IJMMSInternational Journal of Mathematics and Mathematical Sciences1687-04250161-1712Hindawi Publishing Corporation56807810.1155/2009/568078568078Research ArticleOn Some Fractional Stochastic Integrodifferential Equations in Hilbert SpaceAhmedHamdy M.RosalskyAndrewHigher Institute of EngineeringEl-Shrouk AcademyP.O. 3 El-Shorouk CityCairoEgyptelshoroukacademy.edu.eg200914072009200908042009060720092009Copyright © 2009This 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.

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.

1. 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 Lp(0,T;X) will represent the space of X-valued functions that are p-integrable on [0,T]. Let (Ω,Z,P) be a complete probability space equipped with a normal filtration {Zt:0tT}. An H-valued random variable is an Z-measurable function X:ΩH, and a collection of random variables ψ={X(t;ω):ΩH:0tT} is called a stochastic process. The collection of all strongly measurable square integrable H-valued random variables, denoted by L2(Ω;H), is a Banach space equipped with norm X(·)L2(Ω;H)=(EX(·;ω)H2)1/2.

An important subspace is given by L02(Ω;H)={fL2(Ω;H):fisZ0measurable}. Next we define the space γ((0,T);H) to be the set {vC([0,T];L2(Ω;H):visZt-adapted} with norm vγ=sup0tT(Ev(t)H2)1/2 (see in ). In this paper we study the existence and uniqueness of the mild solution of the fractional stochastic integrodifferential equation of the form dαx(t)dtα=Ax(t)+F(x)(t)+0tG(x)(s)dW(s),0tT,x(0)=h(x)+x0, in a real separable Hilbert space H. Here, 1/2α1,A:D(A)HH is a linear closed operator generating semigroup, F:γ([0,T];H)Lp([0,T];L2(Ω;H))(1p<),G:γ([0,T];H)C([0,T];L2(Ω;BL(K;H))) (where K is a real separable Hilbert space), W is a K-valued Wiener process with incremental covariance described by the nuclear operator Q, x0 is an Z0-measurable H-valued random variable independent of W and h:γ([0,T];H)L02(Ω;H).

Definition 1.1.

An Zt-adapted stochastic process x:[0,T]H is called a mild solution of (1.2) if x(t) is measurable, for all t[0,T],0Tx(s)H2ds<,x(t)=0ξα(θ)S(tαθ)(h(x)+x0)dθ+α0t0θ(t-η)α-1ξα(θ)S((t-η)αθ)F(x)(η)dθdη+α0t0θ(t-η)α-1ξα(θ)S((t-η)αθ)[0ηG(x)(τ)dW(τ)]dθdη,0tT, where ξα(θ) is a probability density function defined on (0,), 0ξα(θ)dθ=1 (see ). In the next section, we will prove the existence and uniqueness of the mild solutions to (1.2).

2. Existence and Uniqueness

Consider the initial value problem (1.2) in a real separable Hilbert space H under the following assumptions:

the linear operator A:D(A)HH generates a C0-semigroup{S(t):t0} on H;

F:γ([0,T];H)Lp(0,T;L2(Ω;H)) is such that there exists MF>0 for which F(x)-F(y)LpMFx-yγ,x,yγ([0,T];H);

G:γ([0,T];H)C([0,T];L2(Ω;BL(K;H)))(=γBL) is such that there exists MG>0 for which G(x)-G(y)γBLMGx-yγx,yγ([0,T];H);

h:γ([0,T];H)L02(Ω;H) is such that there exists Mh>0 for which h(x)-h(y)L02Mhx-yγx,yγ([0,T];H);

x0L02(Ω;H).

We can therefore state the following theorem.

Theorem 2.1.

Assume that (I)–(V) hold. Then (1.2) has a unique solution on [0,T], provided that MS[Mh+CFTα+MGCGTα+1/2]<1, where Mh>0, MS>0, and CG>0.

Proof.

Define the solution map J:γ([0,T];H)γ([0,T];H) by (Jx)(t)=0ξα(θ)S(tαθ)(h(x)+x0)dθ+α0t0θ(t-η)α-1ξα(θ)S((t-η)αθ)F(x)(η)dθdη+α0t0θ(t-η)α-1ξα(θ)S((t-η)αθ)×[0ηG(x)(τ)dW(τ)]dθdτ,0tT. From Holder's inequality, we get [E0t0θ(t-η)α-1ξα(θ)S((t-η)αθ)F(x)(η)dθdηH2]1/2MS[0T(T-η)α-1F(x)(η)L2(Ω;H)2dη]1/2MS[0T(T-η)2(α-1)dη]1/2[0TF(x)(η)L2(Ω;H)2dη]1/2MSTα-1/2(2α-1)1/2[0TF(x)(η)L2(Ω;H)2dη]1/2CFMSTα-1/2F(x)Lp, where CF is a constant depending on α.

Subsequently, an application of (II), together with Minkowski's inequality enables us to continue the string of inequalities in (2.6) to conclude that [E0t0θ(t-η)α-1ξα(θ)S((t-η)αθ)F(x)(η)dθdηH2]1/2MSCFTα-1/2[MFxγ+F(0)Lp]. Taking the supermum over [0,T] in (2.7) then implies that 0t0θ(t-η)α-1ξα(θ)S((t-η)αθ)F(x)(η)dθdηγ([0,T];H), for any xγ([0,T];H). Furthermore for such x,G(x)(η)BL(K;H), and h(x)+x0L02(Ω;H) (by (IV) and (V). Consequently, one can argue as in  to conclude that J is well defined.

