Volume 2009, Article ID 785628,27pages doi:10.1155/2009/785628
Research Article
Existence and Stability of
Solutions for Nonautonomous Stochastic
Functional Evolution Equations
Xianlong Fu
Department of Mathematics, East China Normal University, Shanghai 200241, China
Correspondence should be addressed to Xianlong Fu,[email protected]
Received 19 March 2009; Accepted 2 June 2009
Recommended by Jozef Banas
We establish the results on existence and exponent stability of solutions for a semilinear nonautonomous neutral stochastic evolution equation with finite delay; the linear part of this equation is dependent on time and generates a linear evolution system. The obtained results are applied to some neutral stochastic partial differential equations. These kinds of equations arise in systems related to couple oscillators in a noisy environment or in viscoelastic materials under random or stochastic influences.
Copyrightq2009 Xianlong Fu. 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
In this paper we study the existence and asymptotic behavior of mild solutions for the following neutral non-autonomous stochastic evolution equation with finite delay:
dZt−kt, Zt −AtZt−kt, Zt Ft, ZtdtGt, ZtdWt, 0≤t≤T,
Z0 φ∈LpΩ, Cα,
1.1
whereAt generates a linear evolution system, or say linear evolution operator{Ut, s : 0≤s≤t≤T}on a separable Hilbert spaceHwith the inner product·,·and norm · .k;F andGare given functions to be specified later.
discuss these topics is the semigroup approach; see, for example, Da Prato and Zabczyk1, Dawson2, Ichikawa3, and Kotelenez4. In paper5Taniguchi et al.have investigated the existence and asymptotic behavior of solutions for the following stochastic functional differential equation:
dZt −AZt ft, Zt
dtgt, ZtdWt, 0≤t≤T,
Z0φ∈LpΩ, Cα,
1.2
by using analytic semigroups approach and fractional power operator arguments. In this work as well as other related literatures like 6–9, the linear part of the discussed equation is an operator independent of time t and generates a strongly continuous one-parameter semigroup or analytic semigroup so that the semigroup approach can be employed. We would also like to mention that some similar topics to the above for stochastic ordinary functional differential equations with finite delays have already been investigated successfully by various authorscf.6,10–13and references in14among others. Related work on functional stochastic evolution equations of McKean-Vlasov type and of second-order are discussed in15,16.
However, it occurs very often that the linear part of 1.2 is dependent on time t. Indeed, a lot of stochastic partial functional differential equations can be rewritten to semilinear non-autonomous equations having the form of1.2withA At. There exists much work on existence, asymptotic behavior, and controllability for deterministic non-autonomous partial functional differential equations with finite or infinite delays; see, for example,17–20. But little is known to us for non-autonomous stochastic differential equations in abstract space, especially for the case that At is a family of unbounded operators.
Our purpose in the present paper is to obtain results concerning existence, uniqueness, and stability of the solutions of the non-autonomous stochastic differential equations1.1. A motivation example for this class of equations is the following non-autonomous boundary problem:
dZt, x−bZt−r, x
at, x ∂ 2
∂2xZt, x−bZt−r, x ft, Ztr1t, x
dt
gt, Ztr1t, xdβt,
Zt,0 Zt, π 0, t≥0,
Zθ, x φx, θ∈−r,0, 0≤x≤π.
1.3
As stated in paper21, these problems arise in systems related to couple oscillators in a noisy environment or in viscoelastic materials under random or stochastic influencessee also22
et al. have, under coercivity condition in an integral form, investigated the second moment
almost sureexponential stability and ultimate boundedness of solutions to the following non-autonomous semilinear stochastic delay equation:
dXt−kt, Xt−r AtXt Ft, XtdtGt, XtdWt, t≥0,
Xt φt t∈−r,0
1.4
on a Hilbert spaceH, whereA·∈L∞0, T;LV, V∗withV ⊂H⊂H∗⊂V∗.
As we know, non-autonomous evolution equations are much more complicate than autonomous ones to be dealt with. Our approach here is inspired by the work in paper
5,18,19. That is, we assume that{At:t≥0}is a family of unbounded linear operators on Hwithcommondense domain such that it generates a linear evolution system. Thus we will apply the theory of linear evolution system and fractional power operators methods to discuss existence, uniqueness, and pth p > 2 moment exponential stability of mild solutions to the stochastic partial functional differential equation 1.1. Clearly our work can be regarded as extension and development of that in 5,21 and other related papers mentioned above.
