On Some Stochastic Fractional Integro-Differential Equations

Advances in Dynamical Systems and Applications. ISSN 0973-5321 Volume 1 Number 1 (2006), pp. 49–57 c Research India Publications ° http://www.ripublication.com/adsa.htm

On Some Stochastic Fractional Integro-Differential Equations Mahmoud M. El-Borai Faculty of Science, Alexandria University, Alexandria, Egypt E-mail: m− m− [email protected] 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: 34K30, 34K05, 26A33, 35A05. 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 Z t ∂ α u(x, t) = L(x, D)u(x, t) + f (u(x, t)) + g(u(x, s))dW (s), (1.1) ∂tα 0 with the nonlocal condition u(x, 0) = ϕ(x) +

p X

ci u(x, ti ),

(1.2)

i=1

where 0 < α ≤ 1, 0 ≤ t1 < t2 < · · · < tp , x is an element of the n-dimensional ∂ , i = 1, . . . , n, W (t) is a standard Euclidean space Rn , D = (D1 , . . . , Dn ), Di = ∂xi Received May 1, 2006; Accepted July 2, 2006

50

Mahmoud M. El-Borai

Brownian motion defined over the filtered probability space (Ω, F, Ft , P ), u ∈ Rk and {Ft : 0 ≤ 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 Rn such that Z

k X Rn i=1

u2i (x)dx < ∞, u = (u1 , . . . , uk ).

A scalar product (u, v) in H is defined by (u, v)H =

k Z X i=1

ui (x)vi (x)dx. Rn

If S = {V (x, t, ω) : Ω → H|0 ≤ t ≤ T } is a stochastic process, then we shall write for simplicity V (x, t) and Vt : [0, T ] → H in place of S. The collection of all strongly measurable H-valued random variables denoted by 2 L (Ω, H) is a Banach space equipped with the norm Z 2 1 2 kVt kL2 (Ω,H) = [E kVt (ω)kH ] , E(g) = g(ω)dP, Ω

where E(g) is the expectation of g. It is assumed that L(x, D) = L0 (x, D) + L1 (x, D), where L0 (x, D) =

X

Aq (x)Dq ,

L1 (x, D) =

X

Aq (x)Dq ,

|q| 0 for all x ∈ Rn , t ≥ 0 and for any real vector σ, σ12 + · · · + σn2 = 1, [2]. If B is a matrix of order m × n, then we introduce |B| by X |bij |. |B| = i,j

It is assumed that the coefficients of L(x, D) are bounded on Rn and satisfy the H¨older condition (with exponent γ ∈ (0, 1]). It is well known that there exists a fundamental matrix solution Z(x, y, t), which satisfies the system ∂Z(x, y, t) = L(x, D)Z(x, y, t), ∂t

t > 0, x, y ∈ Rn .

On Some Stochastic Fractional Integro-Differential Equations

51

This fundamental matrix satisfies the inequality |Dq Z(x, y, t)| ≤ K1 t−ρ1 exp(−K2 ρ2 ), n X 1 n + |q| |q| ≤ 2m, ρ1 = − , ρ2 = |xi − yi |λ t− 2m−1 , 2m i=1 λ=

(1.3)

2m , (K1 > 0 and K2 > 0 are constants). 2m − 1

The set {vt ∈ C([0, T ]; L2 (Ω, H)) : vt is Ft adapted} is denoted by C([0, T ]; H), where C([0, T ]; H) denotes the space of k-dimensional vectors of continuous functions from [0, T ] into H equipped with norm 1

kvkC = sup [E kvt k2H ] 2 . t∈[0,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 F0 -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 Z t 1 u(x, t) = u(x, 0) + (t − θ)α−1 L(x, D)u(x, θ)dθ Γ(α) 0 Z t 1 + (t − θ)α−1 f (u(x, θ))dθ Γ(α) 0 Z tZ θ 1 (t − θ)α−1 g(u(x, s))dW (s)dθ. (2.1) + Γ(α) 0 0 Let Z(t) be the operator defined on H, for every t > 0, by Z Z(t)ϕ = Z(x, y, t)ϕ(y)dy. Rn

According to condition (1.3), there is a positive constant M such that kZ(t)kH ≤ M.

