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International Journal of Stochastic Analysis
Volume 2011 (2011), Article ID 784638, 17 pages
http://dx.doi.org/10.1155/2011/784638
Research Article

Existence Results for Stochastic Semilinear Differential Inclusions with Nonlocal Conditions

1Department of Mathematics and Computer Applications, PSG College of Technology, Coimbatore, Tamil Nadu 641 004, India
2Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, P.O. Box 5046, Dhabran 31261, Saudi Arabia

Received 31 May 2011; Accepted 6 October 2011

Academic Editor: Jiongmin Yong

Copyright © 2011 A. Vinodkumar and A. Boucherif. 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

We discuss existence results of mild solutions for stochastic differential inclusions subject to nonlocal conditions. We provide sufficient conditions in order to obtain a priori bounds on possible solutions of a one-parameter family of problems related to the original one. We, then, rely on fixed point theorems for multivalued operators to prove our main results.

1. Introduction

We investigate nonlocal stochastic differential inclusions (SDIns) of the form where , , are real numbers, is a single-valued function, and is multivalued map.

The importance of nonlocal conditions and their applications in different field have been discussed in [13]. Existence results for semilinear evolution equations with nonlocal conditions were investigated in [47], and the case of semilinear evolution inclusions with nonlocal conditions and a nonconvex right-hand side was discussed in [8].

Stochastic differential equations (SDEs) play a very important role in formulation and analysis in mechanical, electrical, control engineering and physical sciences, and economic and social sciences. See for instance [912] and the references therein. So far, very few articles have been devoted to the study of stochastic differential inclusions with nonlocal conditions, see [1315] and the references therein. Our objective is to contribute to the study of SDIns with nonlocal conditions. Motivated by the above-mentioned works and using the technique developed in [11, 16, 17], we study the SDIns of the form (1.1). The paper is organized as follows: some preliminaries are presented in Section 2. In Section 3, we investigate the existence of mild solutions for SDIns by using fixed point theorems for Kakutani maps. Finally in Section 4, we give an application to our abstract result.

2. Preliminaries

Let , be real separable Hilbert spaces and be the space of bounded linear operators mapping into . For convenience, we will use to denote inner product of and and to denote norms in , , and without any confusion.

Let be a complete filtered probability space such that contains all -null sets of . An -valued random variable is an -measurable function and the collection of random variables is called a stochastic process. Generally, we just write instead of and is the space of . Let be a complete orthonormal basis of . Suppose that is a cylindrical -valued Wiener process with finite trace nuclear covariance operator , denote , which satisfies . Actually, , where are mutually independent one-dimensional standard Wiener processes. We assume that is the -algebra generated by and . Let and define

If , then is called a -Hilbert-Schmidt operator. Let denote the space of all -Hilbert-Schmidt operators . The completion of with respect to the topology induced by the norm , where is a Hilbert space with the above norm topology.

We now make the system (1.1) precise. Let be the infinitesimal generator of a compact analytic semigroup defined on . Let denote the family of all right continuous functions with left-hand limit from to and is the family of all nonempty measurable subsets of . The functions ; are Borel measurable. The phase space is equipped with the norm . We denote by the family of all almost surely bounded, -measurable, -valued random variables. Further, let be the Banach space of all -adapted process which is almost surely continuous in for fixed , with norm for any . Here the expectation is defined by

We shall assume throughout the remainder of the paper that the initial function .

Some notions from set-valued analysis are in order. Denote by , , , , . A multivalued map is convex valued if for all , closed valued if for all , is compact valued if for all . is bounded on bounded sets if is bounded in , for all ; that is,

is called upper semicontinuous (u.s.c) on , if for each , the set is non-empty, closed subset of , and if for each open set of containing there exists an open neighborhood of such that .

is said to be completely continuous if is relatively compact, for every .

If the multivalued map is completely continuous with nonempty compact values, then is u.s.c if and only if has a closed graph (ie., ).

has a fixed point if there is such that . The fixed point set of the multivalued operator will be denoted by Fix .

The Hausdorff metric on is the function defined by where .

The multivalued map is said to be measurable if for each the function defined by

For more details on multivalued maps see [1820]. Our existence results are based on the following fixed point theorem (nonlinear alternative) for Kakutani maps [21].

Theorem 2.1. Let be a Hilbert space, a closed convex subset of , an open subset of and . Suppose that is an upper semicontinuous compact map. Then either (i) has a fixed point in or (ii) there are and with .

Definition 2.2. The multivalued map is said to be -Carathèodory if(i) is measurable for each ; (ii) is upper semicontinuous for almost all ;(iii)for each , there exists such that for all and for a.e. .

For each define the set of selections of by

Lemma 2.3 (see [22]). Let be a compact interval and be a Hilbert space. Let be an -Carathèodory multivalued map with and let be a linear continuous mapping from . Then the operator is a closed graph operator in .