We will firstly in Section 2 introduce some notations, concepts, and basic results about linear evolution system and stochastic process. The existence and uniqueness of mild solutions are discussed inSection 3by using Banach fixed point theorem. InSection 4, we investigate the exponential stability for the mild solutions obtained in Section 3, and the conditions for stability are somewhat weaker than in5. Finally, inSection 5we apply the obtained results to1.3to illustrate the applications.
2. Preliminaries
In this section we collect some notions, conceptions, and lemmas on stochastic process and linear evolution system which will be used throughout the whole paper.
Let Ω,F, P be a probability space on which an increasing and right continuous family{Ft}t∈0,∞ of complete sub-σ-algebras ofFis defined.H and K are two separable
Hilbert space. Suppose that Wt is a given K-valued Wiener Process with a finite trace nuclear covariance operatorQ > 0. Letβnt,n 1,2, . . ., be a sequence of real-valued
one-dimensional standard Brownian motions mutually independent overΩ,F, P. Set
Wt
∞
n1
λnβnten, t≥0, 2.1
whereλn≥0 n1,2, . . .are nonnegative real numbers, and{en}n1,2, . . .is a complete
orthonormal basis inK. LetQ ∈ LK, Kbe an operator defined byQen λnen with finite
trace trQ n∞1λn < ∞. Then the above K-valued stochastic processWt is called a
Definition 2.1. Letσ∈ LK, Hand define
σ2L0
2 :trσQσ ∗ ∞
n1
λnσen
2
. 2.2
IfσL0
2 <∞, thenσis called aQ-Hilbert-Schmidt operator, and letL 0
2K, Hdenote, the space of allQ-Hilbert-Schmidt operatorsσ:K → H.
In the next section the following lemmasee1, Lemma 7.2plays an important role.
Lemma 2.2. For anyp≥2and for arbitraryL0
2-valued predictable processΦt,t∈0, T, one has
E
sup
s∈0,t
s
0Φ
sdWs
p
≤CpE
t
0
Φs2L0 2ds
p/2
2.3
for some constantCp>0.
Now we turn to state some notations and basic facts from the theory of linear evolution system.
Throughout this paper,{At : 0 ≤ t ≤ T}is a family of linear operators defined on Hilbert spaceH, and for this family we always impose on the following restrictions.
B1The domainDAof{At: 0≤t≤T}is dense inHand independent oft;Atis closed linear operator.
B2For eacht ∈0, T, the resolventRλ, Atexists for allλwith Reλ ≤0,and there existsC >0 so thatRλ, At ≤C/|λ|1.
B3There exists 0< δ≤1 andC >0 such thatAt−AsA−1τ ≤C|t−s|δfor all
t, s, τ∈0, T.
Under these assumptions, the family{At : 0 ≤ t ≤ T} generates a unique linear evolution system, or called linear evolution operators{Ut, s,0≤s≤t≤T}, and there exists a family of bounded linear operators{Rt, τ|0≤τ ≤t≤T}withRt, τ ≤K|t−τ|δ−1such thatUt, shas the representation
Ut, s e−t−sAt
t
s
e−t−τAτRτ, sdτ, 2.4
where exp−τAt denotes the analytic semigroup having infinitesimal generator −At
For the linear evolution system{Ut, s,0 ≤ s ≤ t ≤ T}, the following properties are well known:
aUt, s ∈ LH, the space of bounded linear transformations onH, whenever 0 ≤ s ≤ t ≤ T and mapsHintoDAast > s. For eachx ∈H, the mappingt, s → Ut, sxis continuous jointly insandt;
bUt, sUs, τ Ut, τfor 0≤τ ≤s≤t≤T;
cUt, t I;
d ∂/∂tUt, s −AtUt, s,fors < t.
We also have the following inequalities:
e−tAs ≤K, t≥0, s∈0, T,
Ase−tAs ≤ K
t , t, s∈0, T,
AtUt, s ≤ K
|t−s|, 0≤s≤t≤T.