(2.2)

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Mahmoud M. El-Borai

Let us consider the integrals of the operator-valued functions Z

∞

ψ(t) =

ξα (θ)Z(tα θ)dθ,

0

Z

∞

ψ ∗ (t) = α

θtα−1 ξα (θ)Z(tα θ)dθ,

0

where ξα (θ) is a probability density function defined on [0, ∞] (see [3]). It is supposed that p X cM < 1, where c = |ci |. (2.3) i=1

Theorem 2.1. If u ∈ C([0, T ]; H) is an Ft -adapted stochastic process that satisfies equation (2.1) a.s. [p], then u satisfies the equation (a.s. [p]) u(x, t) = (ψ(t)Λ−1 ϕ)(x) Z p X −1 + ψ(t)Λ ci + ψ(t)Λ

X i=1

Z

(ψ ∗ (ti − η)f (u))(x, η)dη

0

i=1 p

−1

ti

Z

ti

Z

η

ci 0

(ψ ∗ (ti − η)g(u))(x, s)dW (s)dη

0

t

(ψ ∗ (t − η)f (u))(x, η)dη Z0 t Z η (ψ ∗ (t − η)g(u))(x, s)dW (s)dη, +

+

0

where Λ = I −

p X

(2.4)

0

ci ψ(ti ), I is the identity operator.

i=1

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]) Z u(x, t) =

∞

Z

Z(x, y, tα θ)ξα (θ)u(y, 0)dydθ Z Z +α θ(t − η)α−1 ξα (θ)Z(x, y, (t − η)α θ)f (u(y, η))dydθdη n Z0 t Z0 ∞ ZR +α θ(t − η)α−1 ξα (θ)Z(x, y, (t − η)α θ) n R Z η0 0 × g(u(y, s))dW (s)dydθdη. Rn tZ ∞

0

0


53

It is easy to see that (a.s. [p]) Ã ! p p X X ci u(x, ti ) = Λ−1 ci ψ(ti )ϕ (x) i=1

i=1

p µ Z X −1 +Λ ci i=1 p

Xµ Z −1 ci +Λ

¶ ψ (ti − η)f (u) (x, y)dη

ti

∗

0

Z

ti 0

i=1

η

¶ ψ (ti − η)g(u) (x, s)dW (s)dη. ∗

(2.5)

0

Using (1.2) and (2.5), one gets (a.s. [p]) Ã ! p X (ψ(t)ϕ)(x) + ψ(t) ci u (xi , ti ) = (ψ(t)Λ−1 ϕ)(x) i=1

+ ψ(t)Λ

−1

p X

Z ci

+ ψ(t)Λ

X

(ψ ∗ (ti − η)f (u))(x, η)dη

0

i=1 p

−1

ti

Z

ti

Z

η

ci 0

i=1

(ψ ∗ (ti − η)g(u))(x, s)dW (s)dη.

0

¥

Hence the required result follows.

Now we define a mild solution of equation (2.1) to be an Ft - adapted stochastic process u ∈ ([0, T ]; H) which satisfies equation (2.4), a.s. [p]. Theorem 2.2. If kf kC + kgkC ≤ K for all u ∈ Sγ = {u ∈ C([0, T ]; H) kukC ≤ γ} and if ° ° ° ° M E(°Λ−1 ϕ°H ) + KM 2 T α °Λ−1 °H + KM T α + M KT 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 Sγ by (Qu)(x, t) = (ψ(t)Λ−1 ϕ)(x) Z p X −1 + ψ(t)Λ ci + ψ(t)Λ

X i−1

Z

t

+

∗

(ψ ∗ (ti − η)f (u)(x, η)dη

0

i−1 p

−1

ti

Z

ti

Z

η

ci 0

(ψ ∗ (ti − η)g(u)(x, s)dW (s)dη

0

Z tZ

η

(ψ (t − η)f (u))(x, η)dη + 0

0

0

(ψ ∗ (t − η)g(u)(x, s)dW (s)dη.

