On Some Fractional Stochastic Integrodifferential Equations in Hilbert Space

We study a class of fractional stochastic integrodifferential equations considered in a real Hilbert space. The existence and uniqueness of the Mild solutions of the considered problem is also studied. We also give an application for stochastic integropartial differential equations of fractional order.


Introduction
Let H and K denote real Hilbert spaces equipped with norms • H and • K , respectively, and let the space of bounded linear operators from K to H be denoted by BL K; H .For Banach space X and Y , the space of continuous functions from X into Y equipped with the usual sup-norm will be denoted by C X; Y , while L p 0, T; X will represent the space of Xvalued functions that are p-integrable on 0, T .Let Ω, Z, P be a complete probability space equipped with a normal filtration {Z t : 0 ≤ t ≤ T }.An H-valued random variable is an Zmeasurable function X : Ω → H, and a collection of random variables ψ {X t; ω : Ω → H : 0 ≤ t ≤ T } is called a stochastic process.The collection of all strongly measurable square integrable H-valued random variables, denoted by An important subspace is given by where ξ α θ is a probability density function defined on 0, ∞ , ∞ 0 ξ α θ dθ 1 1.4 see 6-12 .In the next section, we will prove the existence and uniqueness of the mild solutions to 1.2 .

Existence and Uniqueness
Consider the initial value problem 1.2 in a real separable Hilbert space H under the following assumptions: International Journal of Mathematics and Mathematical Sciences 3 We can therefore state the following theorem.
Proof.Define the solution map J : γ 0, T ;

2.5
From Holder's inequality, we get where C F is a constant depending on α.

International Journal of Mathematics and Mathematical Sciences
Subsequently, an application of II , together with Minkowski's inequality enables us to continue the string of inequalities in 2.6 to conclude that

2.7
Taking the supermum over 0, T in 2.7 then implies that for any x ∈ γ 0, T ; H . Furthermore for such x, G x η ∈ BL K; H , and h x x 0 ∈ L 2 0 Ω; H by IV and V .Consequently, one can argue as in 13-15 to conclude that J is well defined.
Next we show that J is a strict contraction.
Observe that for x, y ∈ γ 0, T ; H , we infer from 2.5 that

2.9
Squaring both sides and taking the expectation in 2.9 yields, with the help of Young's inequality,

2.10
International Journal of Mathematics and Mathematical Sciences 5 and subsequently,

2.11
Using reasoning similar to that which led to 2.6 , one can show that

2.12
where C F depending on α and M F .We also infer that

6 International Journal of Mathematics and Mathematical Sciences
where C G is a constant depending on α and Tr Q .Using 2.12 and 2.13 in 2.11 enables us to conclude that J is a strict contraction, provided that 2.4 is satisfied, and has a unique fixed point which coincides with a mild solution of 1.2 .This completes the proof.

Application
Let D be a bounded domain in R N with smooth boundary ∂D, and consider the initial boundary value problem: H1 f 1 : 0, T × R × R → R satisfies the Caratheodory conditions as well as , for all x 1 , x 2 , y 1 , y 2 ∈ R and almost t ∈ 0, T for some M f 1 > 0, H2 f 2 : 0, T × R → BL L 2 D where BL L 2 D is the space of bounded linear operator from L 2 D to L 2 D satisfies the Caratheodory conditions as well as for all x, y ∈ R and almost all t ∈ 0, T , for some M f 2 > 0.
H3 f 3 : 0, T × R → R satisfies the Caratheodory conditions as well as

3.3
One can use H1 -H8 to verify that F, G, and h satisfy II -IV in the last section, respectively, with

3.4
Consequently theorem 2.4 can be applied for 3.1 .
In this paper we study the existence and uniqueness of the mild solution of the fractional stochastic integrodifferential equation of the form International Journal of Mathematics and Mathematical Sciences in a real separable Hilbert space H. Here, 1/2 ≤ α ≤ 1, A : D A ⊂ H → H is a linear closed operator generating semigroup, F : γ 0, T ; H → L T 0 ≤ t 1 < t 2 • • • < t n ≤ Tare given and W is an L 2 D -valued Wiener process.We consider the equation 3.1 under the following conditions.
The stochastic integropartial differential equation 3.1 can be written in the abstract form 1.2 , where KH L 2 D , A Δ z , with domain D A H 2 D ∪ H 1 0 D .It is well known that A is a closed linear operator which generates a C 0 -semigroup.We also introduce the mappings F, G, and h defined by, respectively, and almost t, s ∈ Y , H8 g i ∈ L 2 D , i 1, . . ., n.