2.5
Furthermore, AssumptionsB1–B3imply that for eacht∈0, Tthe integral
A−αt 1
Γα
∞
0
sα−1e−sAtds 2.6
exists for eachα ∈ 0,1. The operator defined by2.6is a bounded linear operator and yieldsA−αtA−βt A−αβt. Thus, we can define the fractional power as
Aαt A−αt−1, 2.7
which is a closed linear operator withDAαt dense inH and DAαt ⊂ DAβt for α≥β.DAαtbecomes a Banach space endowed with the normx
α,tAαtx, which is
denoted byHαt.
The following estimates andLemma 2.3are from23, Part II:
fort, τ ∈0, Tand 0≤α < β, and
Aβte−sAt ≤ K
β sβ e
−ws, t >0, β≤0, w >0, 2.9
AβtUt, s ≤ K
β
|t−s|β, 0< β < t1 2.10
for somet >0, whereKα, βandKβindicate their dependence on the constantsα,β.
Lemma 2.3. Assume thatB1–B3hold. If0≤γ≤1,0≤β≤α <1δ,0< α−γ ≤1, then, for any0≤τ < t Δt≤t0,0≤ζ≤T,
AγζUt Δt, τ−Ut, τA−βτ ≤Cβ, γ, αΔtα−γ|t−τ|β−α. 2.11
For more details about the theory of linear evolution system, operator semigroups, and fraction powers of operators, we can refer to23–25.
In the sequel, we denote for brevity thatHα DAαt0for somet0 > 0, andCα
C−r,0, Hα, the space of all continuous functions from−r,0intoHα. Suppose thatZt:
Ω → Hα,t≥ −r, is a continuousFt-adapted,Hα-valued stochastic process, we can associate
with another processZt : Ω → Cα,t ≥ 0 , by settingZtθω Ztθω,θ ∈ −r,0.
Then we say that the processZtis generated by the processZt. LetMCαp,p >2, denote
the space of allFt-measurable functions which belong toLpΩ, Cα; that is,MCαp,p >2,
is the space of allFt-measurableCα-valued functionsψ :Ω → Cαwith the normEψpCα
Esupθ∈−r,0Aαt
0ψθp<∞.
Now we end this section by stating the following result which is fundamental to the work of this note and can be proved by the similar method as that of1, Proposition 4.15.
Lemma 2.4. LetΦt: Ω → L0
2K, H,t≥0be a predictable,Ft-adapted process. IfΦtk∈Hα, t ≥ 0, for arbitraryk ∈K, andt0EΦs2L0
2ds < ∞,
t
0EAαt0Φs 2 L0
2ds < ∞, then there holds
Aαt0
t
0Φ
sdWs
t
0
Aαt0ΦsdWs. 2.12
3. Existence and Uniqueness
In this section we study the existence and uniqueness of mild solutions for 1.1. For this equation we assume that the following conditions holdletp >2.
H1The functionk : 0, T×Cα → Hα satisfies the following Lipschitz conditions:
that is, there is a constantL0 >0 such that, for anyφ1, φ2andφ∈Cα,t∈0, T,
kt, φ2
−kt, φ1
p
α≤L0φ2−φ1 p Cα,
kt, φpα≤L0φpC α,
3.1
H2The function F : 0, T×Cα → H satisfies the following Lipschitz conditions:
that is, there is a constantL1 >0 such that, for anyφ1, φ2∈Cαandt∈0, T,
Ft, φ2−Ft, φ1 p≤L1 φ2−φ1 pCα. 3.2
H3For functionG:0, T×Cα → L02K, H, there exists a constantL2>0 such that
Gt, φ2−Gt, φ1pL0 2≤
L2 φ2−φ1 pC
α 3.3
for anyφ1, φ2∈Cαandt∈0, T.
UnderH2andH3, we may suppose that there exists a constantL3>0 such that
Ft, φ p
Gt, φ pL0 2
≤L3 φ pC α1
3.4
for anyφ∈Cαandt∈0, T.
Similar to the deterministic situation we give the following definition of mild solutions for1.1.
