Existence Results for Stochastic Semilinear Differential Inclusions with Nonlocal Conditions

We discuss existence results of mild solutions for stochastic di ﬀ erential inclusions subject to nonlocal conditions. We provide su ﬃ cient 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.


Introduction
We investigate nonlocal stochastic differential inclusions SDIns of the form dx t ∈ Ax t f t, x t dt G t, x t dw t , t ∈ J 0, T , where T > 0, 0 < t 1 < t 2 < · · · < t m < T, γ i are real numbers, f is a single-valued function, and G is multivalued map. The importance of nonlocal conditions and their applications in different field have been discussed in 1-3 . Existence results for semilinear evolution equations with nonlocal conditions were investigated in 4-7 , 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 9-12 and the references therein. So far, very few articles have been devoted to the study of stochastic differential inclusions with nonlocal conditions, see [13][14][15] 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.

Preliminaries
Let X, Y be real separable Hilbert spaces and L Y, X be the space of bounded linear operators mapping Y into X. For convenience, we will use ·, · to denote inner product of X and Y and · to denote norms in X, Y , and L Y, X without any confusion.
Let Ω, F, P; F F {F t } t≥0 be a complete filtered probability space such that F 0 contains all P -null sets of F. An X-valued random variable is an F-measurable function x t : Ω → X and the collection of random variables H {x t, ω : Ω → X : t ∈ J} is called a stochastic process. Generally, we just write x t instead of x t, ω and x t : i 1 are mutually independent one-dimensional standard Wiener processes. We assume that F t σ{w s : 0 ≤ s ≤ t} is the σ-algebra generated by w and F t F. Let μ ∈ L Y, X and define

2.1
If μ Q < ∞, then μ is called a Q-Hilbert-Schmidt operator. Let L Q Y, X denote the space of all Q-Hilbert-Schmidt operators μ : Y → X. The completion L Q Y, X of L Y, X with respect to the topology induced by the norm · Q , where μ 2 Q μ, μ is a Hilbert space with the above norm topology.
We now make the system 1.1 precise. Let A : X → X be the infinitesimal generator of a compact analytic semigroup {S t , t ≥ 0} defined on X. Let D τ D −∞, 0 , X denote the family of all right continuous functions with left-hand limit ϕ from −∞, 0 to X and P E is the family of all nonempty measurable subsets of E. The functions f : 0 , X the family of all almost surely bounded, F 0 -measurable, D τ -valued random variables. Further, let B T be the Banach space of all F t -adapted process φ t, w which is almost surely continuous in t for fixed w ∈ Ω, with norm International Journal of Stochastic Analysis 3 for any φ ∈ B T . Here the expectation E is defined by We shall assume throughout the remainder of the paper that the initial function ϕ ∈ D b F 0 −∞, 0 , X . Some notions from set-valued analysis are in order. Denote by P cl X {Y ∈ P X : F is said to be completely continuous if F V is relatively compact, for every V ∈ P bd X .
If the multivalued map F is completely continuous with nonempty compact values, then F is u.s.c if and only if F has a closed graph ie., x n → x * , y n → y * , y n ∈ F x n imply y * ∈ F x * .
F has a fixed point if there is x ∈ X such that x ∈ F x . The fixed point set of the multivalued operator F will be denoted by Fix F.
The Hausdorff metric on P bd,cl X is the function H : P bd,cl X × P bd,cl X → R defined by For more details on multivalued maps see 18-20 . Our existence results are based on the following fixed point theorem nonlinear alternative for Kakutani maps 21 .  International Journal of Stochastic Analysis for all u 2 B T ≤ q and for a.e. t ∈ J.
For each x ∈ L 2 L Q Y, X define the set of selections of G by

Lemma 2.3 see 22 .
Let I be a compact interval and X be a Hilbert space. Let G be an L 2 -Carathèodory multivalued map with S G,x / φ and let Γ be a linear continuous mapping from L 2 I, X → C I, X . Then the operator is a closed graph operator in C I, X × C I, X .
Then there exists a bounded operator B on D B X given by the formula

2.12
International Journal of Stochastic Analysis 5 ii x t satisfies the integral equation where g ∈ S G,x .

Existence Results
In this section, we discuss the existence of mild solutions of the system 1.1 . We need the following hypotheses.
Theorem 3.1. Assume that H 1 -H 3 hold. Then the system 1.1 has at least one mild solution on −∞, T , provided that International Journal of Stochastic Analysis Proof. Transform the system 1.1 into a fixed point problem. Consider the multivalued operator M : 3.5 It is clear that the fixed points of M are mild solutions of system 1.1 . Hence we have to find solutions of the inclusion y ∈ M y . We show that the multivalued operator M satisfies all the conditions of Theorem 2.1. The proof will be given in several steps.
Step 1. M x is convex for each x ∈ B T . Since G has convex values it follows that S G,x is convex; so that if g 1 , g 2 ∈ S G,x then αg 1 1 − α g 2 ∈ S G,x , which implies clearly that M x is convex.
Step 2. The operator M is bounded on bounded subsets of B T . For q > 0 let B q {x ∈ B T :

