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:0≤t≤T}. An H-valued random variable is an Z-measurable 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 L2(Ω;H), is a Banach space equipped with norm ∥X(·)∥L2(Ω;H)=(E∥X(·;ω)∥H2)1/2.

An important subspace is given by L02(Ω;H)={f∈L2(Ω;H):fisZ0measurable}. Next we define the space γ((0,T);H) to be the set {v∈C([0,T];L2(Ω;H):visZt-adapted} with norm
∥v∥γ=sup0≤t≤T(E∥v(t)∥H2)1/2
(see in [1–5]). 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),0≤t≤T,x(0)=h(x)+x0,
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)→Lp([0,T];L2(Ω;H))(1≤p<∞),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],∫0T∥x(s)∥H2ds<∞,x(t)=∫0∞ξα(θ)S(tαθ)(h(x)+x0)dθ+α∫0t∫0∞θ(t-η)α-1ξα(θ)S((t-η)αθ)F(x)(η)dθdη+α∫0t∫0∞θ(t-η)α-1ξα(θ)S((t-η)αθ)[∫0ηG(x)(τ)dW(τ)]dθdη,0≤t≤T,
where ξα(θ) is a probability density function defined on (0,∞),
∫0∞ξα(θ)dθ=1
(see [6–12]). 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)⊂H→H generates a C0-semigroup{S(t):t≥0} 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)∥Lp≤MF∥x-y∥γ,∀x,y∈γ([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)∥γBL≤MG∥x-y∥γ∀x,y∈γ([0,T];H);

h:γ([0,T];H)→L02(Ω;H) is such that there exists Mh>0 for which
∥h(x)-h(y)∥L02≤Mh∥x-y∥γ∀x,y∈γ([0,T];H);

x0∈L02(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)=∫0∞ξα(θ)S(tαθ)(h(x)+x0)dθ+α∫0t∫0∞θ(t-η)α-1ξα(θ)S((t-η)αθ)F(x)(η)dθdη+α∫0t∫0∞θ(t-η)α-1ξα(θ)S((t-η)αθ)×[∫0ηG(x)(τ)dW(τ)]dθdτ,0≤t≤T.
From Holder's inequality, we get
[E∥∫0t∫0∞θ(t-η)α-1ξα(θ)S((t-η)αθ)F(x)(η)dθdη∥H2]1/2≤MS[∫0T∥(T-η)α-1F(x)(η)∥L2(Ω;H)2dη]1/2≤MS[∫0T(T-η)2(α-1)dη]1/2[∫0T∥F(x)(η)∥L2(Ω;H)2dη]1/2≤MSTα-1/2(2α-1)1/2[∫0T∥F(x)(η)∥L2(Ω;H)2dη]1/2≤CFMSTα-1/2∥F(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
[E∥∫0t∫0∞θ(t-η)α-1ξα(θ)S((t-η)αθ)F(x)(η)dθdη∥H2]1/2≤MSCFTα-1/2[MF∥x∥γ+∥F(0)∥Lp].
Taking the supermum over [0,T] in (2.7) then implies that
∫0t∫0∞θ(t-η)α-1ξα(θ)S((t-η)αθ)F(x)(η)dθdη∈γ([0,T];H),
for any x∈γ([0,T];H). Furthermore for such x,G(x)(η)∈BL(K;H), and h(x)+x0∈L02(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
(Jx)(t)-(Jy)(t)=∫0∞ξα(θ)S(tαθ)(h(x)-h(y))dθ+α∫0t∫0∞θ(t-η)α-1ξα(θ)S((t-η)αθ)(F(x)(η)-F(y)(η))dθdη+α∫0t∫0∞θ(t-η)α-1ξα(θ)S((t-η)αθ)[∫0η(G(x)(τ)-G(y(τ)))dW(τ)]dθdη,0≤t≤T.
Squaring both sides and taking the expectation in (2.9) yields, with the help of Young's inequality,
E∥(Jx)(t)-(Jy)(t)∥H2≤4E∥∫0∞ξα(θ)S(tαθ)(h(x)+x0)dθ∥H2+4α2[E∥∫0t∫0∞θ(t-η)α-1ξα(θ)S((t-η)αθ)(F(x)(η)-F(y)(η))dθdη∥H2+E∥∫0t∫0∞θ(t-η)α-1ξα(θ)S((t-η)αθ)×[∫0η(G(x)(τ)-G(y)(τ))dW(τ)]dθdη∥H2],
and subsequently,
∥(Jx)(t)-(Jy)(t)∥γ≤∥∫0∞ξα(θ)S(tαθ)(h(x)-h(y))dθ∥γ+4α2[∥∫0t∫0∞θ(t-η)α-1ξα(θ)S((t-η)αθ)(F(x)(η)-F(y)(η))dθdη∥γ+∥∫0t∫0∞θ(t-η)α-1ξα(θ)S((t-η)αθ)×[∫0η(G(x)(τ)-G(y)(τ))dW(τ)]dθdη∥γ].
Using reasoning similar to that which led to (2.6), one can show that
∥∫0∞ξα(θ)S(tαθ)(h(x)-h(y))dθ∥γ=E∥∫0∞ξα(θ)S(tαθ)(h(x)+x0)dθ∥H2≤MsMh∥x-y∥γ,∥∫0t∫0∞θ(t-η)α-1ξα(θ)S((t-η)αθ)(F(x)(η)-F(y)(η))dθdη∥γ=∥∫0t∫0∞θ(t-η)α-1ξα(θ)S((t-η)αθ)(F(x)(η)-F(y)(η))dθdη∥γ≤CFMSTα∥x-y∥γ,
where CF depending on α and MF. We also infer that
∥∫0t∫0∞θ(t-η)α-1ξα(θ)S((t-η)αθ)[∫0η(G(x)(τ)-G(y)(τ))dW(τ)]dθdη∥γ=[E∥∫0t∫0∞θ(t-η)α-1ξα(θ)S((t-η)αθ)[∫0η(G(x)(τ)-G(y)(τ))dW(τ)]dθdη∥H2]1/2≤Tr(Q)Tα-1/2(2α-1)1/2MS[∫0T∫0T∥G(x)(τ)-G(y)(τ)∥L2(Ω;H)2dτdη]1/2≤CGTα+1/2MsMG∥x-y∥γ,
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 0≤t1<t2⋯<tn≤T 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×R→R 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,y2∈R and almost t∈(0,T) for some Mf1>0,

f2:[0,T]×R→BL(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,y∈R and almost all t∈(0,T), for some Mf2>0.

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

f3(·,0)∈L2(0,T),

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

a∈L2((0,T)2),

b∈L∞((0,T)2),

c∈L2((0,T)2),

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

gi∈L2(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=2∑i=1n∥gi∥L2(D)+Mf3m(D)|G|L2(0,T).
Consequently theorem (2.4) can be applied for (3.1).

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