Definition 3.1. A continuous processZ:−r, T×Ω → His said to be a mild solution of1.1
if
iZtis measurable andFt-adapted for eacht∈0, T;
iiT0Zspds <∞, a.s.;
iiiZtverifies the stochastic integral equation
Zt Ut,0φ0−k0, φkt, Zt
t
0
Ut, sFs, Zsds
t
0
Ut, sGs, ZsdWs
3.5
on interval0, T, andZθ φθforθ∈−r,0.
Next we prove the existence and uniqueness of mild solutions for1.1.
Theorem 3.2. Let 0 < α < min{p−2/2p, δ} and p > 2/δ. Suppose that the assumptions
Proof. Denote byDthe Banach space of all the continuous processesZtwhich belong to the spaceC−r, T, LpΩ, HwithZ
D<∞, where
ZD: sup
t∈0,T
EZtpC
1/p
,
EZtpC:E
sup
θ∈−r,0Ztθ p
.
3.6
Define the operatorΨonD:
ΨZt Aαt0Ut,0φ0−k0, φAαt0kt, A−αt0Zt
t
0 Aαt
0Ut, sF
s, A−αt
0Zs
ds
t
0 Aαt
0Ut, sG
s, A−αt
0Zs
dWs.
3.7
Then it is clear that to prove the existence of mild solutions to1.1is equivalent to find a fixed point for the operatorΨ. Next we will show by using Banach fixed point theorem that
Ψhas a unique fixed point. We divide the subsequent proof into three steps.
Step 1. For arbitraryZ∈ D,ΨZtis continuous on the interval0, Tin theLp-sense.
Let 0< t < Tand|h|be sufficiently small. Then for any fixedZ∈ D, we have that
EΨZth−ΨZtp≤4p−1E Aαt0Uth,0−Ut,0φ0−k0, φ p
4p−1EAαt0kth, Aαt0Zth−kt, Aαt0Ztp
4p−1E
th
0 Aαt
0Uth, sF
s, A−αt
0Zs
ds
−
t
0
A−αt0Ut, sFs, A−αt0Zsds
p
4p−1E
th
0
Aαt0Uth, sGs, A−αt0Zs
dWs
−
t
0
Aαt0Ut, sGs, A−αt0ZsdWs
p
:I1I2I3I4.
3.8
Thus, byLemma 2.3we get
I14p−1EAαt0Uth,0−Ut,0A−βt0Aβt0
φ0−k0, φp
≤4p−1Cp|h|pβ−αEAβt0φ0−k0, φp,
whereβ >0 satisfies thatα < β <1 andC > 0 are constant. From ConditionH2it follows that
I3≤8p−1E
t
0
Aαt0Uth, s−Ut, sFs, A−αt0Zsds
p
8p−1E
th
t
Aαt0Uth, sFs, A−αt0Zs
ds
p
I31I32.
3.10
And by2.8–2.10one has that
I32≤8p−1E
th
t
Aαt0A−βthAβthUth, sFs, A−αt0Zs
ds
p
≤8p−1Kα, βKβpE
th
t
1
th−sβ
F
s, A−αt0Zs ds
p
≤8p−1Kα, βKβp
th
t
1
th−sqβds
p/q
E
th
t
Fs, A−αt0Zs
p
ds
≤8p−1Kα, βKβp
1 1−qβ
p−1
L3h1−βp
1ZpD,
3.11
whereq >0 solves 1/p1/q1, and
I31 8p−1E
t
0 Aαt0
e−th−sAth
th
s
e−th−τAτRτ, sdτ
−e−t−sAt−
t
s
e−t−τAτRτ, sdτ
Fs, A−αt0Zsds
p
≤8p−1
⎛
⎝t
0
Aαt0
⎡ ⎣e
−th−sAth
th
s
e−th−τAτRτ, sdτ
−e−t−sAt−
t
s
e−t−τAτRτ, sdτ
