On Some Stochastic Fractional Integro-Differential Equations

Transcription

1 Advances in Dynamical Systems and Applications. ISSN Volume 1 Number 1 (26), pp c Research India Publications On Some Stochastic Fractional Integro-Differential Equations Mahmoud M. El-Borai Faculty of Science, Alexandria University, Alexandria, Egypt m m elborai@yahoo.com Abstract Some classes of stochastic fractional integro-partial differential equations are investigated. Mild solutions of the nonlocal Cauchy problem for the considered classes are studied. The Leray Schauder principle is used to establish the existence of stochastic solutions. The uniqueness of the solution of the considered problem is also studied under suitable conditions. AMS subject classification: 34K3, 34K5, 26A33, 35A5. Keywords: Nonlocal Cauchy problem, stochastic integro-partial differential equations, fractional order. 1. Introduction Many physical models are represented by semi-linear stochastic fractional integro-partial differential systems of the form α u(x, t) t α = L(x, D)u(x, t) + f(u(x, t)) + with the nonlocal condition u(x, ) = ϕ(x) + g(u(x, s))dw (s), (1.1) c i u(x, t i ), (1.2) where < α 1, t 1 < t 2 < < t p, x is an element of the n-dimensional Euclidean space R n, D = (D 1,..., D n ), D i =, i = 1,..., n, W (t) is a standard x i Received May 1, 26; Accepted July 2, 26

2 5 Mahmoud M. El-Borai Brownian motion defined over the filtered probability space (Ω, F, F t, P ), u R k and {F t : t T } is a right-continuous, increasing family of sub σ-algebras of F. Let H be the space of all k-dimensional vectors of functions defined on R n such that R n k u 2 i (x)dx <, u = (u 1,..., u k ). A scalar product (u, v) in H is defined by (u, v) H = k R n u i (x)v i (x)dx. If S = {V (x, t, ω) : Ω H t T } is a stochastic process, then we shall write for simplicity V (x, t) and V t : [, T ] H in place of S. The collection of all strongly measurable H-valued random variables denoted by L 2 (Ω, H) is a Banach space equipped with the norm V t L 2 (Ω,H) = [E V t(ω) 2 H ] 1 2, E(g) = g(ω)dp, where E(g) is the expectation of g. It is assumed that where L (x, D) = L(x, D) = L (x, D) + L 1 (x, D), q =2m A q (x)d q, Ω L 1 (x, D) = q <2m A q (x)d q, D q = D q 1 1 D q n n, q = (q 1,..., q n ) is a multi index, q = q q n, and {A q (x) : q 2m} is a family of deterministic square matrices of order k. Following Petrovsky it is assumed that det{( 1) m L (x, σ) λi} = has roots which satisfy the inequality Reλ < δ, δ > for all x R n, t and for any real vector σ, σ σn 2 = 1, [2]. If B is a matrix of order m n, then we introduce B by B = b ij. i,j It is assumed that the coefficients of L(x, D) are bounded on R n and satisfy the Hölder condition (with exponent γ (, 1]). It is well known that there exists a fundamental matrix solution Z(x, y, t), which satisfies the system Z(x, y, t) t = L(x, D)Z(x, y, t), t >, x, y R n.

