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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
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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
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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
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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
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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
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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
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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.
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