qds ⎞ ⎠ p/q ×E t 0
Fs, A−αt0Zs pds
:8p−1Ip/q31 E
t
0
Fs, A−αt0Zs
p
ds
≤8p−1L 3I
p/q
31 T
1ZpD,
while
I31
t
0
Aαt0
e−th−sAth
th
s
e−th−τAτRτ, sdτ
−e−t−sAt−
t
s
e−t−τAτRτ, sdτ
q
ds
≤2q−1
t
0
Aαt0e−th−sAth−e−t−sAt qds
2q−1
t
0
Aαt0
t
s
e−th−τAτ−e−t−τAτRτ, sdτ
Aαt0
th
t
e−th−τAτRτ, sdτ
q
ds
≤2q−1
t−η
0
Aαt0e−th−sAth−e−t−sAt qds
2q−1
t
t−η
Aαt0e−th−sAth−e−t−sAt qds
8q−1
t
0
t−η
s
Aαt0e−th−τAτ−e−t−τAτRτ, s dτ
q
ds
8q−1
t
0
t
t−η
Aαt0e−th−τAτ−e−t−τAτRτ, s dτ
q
ds
4q−1
t
0
th
t
Aαt0e−th−τAτRτ, sdτ
q
ds,
3.13
whereη >0 is very small. SinceAte−τAsis uniformly continuous int, τ, sfor 0≤t≤T, m≤τ ≤Tand 0≤s≤T, wheremis any positive numbersee23,25, we deduce that
I31 ≤2q−1Kα, βKβqqT−η
4q−1 1−qβq
Kα, βKβqhη1−qβ−h1−qβη1−qβ
8q−1 1δqq
Kα, βKβ
C δ
q
Tη1δqη1δqq
8q−1
Kα, βKβC
1−β1 δ−1q
q
hη1−βh1−βqT1δ−1qη1δ−1q
4q−1
Kα, βKβC
1−β1 δ−1q
q
h1−βqT1δ−1q.
In a similar way, we have that
I4≤6p−1E
t
0
A−αt0Uth, s−Ut−sGs, A−αt0Zs
dWs
p
6p−1E
th
t
A−αt0Uth, sGs, A−αt0Zs
dWs
p
:I41I42.
3.15
By virtue of ConditionH3and by usingLemma 2.2, we infer that
I42≤8p−1E
th
t
Aαt0A−βth AβthUth, s Gs, A−αt0Z s 2L
2ds
p/2
≤8p−1Kα, βKβpEth t
1
th−s2β
Gs, A−αt
0Zs 2L0 2ds
p/2
≤8p−1Kα, βKβp
th
t
1
th−s2pβ/p−2ds
p−2/2 E
th
t
Gs, A−αt0Zs 2L0 2
p
ds
≤8p−1Kα, βKβp
p−2
p−2−2pβ
p−2/2
L3hp−2/p−2−2pβ11ZpD,
I418p−1E
t
0 Aαt0
e−th−sAth
th
s
e−th−τAτRτ, sdτ
−e−t−sAt−
t
s
e−t−τAτRτ, sdτ
Gs, A−αt0ZsdWs
p
≤8p−1CpE
⎛
⎝t
0
Aαt0
e−th−sAth
th
s
e−th−τAτRτ, sdτ
−e−t−sAt−
t
s
e−t−τAτRτ, sdτ
Gs, A−αt
0Zs
2
L0 2
ds
⎞ ⎠
≤8p−1Cp
⎛
⎝t
0
Aαt0
e−th−sAth
th
s
e−th−τAτRτ, sdτ
−e−t−sAt−
t
s
e−t−τAτRτ, sdτ
p/p−2ds
⎞ ⎠
p−2/2
×E
t
0
Gs, A−αt0Zs pds
:8p−1CpIp−
2/2 41 E
t
0
Gs, A−αt0Zs pds
≤8p−1CpIp−
2/2 41 L3T
1ZpD,
3.16
while
I41
t
0
Aαt0
e−th−sAth
th
s
e−th−τAτRτ, sdτ
−e−t−sAt−
t
s
e−t−τAτRτ, sdτ
p/p−2 ds
≤22/p−2
t
0
Aαt0e−th−sAth−e−t−sAt p/p−2ds
22/p−2
t
0
Aαt0
t
s
e−th−τAτ−e−t−τAτRτ, sdτ
Aαt0
⎡
⎣th
t
e−th−τAτRτ, sdτ
p/p−2
ds
≤22/p−2
t−η
0
Aαt
0
e−th−sAth−e−t−sAt p/p−2ds
22/p−2
t
t−η
Aαt
0
e−th−sAth−e−t−sAt p/p−2ds
82/p−2
t
0
t−η
s
Aαt0e−th−τAτ−e−t−τAτRτ, s dτ
p/p−2
82/p−2
t
0
Aαt0
t
t−η
e−th−τAτ−e−t−τAτRτ, s
dτ
p/p−2
ds
42/p−2
t
0
th
t
Aαt0e−th−τAτRτ, sdτ
p/p−2ds.
