] considered the existence and uniqueness problems of a class of neutral stochastic delay differential equations; that is, and in (1) by means of the Banach fixed point theory. Very recently, Liu and Luo in [13] studied the existence and uniqueness problems of a wide class of neutral stochastic delay partial differential equations, that is, in (1) by means of the Banach fixed point theory. However, they all required which is a stronger condition than ours (see Section 3).

In this paper, based on the above papers, we also use the Banach fixed point theory to consider a class of neutral stochastic delay partial differential equations, which is more general than ones in [8, 13]. Some conditions on the existence, uniqueness, and controllability of (1) are obtained. The related known results in Boufoussi and Hajji [8] and Liu and Luo in [13] are improved and generalized.

The contents of this paper are as follows. In Section 2, some notions, definitions, and lemmas which will be needed throughout this paper are introduced. In Section 3, the existence and uniqueness of mild solutions of (1) are proved. In Section 4, the controllability of (1) is investigated by means of the Banach fixed point theory. In Section 5, an example is given to illustrate our main results. At last, in Section 6, our conclusion is presented.

2. Preliminaries

Let be a complete probability space equipped with a normal filtration satisfying standard assumptions; that is, the filtration is right continuous and contains all -null sets. Let ; be the family of all continuous functions from into . Let and , be three real separable Hilbert spaces. We denote by the space of all bounded linear operators from to , . We suppose that is a complete orthonormal basis in . Let be an operator defined by with finite trace , where are nonnegative real numbers. Then there exists a real-valued sequence of one-dimensional standard Brownian motions mutually independent on such that Consider that a -valued stochastic process is defined by the formal infinite sum (see [2]). where the sequence is mutually independent scalar fBms with Hurst parameter .

Let be the space of all -Hilbert-Schmidt operators from to . Now we show the following definitions.

Definition 1. Let and define If , then is called a -Hilbert-Schmidt operator and the space is a real separable Hilbert space equipped with the inner product .

Let , be a Wiener process and be the one-dimensional fBm with Hurst parameter . has the following stochastic integral representation: is the kernel defined as for , where , and denotes the Beta function. Set for . Let denote the reproducing kernel Hilbert space of the fBm. In fact is the closure of the set of indicator functions with respect to the inner product. Then the mapping can be extended to an isometry between and the first Wiener chaos of the fBm . The image of an element by this isometry is called the Wiener integral of with respect to (see [23]).

We recall that for their inner product in is given by is an operator from to defined asThen and is an isometry between the set of indicator functions and that can be extended to (see [1]).

Let for be a mapping with values in . The stochastic integral of with respect to is defined bywhere is the standard Brownian motion which was introduced in (5).

In order to set existence, uniqueness, and controllability problems, we need the following lemmas which are Lemma 2 in [2] and Lemma 7.7 in [24], respectively.

Lemma 2 (see [2]). For any and , if and the series is uniformly convergent for , then one has where and .

Lemma 3 (see [24]). For any and for arbitrary predictable process , one has

We assume that , where is the resolvent set of and the analytic semigroup in is uniformly bounded; that is, for some positive constant . Then, for , it is possible under some circumstance to define the fractional power as a closed linear operator with domain . Moreover, the subspace is dense in and the expression defines a norm on .

Let denote the Banach space equipped with the norm , then the following properties are well known (see [25]).

Lemma 4 (see [25]). Suppose that the preceding conditions are satisfied.(1)If , then the injection is continuous.(2)For every , there exists such that

3. Existence and Uniqueness of Mild Solution

In this section, the Banach fixed point theory is used to investigate the existence and uniqueness problems of (1). Now, we introduce the definition of mild solution of (1).

Definition 5. An -valued stochastic process is called a mild solution of (1) if and for and for satisfies

In order to set the existence and uniqueness problems, we assume that the following conditions hold:(C.1) is the infinitesimal generator of an analytic semigroup of bounded linear operators in and there exists a positive constant such that for .(C.2)The mappings and satisfy the following conditions: for any and , there exist nonnegative constants , , , and , such that(i) ;(ii) .(C.3)The mapping satisfies the following conditions: for any and , there exist nonnegative constants and such that(i) ;(ii) ;(iii)the constants and satisfy the following inequality .(C.4)The mapping is continuous in mean square: for all , .(C.5)The mapping satisfies , .(C.6)The mapping takes values in and is a bounded linear operator from to .

Theorem 6. Let conditions (C.1)–(C.6) be satisfied. Then system (1) has a unique mild solution on .

Proof. Denote by subspace of all continuous functions form into equipped with the norm . Fix and consider the space , for . is a closed subspace of equipped with the norm . Obviously, and are two Banach spaces. Define an operator by for and for ,We first verify that is continuous in mean square on . Let , and let be positive and sufficiently small (similar estimates hold for ), then It is easy to know that as
Further, by using Cauchy-Schwarz inequality, we get Applying conditions (C.1) and (C.3) to , we can obtain as . Applying condition (C.1) and (C.3) to , we have as . That is to say, as .
For the fifth term, by using Lemma 3, we can obtain as
For the sixth term, we get Firstly, apply Lemma 2 to , and we have as . For every fixed, we get as and .
By Lemma 2 and the Lebesgue dominated convergence theorem, we can obtain as . Therefore as . Thus, is continuous in mean square on . So we conclude that .
Moreover, we shall show that is contractive in with some . Let and for any fixed , we get By conditions (C.1)–(C.4) and Lemma 3, we have Hence where By condition (C.3), we have , then there exists such that which shows that is a contraction mapping in ; therefore has a unique fixed point in , which is a unique mild solution of (1) on . Repeat this procedure to extend the mild solution to . The proof is completed.