On Some Fractional Stochastic Integrodifferential Equations in Hilbert Space

Abstract

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 𝐿^𝑝(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 {𝑍_𝑡∶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)‖²_𝐻)^1/2.

An important subspace is given by 𝐿²₀(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)∶𝑣is𝑍_𝑡-adapted} with norm ‖𝑣‖_𝛾=sup_{0≤𝑡≤𝑇}(𝐸‖𝑣(𝑡)‖²_𝐻)^1/2 (1.1) (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 𝑑^𝛼𝑥(𝑡)𝑑𝑡^𝛼=𝐴𝑥(𝑡)+𝐹(𝑥)(𝑡)+^𝑡₀𝐺(𝑥)(𝑠)𝑑𝑊(𝑠),0≤𝑡≤𝑇,𝑥(0)=ℎ(𝑥)+𝑥₀, (1.2) 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^𝑝([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)→𝐿²₀(Ω;𝐻).

Definition 1.1. An 𝑍_𝑡-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],^𝑇₀‖𝑥(𝑠)‖²_𝐻𝑑𝑠<∞,𝑥(𝑡)=^∞₀𝜉_𝛼(𝜃)𝑆(𝑡^𝛼𝜃)(ℎ(𝑥)+𝑥₀)𝑑𝜃+𝛼^𝑡₀^∞₀𝜃(𝑡−𝜂)^𝛼−1𝜉_𝛼(𝜃)𝑆((𝑡−𝜂)^𝛼𝜃)𝐹(𝑥)(𝜂)𝑑𝜃𝑑𝜂+𝛼^𝑡₀^∞₀𝜃(𝑡−𝜂)^𝛼−1𝜉_𝛼(𝜃)𝑆((𝑡−𝜂)^𝛼𝜃)[^𝜂₀𝐺(𝑥)(𝜏)𝑑𝑊(𝜏)]𝑑𝜃𝑑𝜂,0≤𝑡≤𝑇, (1.3) where 𝜉_𝛼(𝜃) is a probability density function defined on (0,∞), ^∞₀𝜉_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).

2. Existence and Uniqueness

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

(I)the linear operator 𝐴∶𝐷(𝐴)⊂𝐻→𝐻 generates a 𝐶₀-semigroup{𝑆(𝑡)∶𝑡≥0} on H;(II)𝐹∶𝛾([0,𝑇];𝐻)→𝐿^𝑝(0,𝑇;𝐿²(Ω;𝐻)) is such that there exists 𝑀_𝐹>0 for which ‖𝐹(𝑥)−𝐹(𝑦)‖_𝐿𝑝≤𝑀_𝐹‖𝑥−𝑦‖_𝛾,∀𝑥,𝑦∈𝛾([0,𝑇];𝐻); (2.1)(III)𝐺∶𝛾([0,𝑇];𝐻)→𝐶([0,𝑇];𝐿²(Ω;BL(𝐾;𝐻)))(=𝛾_BL) is such that there exists 𝑀_𝐺>0 for which ‖𝐺(𝑥)−𝐺(𝑦)‖_𝛾BL≤𝑀_𝐺‖𝑥−𝑦‖_𝛾∀𝑥,𝑦∈𝛾([0,𝑇];𝐻); (2.2)(IV)ℎ∶𝛾([0,𝑇];𝐻)→𝐿²₀(Ω;𝐻) is such that there exists 𝑀_ℎ>0 for which ‖ℎ(𝑥)−ℎ(𝑦)‖_𝐿20≤𝑀_ℎ‖𝑥−𝑦‖_𝛾∀𝑥,𝑦∈𝛾([0,𝑇];𝐻); (2.3)(V)𝑥₀∈𝐿²₀(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 𝑀_𝑆[𝑀_ℎ+𝐶_𝐹𝑇^𝛼+𝑀_𝐺𝐶_𝐺𝑇^𝛼+1/2]<1, (2.4) where 𝑀_ℎ>0, 𝑀_𝑆>0, and 𝐶_𝐺>0.

