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Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2009, Article ID 568078, 8 pages doi:10.1155/2009/568078 Research Article On Some Fractional Stochastic Integrodifferential Equations in Hilbert Space Hamdy M. Ahmed Higher Institute of Engineering, El-Shrouk Academy, P.O. 3 El-Shorouk City, Cairo, Egypt Correspondence should be addressed to Hamdy M. Ahmed, hamdy [email protected] Received 8 April 2009; Accepted 6 July 2009 Recommended by Andrew Rosalsky 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. Copyright q 2009 Hamdy M. Ahmed. This 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. 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 BLK; 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 CX; Y , while Lp 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 {Zt : 0 ≤ t ≤ T }. An H-valued random variable is an Zmeasurable function X : Ω → H, and a collection of random variables ψ {Xt; 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 EX·; ω2H 1/2 . An important subspace is given by L20 Ω; H {f ∈ L2 Ω; H : f is Z0 measurable}. Next we define the space γ0, T ; H to be the set {v ∈ C0, T ; L2 Ω; H : v is Zt -adapted} with norm vγ sup Evt2H 1/2 1.1 0≤t≤T 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 2 International Journal of Mathematics and Mathematical Sciences dα xt Axt dtα t Fxt GxsdWs, 0 ≤ t ≤ T, 0 x0 hx 1.2 x0 , in a real separable Hilbert space H. Here, 1/2 ≤ α ≤ 1, A : DA ⊂ H → H is a linear closed operator generating semigroup, F : γ0, T ; H → Lp 0, T ; L2 Ω; H 1 ≤ p < ∞, G : γ0, T ; H → C0, T ; L2 Ω; BLK; 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 : γ0, T ; H → L20 Ω; H. Definition 1.1. An Zt -adapted stochastic process x : 0, T → H is called a mild solution of 1.2 if xt is measurable, for all t ∈ 0, T , T xt ∞ 0 ξα θSt θhx α x0 dθ xs2H ds < ∞, t ∞ α−1 α θ t−η ξα θS t − η θ Fx η dθ dη α 0 0 0 t ∞ α−1 α η θ t−η ξα θS t − η θ GxτdWτ dθ dη, α 0 0 0 0 ≤ t ≤ T, 1.3 where ξα θ is a probability density function defined on 0, ∞, ∞ ξα θdθ 1 1.4 0 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 : DA ⊂ H → H generates a C0 -semigroup{St : t ≥ 0} on H; II F : γ0, T ; H → Lp 0, T ; L2 Ω; H is such that there exists MF > 0 for which Fx − FyLp ≤ MF x − yγ , ∀x, y ∈ γ0, T ; H; 2.1 III G : γ0, T ; H → C0, T ; L2 Ω; BLK; H γBL is such that there exists MG > 0 for which Gx − GyγBL ≤ MG x − yγ ∀x, y ∈ γ0, T ; H; 2.2 International Journal of Mathematics and Mathematical Sciences 3 IV h : γ0, T ; H → L20 Ω; H is such that there exists Mh > 0 for which hx − h y L2 ≤ Mh x − yγ 0 ∀x, y ∈ γ0, 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 MS Mh CF T α MG CG T α 1/2 2.4 < 1, where Mh > 0, MS > 0, and CG > 0. Proof. Define the solution map J : γ0, T ; H → γ0, T ; H by Jxt ∞ ξα θStα θhx x0 dθ 0 t ∞ α−1 α θ t−η ξα θS t − η θ Fx η dθ dη α 0 0 0 0 2.5 t ∞ α−1 α θ t−η ξα θS t − η θ α × η GxτdWτ dθ dτ, 0 ≤ t ≤ T. 0 From Holder’s inequality, we get ⎡ ⎣E t ∞ 0 θ t−η α−1 α ξα θS t − η θ Fx η dθ dη 0 ≤ MS T ⎦ 1/2 α−1 T − η T T −η 2 FxηL2 Ω;H dη 2α−1 1/2 T dη 0 ≤ MS ⎤1/2 H 0 ≤ MS 2 T α−1/2 0 T 2α − 11/2 0 2 Fx η L2 Ω;H dη ≤ CF MS T α−1/2 FxLp , where CF is a constant depending on α. 2 Fx η L2 Ω;H dη 1/2 1/2 2.6 4 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 ⎡ ⎣E t ∞ θ t−η 0 α−1 α ξα θS t−η θ Fx η dθ dη 0 2 ⎤1/2 ⎦ ≤ MS CF T α−1/2 MF xγ F0Lp . H 2.7 Taking