International Journal of Mathematics and Mathematical SciencesVolume 2012 (2012), Article ID 748590, 12 pagesdoi:10.1155/2012/748590

Research Article

Existence and Uniqueness of Mild Solutions to Some Neutral Stochastic Partial Functional Integrodifferential Equations with Non-Lipschitz Coefficients

Mamadou Abdoul Diop,¹ Khalil Ezzinbi,² and Modou Lo¹

¹Département de Mathématiques, UFR SAT, Université Gaston Berger de Saint-Louis, P.O. Box 234, Saint-Louis, Senegal²Département de Mathématiques, Faculté des Sciences Semlalia, Université Cadi Ayyad, P.O. Box 2390 Marrakech, Morocco

Received 9 November 2011; Revised 17 January 2012; Accepted 25 January 2012

Academic Editor: Charles E. Chidume

Copyright © 2012 Mamadou Abdoul Diop 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 paper presents the result on existence, uniqueness of mild solutions to neutral stochastic partial functional integrodifferential equations under the Carathéodory-type conditions on the coefficients. The results are obtained by using the method of successive approximation. An example is provided to illustrate the results of this work.

1. Introduction

In this paper, our objective is to study the existence of mild solution for the following neutral stochastic partial functional integrodifferential equation in a real separable Hilbert space: where . The mappings , , and are Borel measurable.

Neutral stochastic partial functional differential equations have atracted great interest due to their applications in characterizing many problems in physics, biology, mechanics and so on.

Qualitative properties such as existence, uniqueness, and stability for various stochastic differential and integrodifferential systems have been extensively studied by many researchers; see, for instance, [1–6] and the references therein. The problem of the existence and uniqueness of solution for neutral stochastic partial functional differential equations in the case where the coefficients do not satisfy the global Lipschitz condition was investigated by Taniguchi [7], Turo [8], Cao et al. [9], and recently by Jiang and Shen [10].

Stimulated by the above works, we consider the existence and uniqueness of mild solutions to (1.1) under some carathéodory-type conditions to the Hilbert space with the Lipschitz condition in [11] and the non-Lipschitz condition in [12] being regarded as special cases.

Our main results concerning (1.1) rely essentially on techniques using strongly continuous family of operators, defined on the Hilbert spaceand called the resolvent (precise definition will be given below).

The contents of the paper are as follows. In Section 2, we summarize several important working tools on the Wiener process and deterministic integrodifferential equations that will be used to develop our results. In Section 3, we study the existence of mild solutions for the neutral system (1.1) using the theory of resolvent operators and by means of successive approximation (the Picard iteration). In Section 4, we provide an example to illustrate our main approach.

2. Wiener Process and Deterministic Integrodifferential Equations

2.1. Wiener Process

Throughout this paper, let be a complete probability space with a normal filtration satisfying the usual conditions (i.e., it is increasing and right-continuous while contains all -null sets). Moreover, let and be two real separable Hilbert spaces; we denote by, their inner products and by , their vectors norms, respectively. We denote by the space of all bounded linear operator from into , equipped with the usual operator norm . In this paper, we always use the same symbol to denote norms of operators regardless of the spaces potentially involved when no confusion possibly arises. Let and denote the family of all continuous -valued functions from to with norm . Let be the family of all almost surely bounded, -measurable,-valued random variables.

Let denote a -valued Wiener process defined on the probability space with covariance operator ; that is, , for all , where is a positive, self-adjoint, trace class operator on . In particular, we denote a -valued-Wiener process with respect to . To define stochastic integrals with respect to the -Wiener process, we introduce the subspace endowed with the inner product as a Hilbert space. We assume that there exists a complete orthonormal system in , a bounded sequence of nonnegative real numbers such that , , and a sequence of independent standard Brownian motions such that and , where is the -algebra generated by . Let be the space of all Hilbert-Schmidt operators from . It turns out to be a separable Hilbert space equipped with the norm for any . Obviously, for any bounded operator , this norm reduces to .

2.2. Partial Integrodifferential Equations

In this section, we recall some fundamental results needed to establish our results. Regarding the theory of resolvent operators, we refer the reader to [13, 14]. Throughout the paper, is a Banach space andare closed linear operators on . represents the Banach spaceequipped with the graph norm defined by The notations, stand for the space of all continuous functions frominto, the set of all bounded linear operators frominto, respectively. We consider the following Cauchy problem:

Definition 2.1 (see [13]). A resolvent operator for (2.3) is a bounded linear operator-valued function for , having the following properties. (i) and for some constants and . (ii)For each , is strongly continuous for . (iii) for . For , and In what follows we suppose the following assumptions.(H1) A is the infinitesimal generator of a strongly continuous semigroup on . (H2) For all, is closed linear operator from to and . For any, the mapis bounded and differentiable, and the derivative is bounded uniformly continuous on .

The resolvent operator plays an important role to study the existence of solutions and to give a variation of constants formula for nonlinear systems. We need to know when the linear system 2.1 has a resolvent operator. For more details on resolvent operators, we refer to [13, 14]. The following theorem gives a satisfactory answer to this problem, and it will be used in this work to develop our main results.

