International Journal of Mathematics and Mathematical Sciences Volume 2009, Article ID 568078,8pages
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.
Copyrightq2009 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
LetHandKdenote real Hilbert spaces equipped with norms · Hand · K,respectively,
and let the space of bounded linear operators fromK toH be denoted by BLK;H. For
Banach spaceXandY, the space of continuous functions fromXintoY equipped with the
usual sup-normwill be denoted byCX;Y, whileLp0, T;Xwill represent the space ofX
-valued functions that arep-integrable on0, T. LetΩ, Z, Pbe a complete probability space equipped with a normal filtration{Zt : 0 ≤ t ≤ T}. AnH-valued random variable is anZ
-measurable functionX :Ω → H, and a collection of random variablesψ {Xt;ω:Ω →
H: 0≤t≤T}is called a stochastic process. The collection of all strongly measurable square integrableH-valued random variables, denoted byL2Ω;H, is a Banach space equipped
with normX·L2Ω;H EX·;ω2H1/2. An important subspace is given byL2
0Ω;H {f ∈L2Ω;H:f isZ0measurable}.
Next we define the spaceγ0, T;Hto be the set{v∈C0, T;L2Ω;H:visZ
t-adapted}
with norm
vγ sup
0≤t≤T
Evt2H1/2 1.1
dαxt
dtα Axt Fxt t
0
GxsdWs, 0≤t≤T,
x0 hx x0,
1.2
in a real separable Hilbert spaceH. Here, 1/2≤α≤1, A:DA⊂H → His a linear closed operator generating semigroup, F : γ0, T;H → Lp0, T;L2Ω;H 1 ≤ p < ∞, G :
γ0, T;H → C0, T;L2Ω; BLK;H whereKis a real separable Hilbert space,Wis a
K-valued Wiener process with incremental covariance described by the nuclear operatorQ,
x0is anZ0-measurableH-valued random variable independent ofW andh:γ0, T;H →
L2 0Ω;H.
Definition 1.1. AnZt-adapted stochastic processx :0, T → His called a mild solution of
1.2ifxtis measurable, for allt∈0, T,
T
0
xs2Hds <∞,
xt
∞
0
ξαθStαθhx x0dθ α
t
0
∞
0
θt−ηα−1ξαθS
t−ηαθFxηdθ dη
α
t
0
∞
0
θt−ηα−1ξαθS
t−ηαθ
η
0
GxτdWτ
dθ dη, 0≤t≤T,
1.3
whereξαθis a probability density function defined on0,∞, ∞
0
ξαθdθ1 1.4
see 6–12. In the next section, we will prove the existence and uniqueness of the mild solutions to1.2.
2. Existence and Uniqueness
Consider the initial value problem 1.2 in a real separable Hilbert space H under the following assumptions:
Ithe linear operatorA:DA⊂H → Hgenerates aC0-semigroup{St:t≥0}on H;
IIF:γ0, T;H → Lp0, T;L2Ω;His such that there existsM
F >0 for which
Fx−FyLp ≤MFx−yγ, ∀x, y∈γ0, T;H; 2.1
IIIG:γ0, T;H → C0, T;L2Ω; BLK;H γ
BLis such that there existsMG>
0 for which
Gx−Gyγ
IVh:γ0, T;H → L20Ω;His such that there existsMh>0 for which
hx−hyL2
0 ≤Mhx−yγ ∀x, y∈γ0, T;H; 2.3
Vx0∈L20Ω;H.
We can therefore state the following theorem.
Theorem 2.1. Assume that (I)–(V) hold. Then1.2has a unique solution on0, T, provided that
MS
Mh CFTα MGCGTα 1/2
<1, 2.4
whereMh>0,MS>0, andCG>0.
Proof. Define the solution mapJ:γ0, T;H → γ0, T;Hby
Jxt
∞
0
ξαθStαθhx x0dθ
α
t
0
∞
0
θt−ηα−1ξαθS
t−ηαθFxηdθ dη
α
t
0
∞
0
θt−ηα−1ξαθS
t−ηαθ
×
η
0
GxτdWτ
dθ dτ, 0≤t≤T.
2.5
From Holder’s inequality, we get
⎡ ⎣E
t
0
∞
0
θt−ηα−1ξαθS
t−ηαθFxηdθ dη
2
H ⎤ ⎦
1/2
≤MS T
0
T−ηα−1Fxη2L2Ω;Hdη
1/2
≤MS T
0
T−η2α−1dη
1/2T
0
Fxη2L2Ω;Hdη
1/2
≤MS
Tα−1/2
2α−11/2
T
0
Fxη2L2Ω;Hdη
1/2
≤CFMSTα−1/2FxLp,
2.6
Subsequently, an application ofII, together with Minkowski’s inequality enables us to continue the string of inequalities in2.6to conclude that
⎡ ⎣E t 0 ∞ 0
θt−ηα−1ξαθS
t−ηαθFxηdθ dη
2
H ⎤ ⎦
1/2
≤MSCFTα−1/2
MFxγ F0Lp
.
2.7
Taking the supermum over0, Tin2.7then implies that
t
0
∞
0
θt−ηα−1ξαθS
t−ηαθFxηdθ dη∈γ0, T;H, 2.8
for any x ∈ γ0, T;H. Furthermore for such x, Gxη ∈ BLK;H, and hx x0 ∈
L20Ω;H byIV andV. Consequently, one can argue as in13–15 to conclude that J
is well defined.