3 On Some Stochastic Fractional Integro-Differential Equations 51 This fundamental matrix satisfies the inequality D q Z(x, y, t) K 1 t ρ 1 exp( K 2 ρ 2 ), (1.3) q 2m, ρ 1 = n + q n 2m, ρ 2 = x i y i λ t 1 2m 1, λ = 2m 2m 1, (K 1 > and K 2 > are constants). The set {v t C([, T ]; L 2 (Ω, H)) : v t is F t adapted} is denoted by C([, T ]; H), where C([, T ]; H) denotes the space of k-dimensional vectors of continuous functions from [, T ] into H equipped with norm v C = sup [E v t 2 H ] 1 2. t [,T ] The purpose of this paper is to study the mild solutions of the nonlocal Cauchy problem (1.1), (1.2), assuming that ϕ is an F -measurable H-valued stochastic process. The results in this note may be regarded as a generalization of some recent results developed in [3 11, 14, 16, 17]. In Section 2, we shall find the mild solutions of the nonlocal Cauchy problem (1.1), (1.2). The nonlocal Cauchy problem (1.1), (1.2) has applications in many fields such as viscoelasticity and electromagnetic theory, [1, 12, 13, 15, 18]. 2. Stochastic Integral Equation A slightly modified version of the nonlocal Cauchy problem (1.1), (1.2) is considered. Using the definitions of the fractional derivatives and integrals, it is suitable to rewrite the considered problem in the form u(x, t) = u(x, ) + 1 (t θ) α 1 L(x, D)u(x, θ)dθ Γ(α) + 1 (t θ) α 1 f(u(x, θ))dθ Γ(α) + 1 Γ(α) θ Let Z(t) be the operator defined on H, for every t >, by Z(t)ϕ = Z(x, y, t)ϕ(y)dy. R n (t θ) α 1 g(u(x, s))dw (s)dθ. (2.1) According to condition (1.3), there is a positive constant M such that Z(t) H M. (2.2)

4 52 Mahmoud M. El-Borai Let us consider the integrals of the operator-valued functions ψ(t) = ψ (t) = α ξ α (θ)z(t α θ)dθ, θt α 1 ξ α (θ)z(t α θ)dθ, where ξ α (θ) is a probability density function defined on [, ] (see [3]). It is supposed that cm < 1, where c = c i. (2.3) Theorem 2.1. If u C([, T ]; H) is an F t -adapted stochastic process that satisfies equation (2.1) a.s. [p], then u satisfies the equation (a.s. [p]) where Λ = I u(x, t) = (ψ(t)λ 1 ϕ)(x) i + ψ(t)λ 1 c i (ψ (t i η)f(u))(x, η)dη i + ψ(t)λ 1 c i + + η η (ψ (t η)f(u))(x, η)dη (ψ (t i η)g(u))(x, s)dw (s)dη (ψ (t η)g(u))(x, s)dw (s)dη, (2.4) c i ψ(t i ), I is the identity operator. Proof. Using (2.2) and (2.3), we find that the inverse operator Λ 1 exists. Using the results in [3], the solution of equation (2.1) can be written in the form (a.s. [p]) u(x, t) = + α + α Z(x, y, t α θ)ξ α (θ)u(y, )dydθ R n η θ(t η) α 1 ξ α (θ)z(x, y, (t η) α θ)f(u(y, η))dydθdη R n θ(t η) α 1 ξ α (θ)z(x, y, (t η) α θ) R n g(u(y, s))dw (s)dydθdη.

5 On Some Stochastic Fractional Integro-Differential Equations 53 It is easy to see that (a.s. [p]) ( ) c i u(x, t i ) = Λ 1 c i ψ(t i )ϕ (x) + Λ 1 + Λ 1 ( c i i ( c i i ) ψ (t i η)f(u) (x, y)dη η Using (1.2) and (2.5), one gets (a.s. [p]) ( ) (ψ(t)ϕ)(x) + ψ(t) c i u (x i, t i ) = (ψ(t)λ 1 ϕ)(x) ) ψ (t i η)g(u) (x, s)dw (s)dη. (2.5) i + ψ(t)λ 1 c i (ψ (t i η)f(u))(x, η)dη i + ψ(t)λ 1 c i Hence the required result follows. η (ψ (t i η)g(u))(x, s)dw (s)dη. Now we define a mild solution of equation (2.1) to be an F t - adapted stochastic process u ([, T ]; H) which satisfies equation (2.4), a.s. [p]. Theorem 2.2. If f C + g C K for all u S γ = {u C([, T ]; H) u C γ} and if ME( Λ 1 ϕ H ) + KM 2 T α Λ 1 H + KMT α + MKT 1+α + KM 2 T 1+α γ, (2.6) where K and γ are positive constants, then there exists a mild solution of equation (2.1), a.s. [p]. Proof. Let Q be a map defined on S γ by (Qu)(x, t) = (ψ(t)λ 1 ϕ)(x) i + ψ(t)λ 1 c i (ψ (t i η)f(u)(x, η)dη i 1 i + ψ(t)λ 1 c i + i 1 η (ψ (t η)f(u))(x, η)dη + (ψ (t i η)g(u)(x, s)dw (s)dη η (ψ (t η)g(u)(x, s)dw (s)dη.

