ON NONDENSELY DEFINED SEMILINEAR STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS

where f : J × M̂2([−r,0],H)→H is a given function, A :D(A)⊂H →H is a nondensely defined closed linear operator onH , the functionw(t) is a Hilbert space Q-valued Wiener process, φ ∈ M̂2([−r,0],D(A)), 0 < r <∞, is a suitable initial random function independent of w(t), h : M̂2([−r,0],D(A))→ D(A), H a real separable Hilbert space with inner product 〈·,·〉 and norm | · |, and M̂2 is a class of H-valued stochastic processes that will be specified later (see Section 2). Here yt(·) represents the history of stochastic processes state from time t− r, up to the present time t. The nonlocal conditions were initiated by Byszewski. We refer the readers to [4] and the references cited therein for motivation regarding the nonlocal initial conditions. The nonlocal condition can be applied in physics


Introduction
This paper is concerned with the existence of integral solutions for initial value problems for first-order stochastic semilinear functional differential equations with nonlocal conditions in Hilbert spaces.More precisely in Section 3, we consider first-order stochastic semilinear functional differential equations of the form y (t) = Ay(t) + f t, y t dw(t) dt , t ∈ J := [0,b], where f : J × M 2 ([−r,0],H) → H is a given function, A : D(A) ⊂ H → H is a nondensely defined closed linear operator on H, the function w(t) is a Hilbert space Q-valued Wiener process, φ ∈ M 2 ([−r,0],D(A)), 0 < r < ∞, is a suitable initial random function independent of w(t), h : M 2 ([−r,0],D(A)) → D(A), H a real separable Hilbert space with inner product •, • and norm | • |, and M 2 is a class of H-valued stochastic processes that will be specified later (see Section 2).Here y t (•) represents the history of stochastic processes state from time t − r, up to the present time t.The nonlocal conditions were initiated by Byszewski.We refer the readers to [4] and the references cited therein for motivation regarding the nonlocal initial conditions.The nonlocal condition can be applied in physics 2 Stochastic functional differential equations with better effect than the classical initial condition y(0) = y 0 .For example, h t (y) may be given by where c i , i = 1,..., p, are given constants and 0 < t Random differential and integral equations play an important role in characterizing many social, physical, biological, and engineering problems; see, for instance, the monographs of Da Prato and Zabczyk [6] and Sobczyk [14].For example, a stochastic model for drug distribution in a biological system was described by Tsokos and Padgett [16] to be a closed system with a simplified heat, one organ or capillary bed, and recirculation of blood with a constant rate of flow, where the heart is considered as a mixing chamber of constant volume.The basic theory concerning stochastic differential equations can be found in the monographs of Bharucha-Reid [3], Da Prato and Zabczyk [6], and Tsokos and Padgett [16].For recent results, we refer to the papers of Liu [11], McKibben [12,13], and Taniguchi [15].
Recently, Balasubramaniam and Ntouyas [2] studied the semilinear stochastic evolution delay equations with nonlocal conditions, where A is a densely defined linear operator.Our goal here is to extend the results of Balasubramaniam and Ntouyas [2], where A is nondensely defined.These results can be seen as a contribution to the literature.

Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.
Let K be another real separable Hilbert space and let w(t), t ≥ 0, be a K-valued Wiener process with mean zero and covariance operator Q with trQ < ∞ (trQ denotes the trace of the operator Q) defined by E w(t),g w(s),h = (t ∧ s) Qg,h for every g,h ∈ K, ( where •, • denotes the inner product and E stands for integration with respect to probability measure P. Let L(K,H) denote the space of bounded linear operators from K into H.For g 1 ,g 2 ∈ L(K,H), we define g 1 ,g 2 = tr(g 1 Qg * 2 ), where g * 2 is the adjoint of the operator g 2 and Q is the nuclear operator associated with the Brownian motion, where Q ∈ L + n (K), the space of positive nuclear operator in K. Let L(K Q ,H) denote the completion of L(K,H) with respect to the topology induced by the norm (2. 2) It is easy to verify that M 2 , furnished with the norm topology as defined above, is a Banach space.White noise is usually regarded as informal time derivative w (t) of Brownian motion or Wiener process w(t).In the Itô theory of stochastic integration, an integral with respect to w (t) is rewritten as one with respect to 3) The Itô integral b a ψ(t)dw(t) is defined for any process ψ(t) which satisfies the following conditions: (1) In fact the following equality holds: For more details on Brownian motion and white noise, we refer the reader to the books of Hida [8] and Hida et al. [9].B(H) denotes the Banach space of bounded linear operators from H into H with norm Definition 2.1 (see [1]).Let E be a Banach space.An integrated semigroup is a family of operators (S(t)) t≥0 of bounded linear operators S(t) on E with the following properties: dr, for all t,s ≥ 0. Definition 2.2 (see [10]).An operator A is called a generator of an integrated semigroup if there exists ω ∈ R such that (ω,∞) ⊂ ρ(A) (ρ(A) is the resolvent set of A) and there exists a strongly continuous exponentially bounded family (S(t)) t≥0 of bounded operators such that S(0) = 0 and R(λ,A) := (λI − A) −1 = λ ∞ 0 e −λt S(t)dt exists for all λ with λ > ω.Proposition 2.3 (see [1]).Let A be the generator of an integrated semigroup (S(t)) t≥0 .Then for all x ∈ E and t ≥ 0, Definition 2.4 (see [10]).(i) An integrated semigroup (S(t)) t≥0 is called locally Lipschitz continuous if, for all τ > 0, there exists a constant L such that (ii) An integrated semigroup (S(t)) t≥0 is called nondegenerate if S(t)x = 0, for all t ≥ 0, implies that x = 0. Definition 2.5.We say that the linear operator A satisfies the Hille-Yosida condition if there exist M ≥ 0 and (2.8) Theorem 2.6 (see [10]).The following assertions are equivalent: (H0) A is the generator of a nondegenerate, locally Lipschitz continuous integrated semigroup; (H1) A satisfies the Hille-Yosida condition.
If A is the generator of an integrated semigroup (S(t)) t≥0 which is locally Lipschitz, then from [1], S(•)x is continuously differentiable if and only if (iii) for each q > 0, there exists h q ∈ L 1 (J,R + ) such that f (t,u) 2 ≤ h q (t) ∀ u 2 M2 ≤ q and for almost all t ∈ J. (2.9) In what follows, we will assume that f is an L 2 -Carathéodory function.