3.7
Hence for each h ∈ M B q , we get Step 3. M sends bounded sets into equicontinuous sets in B T . For each x ∈ B q let h ∈ M x be given by 3.6 . Let τ 1 , τ 2 ∈ J with 0 < τ 1 < τ 2 ≤ T . Then

3.11
Since there is δ > 0 such that see 23, proposition 1 and the compactness of S t for t > 0 implies the continuity in the uniform operator topology, we have

3.14
International Journal of Stochastic Analysis 9 When τ 1 0 we have 3.15 so that, similar to the previous situation, we have Step 4. M sends bounded sets into relatively compact sets in B T . Let 0 < < t, for t ∈ J. For ω q s ds.

3.18
Since ω q ∈ L 1 J and meas t − , t it follows that As a consequence of Step 1 through Step 4, together with Ascoli-Arzela theorem, we can conclude that the multivalued operator M is compact.
Step 5. M has a closed graph. Let x n → x * and h n ∈ M x n with h n → h * . We shall show that h * ∈ M x * . There exists g n ∈ S G,x n such that

3.20
International Journal of Stochastic Analysis We must prove that there exists g * ∈ S G,x * such that

3.21
Consider the linear continuous operator Γ : Clearly, Γ is linear and continuous. Indeed, one has

3.23
Let S t − s f s, x n,s ds, S t − s f s, x * s ds.

3.24
We have Since f is continuous see H 1 Lemma 2.3 implies that Γ • S G has a closed graph. Hence there exists g * ∈ S G,x * such that

3.27
Hence h * ∈ M x * , which shows that graph M is closed.
International Journal of Stochastic Analysis

11
Step 6. Let λ ∈ 0, 1 and let x ∈ λM x . Then there exists g ∈ S G,x such that

3.30
The function defined on 0, T by

3.32
This yields Since it follows that Now, by 3.3 there exists ρ 0 > 0 such that Then x B T 0 satisfies 3.36 , which contradicts 3.37 . So, alternative ii in Theorem 2.1. does not hold, and consequently, the multivalued operator M 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 f and the multivalued G satisfy a Wintner-type growth condition with respect to their second variable.

Theorem 3.2. Assume that H 2 and the following condition hold.
H fG : There exists ∈ L 1 0, T , R such that 3.38 then the system 1.1 has at least one mild solution on −∞, T .
Proof. The multivalued operator M defined in the proof of the previous theorem is completely continuous and upper semicontinuous. Now, we prove that for some λ ∈ 0, 1 . Then

3.43
Using the function t , defined by 3.31 , we obtain

Example
Consider the following stochastic partial differential inclusion with infinite delay where a 0 , r, and are positive constants, J 0, T , Δ is an open bounded set in R n with a smooth boundary ∂Δ, β 1 : −∞, 0 → R is a positive function, β t stands for a standard cylindrical Wiener process in L 2 Δ defined on a stochastic basis Ω, F, P , and The coefficients a ij ∈ L ∞ Δ are symmetric and satisfy the ellipticity condition Here H 1 Δ is the Sobolev space of functions u ∈ L 2 Δ with distributional derivative u ∈ L 2 Δ , H 1 0 Δ {u ∈ H 1 Δ ; u 0 on ∂Δ} and H 2 Δ {u ∈ L 2 Δ ; u , u ∈ L 2 Δ }. Then A generates a symmetric compact analytic semigroup e −tA in X, and there exists a constant M 1 > 0 such that e −tA ≤ M 1 . Also, note that there exists a complete orthonormal set {ξ n }, n 1, 2, . . . of eigenvectors of A with ξ n x 2/n sin nx . We assume the following conditions hold.
i The function β 1 · is continuous in J with iii The multifunction G 1 · is an L 2 -Carathèodory multivalued function with compact and convex values and where ψ 0 · : 0, ∞ → 0, ∞ is continuous and nondecreasing.
Assuming that conditions i -iii are verified, then the problem 4.1 can be modeled as the abstract stochastic partial functional differential inclusions of the form 1.1 , with

4.7
The next result is a consequence of Theorem 3.1. Proof. Condition i implies that H 1 holds with C 1 0 −∞ β 2 1 θ dθ and C 2 0. H 2 and H 3 follow from conditions ii and iii with p t Δ p 2 1 t, x dx 1/2 and ψ ψ 0 .