Next we show thatJis a strict contraction.
Observe that forx, y∈γ0, T;H,we infer from2.5that Jxt−Jyt
∞
0
ξαθStαθ
hx−hydθ
α
t
0
∞
0
θt−ηα−1ξαθS
t−ηαθFxη−Fyηdθ dη
α
t
0
∞
0
θt−ηα−1ξαθS
t−ηαθ
η
0
Gxτ−GyτdWτ
dθ dη, 0≤t≤T.
2.9
Squaring both sides and taking the expectation in 2.9 yields, with the help of Young’s inequality,
EJxt−Jyt2H
≤4E
∞
0
ξαθStαθhx x0dθ
2
H
4α2
⎡ ⎣E t 0 ∞ 0
θt−ηα−1ξαθS
t−ηαθFxη−Fyηdθ dη
2 H E t 0 ∞ 0
θt−ηα−1ξαθS
t−ηαθ
×
η
0
Gxτ−GyτdWτ
and subsequently, Jxt−Jytγ
≤
∞
0
ξαθStαθhx−hydθ γ
4α2
⎡ ⎣ t 0 ∞ 0
θt−ηα−1ξαθS
t−ηαθFxη−Fyηdθ dη
γ t 0 ∞ 0
θt−ηα−1ξαθS
t−ηαθ
×
η
0
Gxτ−GyτdWτ
dθ dη γ . 2.11
Using reasoning similar to that which led to2.6, one can show that
∞
0
ξαθStαθ
hx−hydθ
γ
E
∞
0
ξαθStαθhx x0dθ
2
H
≤MsMhx−yγ, t 0 ∞ 0
θt−ηα−1ξαθS
t−ηαθFxη−Fyηdθ dη
γ t 0 ∞ 0
θt−ηα−1ξαθSt−ηαθFxη−Fyηdθ dη γ
≤CFMSTαx−yγ,
2.12
whereCFdepending onαandMF. We also infer that t 0 ∞ 0
θt−ηα−1ξαθS
t−ηαθ
η
0
Gxτ−GyτdWτ
dθ dη γ ⎡ ⎣E t 0 ∞ 0
θt−ηα−1ξαθS
t−ηαθ
η
0
Gxτ−GyτdWτ
dθ dη 2 H ⎤ ⎦
1/2
≤TrQ T
α−1/2
2α−11/2MS
T
0
T
0
Gxτ−Gyτ2L2Ω;Hdτ dη
1/2
≤CGTα 1/2MsMGx−yγ,
whereCGis a constant depending onαand TrQ. Using2.12and2.13in2.11enables
us to conclude thatJ is a strict contraction, provided that2.4is satisfied, and has a unique fixed point which coincides with a mild solution of1.2. 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:
∂αt, z
∂tα Δzxt, z T
0
at, sf1
s, xs, z,
s
0
ks, τ, xτ, zdτ
ds
T
0
bt, sf2s, xs, zdWs, on0, T×D,
3.1
x0, z
n
i1
gizxti, z T
0
csf3s, xs, zds, onD,
xt, z 0, on0, T×∂D,
3.2
where 0≤t1< t2· · ·< tn≤Tare given andWis anL2D-valued Wiener process. We consider
the equation3.1under the following conditions.
H1f1:0, T×R×R → Rsatisfies the Caratheodory conditions as well as
if1·,0,0∈L20, T,
ii|f1t, x1, y1−f1t, x2, y2| ≤Mf1|x1−x2| |y1−y2|, for allx1, x2, y1, y2 ∈R and almostt∈0, Tfor someMf1>0,
H2f2 : 0, T× R → BLL2D where BLL2D is the space of bounded linear
operator fromL2DtoL2Dsatisfies the Caratheodory conditions as well as
if2·,0∈L20, T,
ii|f2t, x−f2t, y|BLH ≤ Mf2|x−y|, for allx, y ∈Rand almost allt∈ 0, T, for someMf2>0.
H3f3:0, T×R → Rsatisfies the Caratheodory conditions as well as
if3·,0∈L20, T,
ii|f3t, x−f3t, y| ≤Mf3|x−y|, for allx, y∈Rand almostt∈0, Tfor some
Mf3 >0, H4a∈L20, T2,
H5b∈L∞0, T2,
H6c∈L20, T2,
H7k :Y×R → R, whereY {t, s: 0< s < t < T}, satisfies|kt, s, x1−kt, s, x2| ≤
Mk|x1−x2|, for allx1, x2∈R, and almostt, s∈Y,
The stochastic integropartial differential equation3.1can be written in the abstract form 1.2, whereK H L2D,A Δ
z, with domainDA H2D∪H01D. It is well
known thatAis a closed linear operator which generates aC0-semigroup. We also introduce
the mappingsF,G, andhdefined by, respectively,
Fxt,·
T
0
at, sf1
s, xs, z,
s
0
ks, τ, xτ, zdτ
ds,
Gxt,· bt, sf2s, xs,·,
hx· x0, z
n
i1
gi·xti,· T
0
csf3s, xs,·ds.
3.3
One can use H1–H8 to verify that F, G, and h satisfy II–IV in the last section, respectively, with
MF 2Mf1T|a|L20,T2
1 MkT3 1/2
,
MGMf2,
Mh2 n
i1
giL2D Mf3
mD|G|L20,T.
3.4
Consequently theorem2.4can be applied for3.1.
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