the supermum over 0, T in 2.7 then implies that t ∞ 0 α−1 α θ t−η ξα θS t − η θ Fx η dθ dη ∈ γ0, T ; H, 2.8 0 for any x ∈ γ0, T ; H. Furthermore for such x, Gxη ∈ BLK; H, and hx 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 ∈ γ0, T ; H, we infer from 2.5 that Jxt − Jy t ∞ ξα θStα θ hx − h y dθ 0 t ∞ α−1 α θ t−η ξα θS t − η θ Fx η − F y η dθ dη α 0 0 0 0 t ∞ η α−1 α θt − η ξα θS t − η θ α Gxτ − G yτ dWτ dθ dη, 0 ≤ t ≤ T. 0 2.9 Squaring both sides and taking the expectation in 2.9 yields, with the help of Young’s inequality, EJxt − Jyt2H ≤ 4E ∞ 2 ξα θStα θhx 0 ⎡ 2⎣ 4α x0 dθ H t ∞ E 0 θ t−η α ξα θS t − η θ Fx η − F y η dθ dη 0 E 0 2 H t ∞ × α−1 α−1 α θ t−η ξα θS t − η θ 0 η 0 Gxτ − G y τ dWτ dθ dη 2 , H 2.10 International Journal of Mathematics and Mathematical Sciences 5 and subsequently, Jxt − Jytγ ∞ ≤ ξα θStα θhx − hydθ 0 γ ⎡ t ∞ 2⎣ 4α 0 α−1 α θ t−η ξα θS t − η θ Fx η − F y η dθ dη 0 γ t ∞ 0 × 2.11 θt − ηα−1 ξα θS t − ηα θ 0 η Gxτ − G y τ dWτ dθ dη 0 . γ Using reasoning similar to that which led to 2.6, one can show that ∞ ξα θStα θ hx − h y dθ 0 γ E ∞ 2 ξα θStα θhx x0 dθ 0 t ∞ 0 H ≤ Ms Mh x − yγ , α−1 α θ t−η ξα θS t − η θ Fx η − F y η dθ dη 0 2.12 γ t ∞ 0 θt − ηα−1 ξα θSt − ηα θFxη − Fyηdθ dη 0 γ ≤ CF MS T α x − yγ , where CF depending on α and MF . We also infer that t ∞ 0 θ t−η α−1 η α ξα θS t − η θ 0 Gxτ − G y τ dWτ dθ dη 0 ⎡ ⎣E t ∞ 0 ≤ TrQ ≤ CG T α θ t−η α−1 α ξα θS t − η θ 0 T α−1/2 2α − 11/2 1/2 γ η Gxτ − G y τ dWτ dθ dη 0 T T MS 0 0 2 Gxτ − G y τL2 Ω;H dτ dη 2 ⎤1/2 ⎦ H 1/2 Ms MG x − yγ , 2.13 6 International Journal of Mathematics and Mathematical Sciences where CG is a constant depending on α and TrQ. 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: s T ∂α t, z s, xs, z, Δ xt, z at, sf ks, τ, xτ, zdτ ds z 1 ∂tα 0 0 T bt, sf2 s, xs, zdWs, on 0, T × D, 0 x0, z n gi zxti , z T csf3 s, xs, zds, on D, 0 i1 xt, z 0, 3.1 3.2 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. 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 | ≤ Mf1 |x1 − x2 | and almost t ∈ 0, T for some Mf1 > 0, |y1 − y2 |, for all x1 , x2 , y1 , y2 ∈ R H2 f2 : 0, T × R → BLL2 D where BLL2 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|BLH ≤ Mf2 |x − y|, for all x, y ∈ R and almost all t ∈ 0, T , for some Mf2 > 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| ≤ Mf3 |x − y|, for all x, y ∈ R and almost t ∈ 0, T for some Mf3 > 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 |kt, s, x1 − kt, s, x2 | ≤ Mk |x1 − x2 |, for all x1 , x2 ∈ R, and almost t, s ∈ Y , H8 gi ∈ L2 D, i 1, . . . , n. International Journal of Mathematics and Mathematical Sciences 7 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 DA H 2 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, Fxt, · T s at, sf1 s, xs, z, 0 ks, τ, xτ, zdτ ds, 0 Gxt, · bt, sf2 s, xs, ·, T n gi ·xti , · csf3 s, xs, ·ds. hx· x0, z 3.3 0 i1 One can use H1–H8 to verify that F, G, and h satisfy II–IV in the last section, respectively, with MF 2Mf1 T |a|L2 0,T 2 1 Mk T 3 1/2 , MG Mf2 , n Mh 2 i1 gi L2 D Mf3 mD|G|L2 0,T . 3.4 Consequently theorem 2.4 can be applied for 3.1. References 1 D. N. Keck and M. A. McKibben, “Functional integro-diﬀerential stochastic evolution equations in Hilbert space,” Journal of Applied Mathematics and Stochastic Analysis, vol. 16, no. 2, pp. 141–161, 2003. 2 M. M. El-Borai, K. El-Said El-Nadi, O. L. 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