3.17
Again by the uniform continuity ofAt−τAsand2.9and2.10, we can compute that
I41 ≤2q1−1α, βq1Kβq1q1T−η
4q1−1 1−q1βq1
Kα, βKβq1hη1−q1β−h1−q1β
η1−q1β
8q1−1 1δq1q1
Kα, βKβ
C δ
q1
Tη1δq1η1δq1
q1
8q1−1
Kα, βKβC
1−β1 δ−1q1
q1
hη1−βh1−βq1T1δ−1q1η1δ−1q1
4q1−1
Kα, βKβC
1−β1 δ−1q1
q1
h1−βq1T1δ−1q1,
3.18
whereq1 p/p−2.
The above arguments show thatI1,I3 I31I32 andI4 I41I42are all tend to 0 as
|h| → 0 andη → 0, andI2also clearly tends to 0 from ConditionH1. Therefore,ΨZtis continuous on the interval0, Tin theLp-sense.
Step 2. We prove thatΨD⊂ D.
To this end, letZ∈ D. Then we have that
EΨZtpC≤4p−1Esup −r≤θ≤0
Aαt0Utθ,0φ0−k0, φ p
4p−1E sup −r≤θ≤0
Aαt0ktθ, A−αt0Z tθ p
4p−1E sup −r≤θ≤0
tθ
0
Aαt0Utθ, sFs, A−αt0Zs
ds
p
4p−1E sup −r≤θ≤0
tθ
0
Aαt0Utθ, sGs, A−αt0Zs
dWs
p
:I5I6I7I8.
Again byLemma 2.3, we get that
I5≤4p−1Esup −r≤θ≤0
Aαt0Utθ,0−U0,0A−βt0Aβt0φ0−k0, φ
−Aαt0φ0−k0, φ p
≤8p−1
E sup −r≤θ≤0
Aαt
0Utθ,0−U0,0A−βt0Aβt0φ0−k0, φ
p
E Aαt0
φ0−k0, φ p
≤8p−1CpTpβ−αE φ0−k0, φ pαE φ0−k0, φ pα.
3.20
By ConditionH1one easily has
I6≤4p−1L0ZpD. 3.21
And2.9and2.10imply that
I7≤4p−1E sup −r≤θ≤0
tθ
0
Aαt0Utθ, sFs, A−αt0Zsds
p
≤4p−1E sup −r≤θ≤0
tθ
0
Aαt0A−βtθ AβtθUtθ, s Fs, A−αt0Z s ds
p
≤4p−1Kα, βKβpE sup −r≤θ≤0
tθ
0
1
tθ−sβ
Fs, A−αt0Z s ds
p
≤4p−1Kα, βKβpE sup −r≤θ≤0
tθ
0
1
tθ−sqβds
p/qtθ
0
Fs, A−αt0Zs pds
≤4p−1L3Kα, βKβp
p−1
p−pβ−1
p−1
Tp−pβ1ZpD.
3.22
From the inequality
t
σ
t−sα−1s−σ−αds π
established in26, it follows that
I83p−1
sinππγpE sup −r≤θ≤0
tθ
0
Aαt0Utθ, stθ−sγ−1Rsds
p,
3.24
where
Rs
s
0
s−σ−γUs, σGσ, A−αt0Zσ
dWσ, 3.25
and 0< γ <p−2/2p. Thus,
I8≤3p−1
Kα, βKβsinπγ π
pE sup
−r≤θ≤0
tθ
0
tθ−sγ−β−1Rsds
p
≤3p−1L3Cp
Kα, βKβM0p
p−1 pβ−γ−1
p−1
Tpβ−γ−1
· 1
1−2pγ/p−2T
4−2pγ/p−21Zp
D
,
3.26
whereM0sup0≤σ≤s≤TUs, σ. HenceΨZ p
D<∞and soΨD⊂ D.