Next we show that J is a strict contraction.

Observe that for x,yγ([0,T];H), we infer from (2.5) that (Jx)(t)-(Jy)(t)=0ξα(θ)S(tαθ)(h(x)-h(y))dθ+α0t0θ(t-η)α-1ξα(θ)S((t-η)αθ)(F(x)(η)-F(y)(η))dθdη+α0t0θ(t-η)α-1ξα(θ)S((t-η)αθ)[0η(G(x)(τ)-G(y(τ)))dW(τ)]dθdη,0tT. Squaring both sides and taking the expectation in (2.9) yields, with the help of Young's inequality, E(Jx)(t)-(Jy)(t)H24E0ξα(θ)S(tαθ)(h(x)+x0)dθH2+4α2[E0t0θ(t-η)α-1ξα(θ)S((t-η)αθ)(F(x)(η)-F(y)(η))dθdηH2+E0t0θ(t-η)α-1ξα(θ)S((t-η)αθ)×[0η(G(x)(τ)-G(y)(τ))dW(τ)]dθdηH2], and subsequently, (Jx)(t)-(Jy)(t)γ0ξα(θ)S(tαθ)(h(x)-h(y))dθγ+4α2[0t0θ(t-η)α-1ξα(θ)S((t-η)αθ)(F(x)(η)-F(y)(η))dθdηγ+0t0θ(t-η)α-1ξα(θ)S((t-η)αθ)×[0η(G(x)(τ)-G(y)(τ))dW(τ)]dθdηγ]. Using reasoning similar to that which led to (2.6), one can show that 0ξα(θ)S(tαθ)(h(x)-h(y))dθγ=E0ξα(θ)S(tαθ)(h(x)+x0)dθH2MsMhx-yγ,0t0θ(t-η)α-1ξα(θ)S((t-η)αθ)(F(x)(η)-F(y)(η))dθdηγ=0t0θ(t-η)α-1ξα(θ)S((t-η)αθ)(F(x)(η)-F(y)(η))dθdηγCFMSTαx-yγ, where CF depending on α and MF. We also infer that 0t0θ(t-η)α-1ξα(θ)S((t-η)αθ)[0η(G(x)(τ)-G(y)(τ))dW(τ)]dθdηγ=[E0t0θ(t-η)α-1ξα(θ)S((t-η)αθ)[0η(G(x)(τ)-G(y)(τ))dW(τ)]dθdηH2]1/2Tr(Q)Tα-1/2(2α-1)1/2MS[0T0TG(x)(τ)-G(y)(τ)L2(Ω;H)2dτdη]1/2CGTα+1/2MsMGx-yγ, where CG 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.

3. Application

Let D be a bounded domain in RN with smooth boundary D, and consider the initial boundary value problem: α(t,z)tα=Δzx(t,z)+0Ta(t,s)f1(s,x(s,z),0sk(s,τ,x(τ,z))dτ)ds+0Tb(t,s)f2(s,x(s,z))dW(s),on(0,T)×D,x(0,z)=i=1ngi(z)x(ti,z)+0Tc(s)f3(s,x(s,z))ds,onD,x(t,z)=0,on(0,T)×D, where 0t1<t2<tnT are given and W is an L2(D)-valued Wiener process. We consider the equation (3.1) under the following conditions.

f1:[0,T]×R×RR satisfies the Caratheodory conditions as well as

f1(·,0,0)L2(0,T),

|f1(t,x1,y1)-f1(t,x2,y2)|Mf1[|x1-x2|+|y1-y2|], for all x1,x2,y1,y2R and almost t(0,T) for some Mf1>0,

f2:[0,T]×RBL(L2(D)) where BL(L2(D)) is the space of bounded linear operator from L2(D) to L2(D) satisfies the Caratheodory conditions as well as

f2(·,0)L2(0,T),

|f2(t,x)-f2(t,y)|BL(H)Mf2|x-y|, for all x,yR and almost all t(0,T), for some Mf2>0.

f3:[0,T]×RR satisfies the Caratheodory conditions as well as

f3(·,0)L2(0,T),

|f3(t,x)-f3(t,y)|Mf3|x-y|, for all x,yR and almost t(0,T) for some Mf3>0,

aL2((0,T)2),

bL((0,T)2),

cL2((0,T)2),

k:Y×RR, where Y={(t,s):0<s<t<T}, satisfies |k(t,s,x1)-k(t,s,x2)|Mk|x1-x2|, for all x1,x2R, and almost (t,s)Y,

giL2(D),i=1,,n.

The stochastic integropartial differential equation (3.1) can be written in the abstract form (1.2), where K=H=L2(D), A=Δz, with domain D(A)=H2(D)H01((D)). It is well known that A is a closed linear operator which generates a C0-semigroup. We also introduce the mappings F, G, and h defined by, respectively, F(x)(t,·)=0Ta(t,s)f1(s,x(s,z),0sk(s,τ,x(τ,z))dτ)ds,G(x)(t,·)=b(t,s)f2(s,x(s,·)),h(x)(·)=x(0,z)=i=1ngi(·)x(ti,·)+0Tc(s)f3(s,x(s,·))ds. One can use (H1)–(H8) to verify that F, G, and h satisfy (II)–(IV) in the last section, respectively, with MF=2Mf1T|a|L2((0,T)2)(1+MkT3)1/2,MG=Mf2,Mh=2i=1ngiL2(D)+Mf3m(D)|G|L2(0,T). Consequently theorem (2.4) can be applied for (3.1).

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