Proof. Define the solution map 𝐽∶𝛾([0,𝑇];𝐻)→𝛾([0,𝑇];𝐻) by (𝐽𝑥)(𝑡)=^∞₀𝜉_𝛼(𝜃)𝑆(𝑡^𝛼𝜃)(ℎ(𝑥)+𝑥₀)𝑑𝜃+𝛼^𝑡₀^∞₀𝜃(𝑡−𝜂)^𝛼−1𝜉_𝛼(𝜃)𝑆((𝑡−𝜂)^𝛼𝜃)𝐹(𝑥)(𝜂)𝑑𝜃𝑑𝜂+𝛼^𝑡₀^∞₀𝜃(𝑡−𝜂)^𝛼−1𝜉_𝛼(𝜃)𝑆((𝑡−𝜂)^𝛼𝜃)×[^𝜂₀𝐺(𝑥)(𝜏)𝑑𝑊(𝜏)]𝑑𝜃𝑑𝜏,0≤𝑡≤𝑇. (2.5) From Holder's inequality, we get [𝐸‖^𝑡₀^∞₀𝜃(𝑡−𝜂)^𝛼−1𝜉_𝛼(𝜃)𝑆((𝑡−𝜂)^𝛼𝜃)𝐹(𝑥)(𝜂)𝑑𝜃𝑑𝜂‖²_𝐻]^1/2≤𝑀_𝑆[^𝑇₀‖(𝑇−𝜂)^𝛼−1𝐹(𝑥)(𝜂)‖²_{𝐿2(Ω;𝐻)}𝑑𝜂]^1/2≤𝑀_𝑆[^𝑇₀(𝑇−𝜂)^2(𝛼−1)𝑑𝜂]^1/2[^𝑇₀‖𝐹(𝑥)(𝜂)‖²_{𝐿2(Ω;𝐻)}𝑑𝜂]^1/2≤𝑀_𝑆𝑇^𝛼−1/2(2𝛼−1)^1/2[^𝑇₀‖𝐹(𝑥)(𝜂)‖²_{𝐿2(Ω;𝐻)}𝑑𝜂]^1/2≤𝐶_𝐹𝑀_𝑆𝑇^𝛼−1/2‖𝐹(𝑥)‖_𝐿^𝑝, (2.6) where 𝐶_𝐹 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‖^𝑡₀^∞₀𝜃(𝑡−𝜂)^𝛼−1𝜉_𝛼(𝜃)𝑆((𝑡−𝜂)^𝛼𝜃)𝐹(𝑥)(𝜂)𝑑𝜃𝑑𝜂‖²_𝐻]^1/2≤𝑀_𝑆𝐶_𝐹𝑇^𝛼−1/2[𝑀_𝐹‖𝑥‖_𝛾+‖𝐹(0)‖_𝐿^𝑝]. (2.7) Taking the supermum over [0,𝑇] in (2.7) then implies that ^𝑡₀^∞₀𝜃(𝑡−𝜂)^𝛼−1𝜉_𝛼(𝜃)𝑆((𝑡−𝜂)^𝛼𝜃)𝐹(𝑥)(𝜂)𝑑𝜃𝑑𝜂∈𝛾([0,𝑇];𝐻), (2.8) for any 𝑥∈𝛾([0,𝑇];𝐻). Furthermore for such 𝑥,𝐺(𝑥)(𝜂)∈BL(𝐾;𝐻), and ℎ(𝑥)+𝑥₀∈𝐿²₀(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)_∞0𝜉_𝛼(𝜃)𝑆(𝑡^𝛼𝜃)(ℎ(𝑥)−ℎ(𝑦))𝑑𝜃+𝛼^𝑡₀^∞₀𝜃(𝑡−𝜂)^𝛼−1𝜉_𝛼(𝜃)𝑆((𝑡−𝜂)^𝛼𝜃)(𝐹(𝑥)(𝜂)−𝐹(𝑦)(𝜂))𝑑𝜃𝑑𝜂+𝛼^𝑡₀^∞₀𝜃(𝑡−𝜂)^𝛼−1𝜉_𝛼(𝜃)𝑆((𝑡−𝜂)^𝛼𝜃)[^𝜂₀(𝐺(𝑥)(𝜏)−𝐺(𝑦(𝜏)))𝑑𝑊(𝜏)]𝑑𝜃𝑑𝜂,0≤𝑡≤𝑇. (2.9) Squaring both sides and taking the expectation in (2.9) yields, with the help of Young's inequality, 𝐸‖(𝐽𝑥)(𝑡)−(𝐽𝑦)(𝑡)‖²_𝐻≤4𝐸‖^∞₀𝜉_𝛼(𝜃)𝑆(𝑡^𝛼𝜃)(ℎ(𝑥)+𝑥₀)𝑑𝜃‖²_𝐻+4𝛼²[𝐸‖^𝑡₀^∞₀𝜃(𝑡−𝜂)^𝛼−1𝜉_𝛼(𝜃)𝑆((𝑡−𝜂)^𝛼𝜃)(𝐹(𝑥)(𝜂)−𝐹(𝑦)(𝜂))𝑑𝜃𝑑𝜂‖²_𝐻+𝐸‖^𝑡₀^∞₀𝜃(𝑡−𝜂)^𝛼−1𝜉_𝛼(𝜃)𝑆((𝑡−𝜂)^𝛼𝜃)×[^𝜂₀(𝐺(𝑥)(𝜏)−𝐺(𝑦)(𝜏))𝑑𝑊(𝜏)]𝑑𝜃𝑑𝜂‖²_𝐻], (2.10) and subsequently, ‖(𝐽𝑥)(𝑡)−(𝐽𝑦)(𝑡)‖_𝛾≤‖^∞₀𝜉_𝛼(𝜃)𝑆(𝑡^𝛼𝜃)(ℎ(𝑥)−ℎ(𝑦))𝑑𝜃‖_𝛾+4𝛼²[‖^𝑡₀^∞₀𝜃(𝑡−𝜂)^𝛼−1𝜉_𝛼(𝜃)𝑆((𝑡−𝜂)^𝛼𝜃)(𝐹(𝑥)(𝜂)−𝐹(𝑦)(𝜂))𝑑𝜃𝑑𝜂‖_𝛾+‖^𝑡₀^∞₀𝜃(𝑡−𝜂)^𝛼−1𝜉_𝛼(𝜃)𝑆((𝑡−𝜂)^𝛼𝜃)×[^𝜂₀(𝐺(𝑥)(𝜏)−𝐺(𝑦)(𝜏))𝑑𝑊(𝜏)]𝑑𝜃𝑑𝜂‖_𝛾]. (2.11) Using reasoning similar to that which led to (2.6), one can show that ‖^∞₀𝜉_𝛼(𝜃)𝑆(𝑡^𝛼𝜃)(ℎ(𝑥)−ℎ(𝑦))𝑑𝜃‖_𝛾=𝐸‖^∞₀𝜉_𝛼(𝜃)𝑆(𝑡^𝛼𝜃)(ℎ(𝑥)+𝑥₀)𝑑𝜃‖²_𝐻≤𝑀_𝑠𝑀_ℎ‖𝑥−𝑦‖_𝛾,‖^𝑡₀^∞₀𝜃(𝑡−𝜂)^𝛼−1𝜉_𝛼(𝜃)𝑆((𝑡−𝜂)^𝛼𝜃)(𝐹(𝑥)(𝜂)−𝐹(𝑦)(𝜂))𝑑𝜃𝑑𝜂‖_𝛾=‖^𝑡₀^∞₀𝜃(𝑡−𝜂)^𝛼−1𝜉_𝛼(𝜃)𝑆((𝑡−𝜂)^𝛼𝜃)(𝐹(𝑥)(𝜂)−𝐹(𝑦)(𝜂))𝑑𝜃𝑑𝜂‖_𝛾≤𝐶_𝐹𝑀_𝑆𝑇^𝛼‖𝑥−𝑦‖_𝛾, (2.12) where 𝐶_𝐹 depending on 𝛼 and 𝑀_𝐹. We also infer that ‖^𝑡₀^∞₀𝜃(𝑡−𝜂)^𝛼−1𝜉_𝛼(𝜃)𝑆((𝑡−𝜂)^𝛼𝜃)[^𝜂₀(𝐺(𝑥)(𝜏)−𝐺(𝑦)(𝜏))𝑑𝑊(𝜏)]𝑑𝜃𝑑𝜂‖_𝛾=[𝐸‖^𝑡₀^∞₀𝜃(𝑡−𝜂)^𝛼−1𝜉_𝛼(𝜃)𝑆((𝑡−𝜂)^𝛼𝜃)[^𝜂₀(𝐺(𝑥)(𝜏)−𝐺(𝑦)(𝜏))𝑑𝑊(𝜏)]𝑑𝜃𝑑𝜂‖²_𝐻]^1/2𝑇≤Tr(𝑄)^𝛼−1/2(2𝛼−1)^1/2𝑀_𝑆[^𝑇₀^𝑇₀‖𝐺(𝑥)(𝜏)−𝐺(𝑦)(𝜏)‖²_{𝐿2(Ω;𝐻)}𝑑𝜏𝑑𝜂]^1/2≤𝐶_𝐺𝑇^𝛼+1/2𝑀_𝑠𝑀_𝐺‖𝑥−𝑦‖_𝛾, (2.13) 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.