Main result
The aim of this section is to study the existence of integral solutions for the nonlocal problem (1.1).
Definition 3.1.For any H-valued Ᏺ 0 -measurable stochastic processes φ satisfying the con- From the definition it follows that y(t) ∈ D(A), t ≥ 0.Moreover, y satisfies the following variation of constant formula: We are now in a position to state and prove our existence result for the problem (1.1).
M. Benchohra et al. 5 Theorem 3.2.Assume (H1) and (H2) w is an H-valued Wiener process defined on Hilbert space K; (H3) S (t), t > 0, is compact and there exist M > 0, ω ∈ R such that (H4) the function h is continuous with respect to t and there exists a constant β > 0 such that and for each k > 0, the set  In order to use the Leray-Schauder alternative, we will obtain a priori estimates for the solutions of the integral equation and y(t Let us take the right-hand side of the above inequality as v(t).Then we have From (H5), there exists a constant K * such that e ωt v(t) ≤ K * , t ∈ [0,b], and there exists M * such that y M2 ≤ M * .
In the next steps, we will prove that N is continuous and completely continuous.
Step 1. N is continuous.
Let {y n } be a sequence such that Then (3.17) (3.18) Step 2. N maps bounded sets into bounded sets in Indeed, it is enough to show that there exists a positive constant such that for each 2Mb max e |ω|b ,1 h q L 2 := . (3.20) Step 3. N maps bounded sets into equicontinuous sets in The right-hand side tends to zero as t 2 − t 1 → 0. Now we will show that NᏮ q (t) is relatively compact for every t ∈ [0,b].In the case where t = 0, we have NᏮ q (0) = {φ(0) − h 0 (y)} which is precompact from (H4).Let 0 < t ≤ b and < t ≤ b.For y ∈ Ꮾ q , N (y)(t) = S (t) φ(0) − h 0 (y) + lim  Since S (t) is a compact operator, the set H (t) = {N (y)(t) : y ∈ Ꮾ q } is precompact in D(A) for every , 0 < < t.Moreover, for every y ∈ Ꮾ q , we have  Therefore, there are precompact sets arbitrarily close to the set {N (y)(t) : y ∈ Ꮾ q }.Hence the set {N (y)(t) : y ∈ Ꮾ q } is precompact in D(A).
10 Stochastic functional differential equations The cases when t 1 ,t 2 < 0 or t 1 < 0 < t 2 are obvious.Set From the choice of U, there is no y ∈ ∂U such that y = λN(y), for some λ ∈ (0,1).As a consequence of the nonlinear alternative of Leray-Schauder type [7], we deduce that N has a fixed point y in U which is an integral solution of the problem (1.1).
Remark 3.4.We can replace (H5) by the following condition.(H5) * There exists a continuous nondecreasing function ψ Then the step on a priori estimates will be modified as follows.
Let y be solution of the problem (1.1), then we have  and proceed as in Theorem 3.2.

An example
To apply the previous result, we consider the following partial stochastic differential equation:

. 6 ) 3 . 3 .
Remark It is clear that the fixed points of N are integral solutions to (1.1).

SS
(t − s)B λ f s, y s dw(s) (t − − s)B λ f s, y s dw(s).