Definition 2.4. A semigroup is said to be uniformly bounded if there exists a constant such that

Assume that Then there exists a bounded operator on given by the formula

Definition 2.5. A stochastic process is called a mild solution of system (1.1) if(i) is -adapted with almost surely;(ii) satisfies the integral equation where .

3. Existence Results

In this section, we discuss the existence of mild solutions of the system (1.1). We need the following hypotheses.:The function is continuous and there exist two positive constants such that is an -Carathéodory multivalued function with compact and convex values. :There exists a continuous nondecreasing function and such that

Theorem 3.1. Assume that hold. Then the system (1.1) has at least one mild solution on , provided that where

Proof. Transform the system (1.1) into a fixed point problem. Consider the multivalued operator defined by It is clear that the fixed points of are mild solutions of system (1.1). Hence we have to find solutions of the inclusion . We show that the multivalued operator satisfies all the conditions of Theorem 2.1. The proof will be given in several steps.Step 1. is convex for each . Since has convex values it follows that is convex; so that if then , which implies clearly that is convex.Step 2. The operator is bounded on bounded subsets of . For let be a bounded subset of . We show that is a bounded subset of . For each let . Then there exists such that for each we have Hence for each , we get Then, for each , we have , where .Step 3. sends bounded sets into equicontinuous sets in . For each let be given by (3.6). Let with . Then This implies that It follows that Since there is such that (see [23, proposition  1]) and the compactness of for implies the continuity in the uniform operator topology, we have Therefore When we have so that, similar to the previous situation, we have Step 4. sends bounded sets into relatively compact sets in . Let , for . For define a function by where . Since is a compact operator, the set is relatively compact in for every in . Moreover, for every we have Since and meas it follows that As a consequence of Step 1 through Step 4, together with Ascoli-Arzela theorem, we can conclude that the multivalued operator is compact.Step 5. has a closed graph. Let and with . We shall show that .
There exists such that We must prove that there exists such that Consider the linear continuous operator defined by Clearly, is linear and continuous. Indeed, one has Let We have Since is continuous (see ()) Lemma 2.3 implies that has a closed graph. Hence there exists such that Hence , which shows that graph is closed.
Step 6. Let and let . Then there exists such that Thus Conditions imply that for each The function defined on by satisfies This yields Since it follows that Therefore Now, by (3.3) there exists such that Let . Suppose that there is such that for . Then satisfies (3.36), which contradicts (3.37). So, alternative (ii) in Theorem 2.1. does not hold, and consequently, the multivalued operator has a fixed point, which is a solution of (1.1).

We now present another existence result for system (1.1). We shall assume that the single-valued and the multivalued satisfy a Wintner-type growth condition with respect to their second variable.

Theorem 3.2. Assume that and the following condition hold.:There exists such that then the system (1.1) has at least one mild solution on .

Remark 3.3. .

Proof. The multivalued operator defined in the proof of the previous theorem is completely continuous and upper semicontinuous. Now, we prove that is bounded. Let . Then there exists such that for each for some . Then Thus where Using the function , defined by (3.31), we obtain Gronwall’s inequality gives Therefore there exists such that which implies that This shows that is bounded. Theorem 2.1. shows that has a fixed point, which is a solution of (1.1), and this completes the proof.

4. Example

Consider the following stochastic partial differential inclusion with infinite delay where , and are positive constants, , is an open bounded set in with a smooth boundary is a positive function, stands for a standard cylindrical Wiener process in defined on a stochastic basis , and .

The coefficients are symmetric and satisfy the ellipticity condition for a positive constant .

In order to rewrite (4.1) in the abstract form, we introduce and we define the linear operator by

Here is the Sobolev space of functions with distributional derivative , and .

Then generates a symmetric compact analytic semigroup in , and there exists a constant such that . Also, note that there exists a complete orthonormal set , of eigenvectors of with .

We assume the following conditions hold.(i)The function is continuous in with (ii)The function is continuous in with (iii)The multifunction is an -Carathèodory multivalued function with compact and convex values and where is continuous and nondecreasing.

Assuming that conditions are verified, then the problem (4.1) can be modeled as the abstract stochastic partial functional differential inclusions of the form (1.1), with

The next result is a consequence of Theorem 3.1.

Proposition 4.1. Assume that the conditions hold. Then there exists at least one mild solution for the system (4.1) provided that where .

Proof. Condition (i) implies that holds with and .   and follow from conditions (ii) and (iii) with and .

Acknowledgments

A. Boucherif is grateful to King Fahd University of Petroleum and Minerals for its constant support. The authors sincerely thank the anonymous reviewer for his/her constructive comments and suggestions to improve the quality of the paper.

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