3. Application

Let 𝐷 be a bounded domain in 𝑅^𝑁 with smooth boundary 𝜕𝐷, and consider the initial boundary value problem: 𝜕^𝛼(𝑡,𝑧)𝜕𝑡^𝛼=Δ_𝑧𝑥(𝑡,𝑧)+^𝑇₀𝑎(𝑡,𝑠)𝑓₁(𝑠,𝑥(𝑠,𝑧),^𝑠₀+𝑘(𝑠,𝜏,𝑥(𝜏,𝑧))𝑑𝜏)𝑑𝑠_𝑇0𝑏(𝑡,𝑠)𝑓₂(𝑠,𝑥(𝑠,𝑧))𝑑𝑊(𝑠),on(0,𝑇)×𝐷,𝑥(0,𝑧)=^𝑛_𝑖=1𝑔_𝑖(𝑧)𝑥(𝑡_𝑖,𝑧)+^𝑇₀𝑐(𝑠)𝑓₃(𝑠,𝑥(𝑠,𝑧))𝑑𝑠,on𝐷,𝑥(𝑡,𝑧)=0,on(0,𝑇)×𝜕𝐷, (3.1)(3.2) where 0≤𝑡₁<𝑡₂⋯<𝑡_𝑛≤𝑇 are given and 𝑊 is an 𝐿²(𝐷)-valued Wiener process. We consider the equation (3.1) under the following conditions.

(H1)𝑓₁∶[0,𝑇]×𝑅×𝑅→𝑅 satisfies the Caratheodory conditions as well as(i)𝑓₁(⋅,0,0)∈𝐿²(0,𝑇),(ii)|𝑓₁(𝑡,𝑥₁,𝑦₁)−𝑓₁(𝑡,𝑥₂,𝑦₂)|≤𝑀_𝑓1[|𝑥₁−𝑥₂|+|𝑦₁−𝑦₂|], for all 𝑥₁,𝑥₂,𝑦₁,𝑦₂∈𝑅 and almost 𝑡∈(0,𝑇) for some 𝑀_𝑓1>0,(H2)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(i)𝑓₂(⋅,0)∈𝐿²(0,𝑇),(ii)|𝑓₂(𝑡,𝑥)−𝑓₂(𝑡,𝑦)|_BL(𝐻)≤𝑀_𝑓2|𝑥−𝑦|, for all 𝑥,𝑦∈𝑅 and almost all 𝑡∈(0,𝑇), for some 𝑀_𝑓2>0.(H3)𝑓₃∶[0,𝑇]×𝑅→𝑅 satisfies the Caratheodory conditions as well as(i)𝑓₃(⋅,0)∈𝐿²(0,𝑇),(ii)|𝑓₃(𝑡,𝑥)−𝑓₃(𝑡,𝑦)|≤𝑀_𝑓3|𝑥−𝑦|, for all 𝑥,𝑦∈𝑅 and almost 𝑡∈(0,𝑇) for some 𝑀_𝑓3>0,(H4)𝑎∈𝐿²((0,𝑇)²),(H5)𝑏∈𝐿^∞((0,𝑇)²),(H6)𝑐∈𝐿²((0,𝑇)²),(H7)𝑘∶𝑌×𝑅→𝑅, where 𝑌={(𝑡,𝑠)∶0<𝑠<𝑡<𝑇}, satisfies |𝑘(𝑡,𝑠,𝑥₁)−𝑘(𝑡,𝑠,𝑥₂)|≤𝑀_𝑘|𝑥₁−𝑥₂|, for all 𝑥₁,𝑥₂∈𝑅, and almost (𝑡,𝑠)∈𝑌,(H8)𝑔_𝑖∈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=Δ_𝑧, with domain 𝐷(𝐴)=𝐻²(𝐷)∪𝐻¹₀((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,⋅)=^𝑇₀𝑎(𝑡,𝑠)𝑓₁(𝑠,𝑥(𝑠,𝑧),^𝑠₀𝑘(𝑠,𝜏,𝑥(𝜏,𝑧))𝑑𝜏)𝑑𝑠,𝐺(𝑥)(𝑡,⋅)=𝑏(𝑡,𝑠)𝑓₂(𝑠,𝑥(𝑠,⋅)),ℎ(𝑥)(⋅)=𝑥(0,𝑧)=^𝑛_𝑖=1𝑔_𝑖(⋅)𝑥(𝑡_𝑖,⋅)+^𝑇₀𝑐(𝑠)𝑓₃(𝑠,𝑥(𝑠,⋅))𝑑𝑠. (3.3) One can use (H1)–(H8) to verify that 𝐹, 𝐺, and ℎ satisfy (II)–(IV) in the last section, respectively, with 𝑀_𝐹=2𝑀_𝑓1𝑇|𝑎|_{𝐿2((0,𝑇)}²₎(1+𝑀_𝑘𝑇³)^1/2,𝑀_𝐺=𝑀_𝑓2,𝑀_ℎ=2^𝑛_𝑖=1‖𝑔_𝑖‖_𝐿2(𝐷)+𝑀_𝑓3𝑚(𝐷)|𝐺|_𝐿²_(0,𝑇). (3.4) Consequently theorem (2.4) can be applied for (3.1).