Theorem 2.2 (see [13]). Assume that (H1)-(H2) hold. Then there exists a unique resolvent operator of the Cauchy problem (2.3).

In the following, we give some results for the existence of solutions for the following integrodifferential equation: where is a continuous function.

Definition 2.3 (see [13]). A continuous function is said to be a strict solution of (2.5) if(i), (ii) satisfies (2.5), for .

Remark 2.4. From this definition, we deduce that , the functionis integrable, for all, and.

Theorem 2.5 (see [13]). Assume that (H1)-(H2) hold. If v is a strict solution of (2.5), then

Accordingly, we make the following definition.

Definition 2.6 (see [13]). For , a function is called a mild solution of (2.5) if satisfies (2.6).

The next theorem provides sufficient conditions for the regularity of solutions of (2.5).

Theorem 2.7 (see [13]). Let and be defined by (2.6). If , then is a strict solution of (2.5).

3. Existence of Mild Solutions of (1.1)

Definition 3.1. A process , , is called a mild solution of (1.1) if(i) is -adapted,with a.s.;(ii)has continuous paths on a.s., and, for each , satisfies the integral equation for any.

In the rest of this paper we replace by in (H1) and (H2).

To guarantee the existence and uniqueness of a mild solution to (1.1), the following much weaker conditions, instead of the non-Lipschitz condition, are used.(H3)(a) There exists a function such that is locally integrable in and is continuous monotone nondecreasing and concave in for any fixed. Moreover, for any fixed and , the following inequality is satisfied: (b) For any constant, the differential equation has a global solution for any initial value.(H4)(a) There exists a function such that is locally integrable in for any fixed and is continuous monotone nondecreasing and concave in for any fixed . for any fixed. Moreover, for any fixed and , the following inequality is satisfied: (b) For any constant, if a nonnegative function satisfies that then holds for any .(H5) The mapping satisfies that there exists a positive such that, for any and ,

Remark 3.2. Let, , where is locally integrable andis a concave nondecreasing function fromtosuch that, for and. Then, by the comparison theorem of differential equations we know that assumption (H4-(b)) holds.Now let us give some concrete examples of the function. Letand letbe sufficient small. Define wheredenotes the derivative of function. They are all concave nondecreasing functions satisfying.

In the following, we establish the existence and uniqueness theorem of the mild solution.

Theorem 3.3. If (H1)–(H5) hold, then there exists a unique mild solution to (1.1), provided that

Proof. To obtain the existence of the solution to (1.1), we consider the Picard iteration which is defined by andforis defined by The proof is divided into the following three steps.Step 1. We claim that the sequence is bounded. From (3.11), for, By (H5), where .Note that By (H3) and the Jensen inequality, we obtain By (H3), Liu and Hu [15, Theorem 1.2.5, page 14], and the Jensen inequality, there exists a positive constantsuch that Since is continuous monotone nondecreasing in, we obtain Recall that, for, , . Hence, substituting (3.13)−(3.16) into (3.12) yields Assumption (H3-(b)) indicates that there is a solutionthat satisfies where , .Since, from (3.17), we have , which shows the boundedness of the .Step 2. We claim that is a Cauchy sequence. For all and , from (3.11), (H4), and Step 1, we have where is a generic constant used by Liu and Hu [15,Theorem 1.2.6, page 14]. Therefore, Let By assumption (H4-(b)) and the Fatou lemma, we have where. By assumption (H4-(b)) we obtain. This shows that is Cauchy.Step 3. We claim the existence and uniqueness of the solution to (1.1). The Borel-Cantelli lemma shows that, as , holds uniformly for. Hence, taking limits on both sides of (3.11), we obtain that is a solution to (1.1). This shows the existence. The uniqueness of the solution could be obtained by the same procedure as Step 2. The proof is complete.

Remark 3.4. Iffor some constant, then condition (H4) implies a global Lipschitz condition.

4. Application

We conclude this work with an example of the form where denotes an -valued Brownian motion, are continuous functions,is continuous, and is a given continuous function such that is -measurable and satisfies .

Let with the norm and, denote the completed orthonormal basis in. Let, whereare one-dimensional standard Brownian motion mutually independent on a usual complete probability space.

Defineby, with domain.

Then, , where, , is also the orthonormal set of eigenvectors of. It is wellknown that is the infinitesimal generator of a strongly continuous semigroup on; thus, is true.

Letbe the operator defined by forand .

We suppose that (i)there exists a positive constant , , such that (ii)there exists a constant , , such that (iii)for and, .

Let and define the operators by If we put then (4.1) takes the following abstract form: Moreover, if is bounded and function such that is bounded and uniformly continuous, then (H1) and (H2) are satisfied, and hence, by Theorem 2.2, (2.3) has a resolvent operator on . Thus, all the assumptions of Theorem 3.3 are fulfilled. Therefore, the existence of a unique mild solution of (4.1) follows.

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