54

Mahmoud M. El-Borai Z

∞

θξα (θ)dθ = 1, one gets,

Using (2.6) and noting that 0

E(kQut kH ) ≤ γ, where

Z kvt k2H

k X

= Rn

vi2 (x, t)dx.

i=1

Thus Q maps C([0, T ]; H) into itself. For 0 ≤ t1 < t2 ≤ T , we have E kQut1 − Qut2 kH ° ° ° ° ≤ [E(°Λ−1 ϕ°H ) + 2cKT α °Λ−1 °H ] kψ(t1 ) − ψ(t2 )kH Z t2 Z t1 ∗ + 2K kψ (t − η)kH dη + 2M kψ ∗ (t1 − η) − ψ ∗ (t2 − η)kH dη. (2.7) t1

0

It can be proved that k[ψ(t2 ) − ψ(t1 )]wkH ≤ αM [log t2 − log t1 ] kwkH .

(2.8)

Consequently kψ(t2 ) − ψ(t1 )kH tends to zero as s tends to t. Similarly Z

t1

lim

t1 →t2

Z

0 t2

lim

t1 →t2

t1

kψ ∗ (t2 − η) − ψ ∗ (t1 − η)kH dη = 0,

kψ ∗ (t − η)kH dη = 0,

(see [7]).

(2.9) (2.10)

The right-hand side of inequality (2.7) is independent of u and by using (2.8), (2.9) and (2.10), one gets that [kQut1 − Qut2 kH ] tends to zero as t1 → t2 . Now it is clear that by Arzela–Ascoli’s theorem, {(Qu)(x, t) : u ∈ C([0, T ]; H)} is precompact. Hence by Leray–Schauder’s fixed point theorem, Q has a fixed point in C([0, 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[kf (u) − f (v)kH ] + E[kg(u) − g(v)kH ] ≤ K1 E(ku − vkH ) is satisfied, where K1 > 0 is a constant. If Equation (2.1) has a mild solution u ∈ C([0, T ]; H), then that mild solution is unique on [0, T ].


55

Proof. For u, v, ∈ C([0, T ]; H) we infer from (2.4) that Z K2 T α t 2 E kut − vt kH ≤ (t − η)α−1 E kuη − vη k2H dη α 0 Z ti p α X K3 T |ci | (ti − η)α−1 E kuη − vη k2H dη + α i=1 0 Z t + K2 (t − s)α E kus − vs k2H ds 0 p

X

+ K3

Z

ti

|ci | 0

i=1

(ti − s)α E kus − vs k2H ds.

(2.11)

Let ρ(u, v) = sup (e−λt E kut − vt k2H ), where λ > 1. It is easy to see that t∈[0,T ]

Z

ti 0

(ti − η)α−1 E kuη − vη k2H dη "

Z

1 ti − λ

≤ λ1−α

# dη ρ(u, v)

0 λti

≤e

µ ¶ µ ¶α 1 1 1+ ρ(u, v). α λ

(2.12)

Let us consider the following two cases. The first one is t ≥ tp and the second is t ≤ tp . For the first case t ≥ tp , we deduce from (2.11) and (2.12) that ¶α ¸ ·µ 1 1 2 −λt e E kut − vt kH ≤ K4 1 + + ρ(u, v). λ λ Thus ρ(u, v) = 0, for sufficiently large λ, (K4 > 0 is a constant). For the second case t ≤ tp , one gets E kut − vt k2H ≤ K5 tαp sup E kut − vt k2H , (K5 > 0 is a constant). t∈[0,T ]

Now if tp is sufficiently small such that K5 tαp < 1, we get sup E kut − vt k2H = 0. 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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Mahmoud M. El-Borai

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