6 54 Mahmoud M. El-Borai Using (2.6) and noting that θξ α (θ)dθ = 1, one gets, E( Qu t H ) γ, where v t 2 H = R n k vi 2 (x, t)dx. Thus Q maps C([, T ]; H) into itself. For t 1 < t 2 T, we have E Qu t1 Qu t2 H [E( Λ 1 ϕ H ) + 2cKT α Λ 1 H ] ψ(t 1 ) ψ(t 2 ) H + 2K 2 ψ (t η) H dη + 2M 1 t 1 ψ (t 1 η) ψ (t 2 η) H dη. (2.7) It can be proved that [ψ(t 2 ) ψ(t 1 )]w H αm[log t 2 log t 1 ] w H. (2.8) Consequently ψ(t 2 ) ψ(t 1 ) H tends to zero as s tends to t. Similarly 1 lim ψ (t 2 η) ψ (t 1 η) t 1 t H dη =, (2.9) 2 2 lim ψ (t η) t 1 t H dη =, (see [7]). (2.1) 2 t 1 The right-hand side of inequality (2.7) is independent of u and by using (2.8), (2.9) and (2.1), one gets that [ Qu t1 Qu t2 H ] tends to zero as t 1 t 2. Now it is clear that by Arzela Ascoli s theorem, {(Qu)(x, t) : u C([, T ]; H)} is precompact. Hence by Leray Schauder s fixed point theorem, Q has a fixed point in C([, T ]; H) and any fixed point of Q represents a mild solution of equation (2.1), a.s. [p]. This completes the proof of the theorem. Theorem 2.3. Suppose the Lipschitz condition E[ f(u) f(v) H ] + E[ g(u) g(v) H ] K 1 E( u v H ) is satisfied, where K 1 > is a constant. If Equation (2.1) has a mild solution u C([, T ]; H), then that mild solution is unique on [, T ].

7 On Some Stochastic Fractional Integro-Differential Equations 55 Proof. For u, v, C([, T ]; H) we infer from (2.4) that E u t v t 2 H K 2T α α + K 3T α α (t η) α 1 E u η v η 2 H dη c i i + K 2 (t s) α E u s v s 2 H ds + K 3 c i i (t i η) α 1 E u η v η 2 H dη (t i s) α E u s v s 2 H ds. (2.11) Let ρ(u, v) = sup (e λt E u t v t 2 H ), where λ > 1. It is easy to see that t [,T ] i (t i η) α 1 E u η v η 2 H dη [ e λt i λ 1 α i 1 λ dη ] ρ(u, v) ( ) ( ) α 1 ρ(u, v). (2.12) α λ Let us consider the following two cases. The first one is t t p and the second is t t p. For the first case t t p, we deduce from (2.11) and (2.12) that [( e λt E u t v t 2 H K ) α + 1 ] ρ(u, v). λ λ Thus ρ(u, v) =, for sufficiently large λ, (K 4 > is a constant). For the second case t t p, one gets E u t v t 2 H K 5t α p sup E u t v t 2 H, (K 5 > is a constant). t [,T ] Now if t p is sufficiently small such that K 5 t α p < 1, we get sup E u t v t 2 H =. This completes the proof of the theorem. Acknowledgement The author is very grateful to the anonymous referees and to Professor Martin Bohner for their careful reading of the paper and their valuable comments.

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