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.
Let and denote real Hilbert spaces equipped with norms and respectively, and let the space of bounded linear operators from to be denoted by . For Banach space and , the space of continuous functions from into (equipped with the usual sup-norm) will be denoted by , while will represent the space of -valued functions that are -integrable on . Let be a complete probability space equipped with a normal filtration . An -valued random variable is an -measurable function , and a collection of random variables is called a stochastic process. The collection of all strongly measurable square integrable -valued random variables, denoted by , is a Banach space equipped with norm .
An important subspace is given by . Next we define the space to be the set with norm (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 in a real separable Hilbert space . Here, is a linear closed operator generating semigroup, (where is a real separable Hilbert space), is a -valued Wiener process with incremental covariance described by the nuclear operator , is an -measurable -valued random variable independent of and
Definition 1.1. An stochastic process is called a mild solution of (1.2) if is measurable, for all where is a probability density function defined on , (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 under the following assumptions:(I)the linear operator generates a on H;(II) is such that there exists for which (III) is such that there exists for which (IV) is such that there exists for which (V)
We can therefore state the following theorem.
Theorem 2.1. Assume that (I)–(V) hold. Then (1.2) has a unique solution on , provided that where , , and .
Proof. Define the solution map by
From Holder's inequality, we get
where is a constant depending on .
Subsequently, an application of (II), together with Minkowski's inequality enables us to continue the string of inequalities in (2.6) to conclude that Taking the supermum over in (2.7) then implies that for any Furthermore for such , and (by (IV) and (V). Consequently, one can argue as in [13–15] to conclude that is well defined.
Next we show that is a strict contraction.
Observe that for we infer from (2.5) that Squaring both sides and taking the expectation in (2.9) yields, with the help of Young's inequality, and subsequently, Using reasoning similar to that which led to (2.6), one can show that where depending on and . We also infer that where is a constant depending on ( and Tr(). Using (2.12) and (2.13) in (2.11) enables us to conclude that 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.
Let be a bounded domain in with smooth boundary , and consider the initial boundary value problem: where are given and is an -valued Wiener process. We consider the equation (3.1) under the following conditions.(H1) satisfies the Caratheodory conditions as well as(i),(ii), for all and almost for some ,(H2) where is the space of bounded linear operator from to satisfies the Caratheodory conditions as well as(i)(ii), for all and almost all , for some .(H3) satisfies the Caratheodory conditions as well as(i),(ii), for all and almost for some ,(H4)(H5)(H6)(H7), where , satisfies , for all , and almost ,(H8)
The stochastic integropartial differential equation (3.1) can be written in the abstract form (1.2), where , , with domain . It is well known that is a closed linear operator which generates a . We also introduce the mappings , , and defined by, respectively, One can use (H1)–(H8) to verify that , , and satisfy (II)–(IV) in the last section, respectively, with Consequently theorem (2.4) can be applied for (3.1).
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