Step 3. It remains to verify thatΨis a contraction onD. Suppose thatZ1,Z2∈ D, then for any fixedt∈0, T,
EΨZ2t−ΨZ1tpC
≤3p−1Esup −r≤θ≤0
Aαt0ktθ, A−αt0Z2
,tθ−Aαt0ktθ, A−αt0Z1,tθ p
3p−1E sup −r≤θ≤0
tθ
0
Aαt0Utθ, s
Fs, A−αt0Z2s
−Fs, A−αt0Z1s
ds
p
3p−1E sup −r≤θ≤0
tθ
0
Aαt0Utθ, s
Gs, A−αt0Z2s
−Gs, A−αt0Z1s
dWs
p
:I9I10I11.
Clearly,
I9≤3p−1L0Z2−Z1pD,
I10≤
Kα, βKβpE sup −r≤θ≤0
tθ
0
1
tθ−sqβds
p/q
×
tθ
0
Fs, A−αt0Z2s
−Fs, A−αt0Z1s pds
≤L1Kα, βKβp
p−1
p−1pβ
p−1
TppβZ2−Z1pD.
3.28
Next, let
R1u
s
0
s−σ−γUs, σGs, A−αt0Z2s−Gs, A−αt0Z1sdWσ, 3.29
then there holds that
I11 ≤3p−1
Kα, βKβsinπγ π
pEsup
−r≤θ≤0
tθ
0
tθ−sγ−β−1R1sds
p
≤3p−1L2Cp
Kα, βKβM0
p p−1
pγ−β−1
p−1
Tpγ−β−1· 1
1−2pγ/p−2T
4−2pγ/p−2
· Z2−Z1pD.
3.30
Therefore,
ΨZ2−ΨZ1pD≤ΘTZ2−Z1pD 3.31
with
ΘT 1
1−3p−1L 0
Kα, βKβp
p−1
p−1pβ
p−1 Tppβ
3p−1L2Cp
Kα, βKβM0sinπγ π
ppγp−−β1−1p−1Tpγ−β−1
· 1
1−2pγ/p−2T
4−2pγ/p−2
.
3.32
Banach fixed point theorem we obtain a unique fixed pointZ∗∈ DT1for operatorΨ, and hence Zt A−αt
0Z∗tis a mild solution of1.1. This procedure can be repeated to extend the solution to the entire interval0, Tin finitely many similar steps, thereby completing the proof for the existence and uniqueness of mild solutions on the whole interval0, T.
For the globe existence of mild solutions for 1.1, it is easy to prove the following result.
Theorem 3.3. Suppose that the family Att≥0 satisfiesB1–B3 on interval 0,∞ such that Ut, sis defined for all0≤s≤t <∞. Let the functionsk:0,∞×Cα → Hα,F:0,∞×Cα →
HandG:0,∞×Cα → L02K, Hsatisfy the AssumptionsH1–H3,respectively. Then there exists a unique, global, continuous solutionZ : −r,∞×Ω → Hto1.1for any initial value φ∈MCαp.
Proof. Since T is arbitrary in the proof of the previous theorem, this assertion follows immediately.
4. Exponential Stability
Now, we consider the stability result of mild solutions to1.1. For this purpose we need to assume further that the familyAtt≥0verifies additionally the following.
B4sup0<t,s<∞AtA−1s < ∞, and there exists a closed operator A∞ with bounded inverse and domainDAsuch that
At−A∞A−10 −→0 4.1
ast → ∞. Then the following inequalities are true:
Ut, s ≤Me−ωt−s,
AαtUt, s ≤M1e−ωt−st−s−α,
Aαt0Ut, s ≤M1e−ωt−st−s−α
4.2
for allt≥s≥0see23,25.
Theorem 4.1. Let the functionsk:0,∞×Cα → Hα,F :0,∞×Cα → H,G:0,∞×
Cα → L02K, Hsatisfy the Lipschitz conditionsH1–H3, respectively. Furthermore, assume that there exist nonnegative real numbersNi ≥ 0and continuous functions ci : 0,∞ → R with
cit≤Pie−ωt(i0,1,2),Pi>0, such that
Ekt, ZtpCα ≤N0EZtpCαc0t, t≥0,
EFt, Ztp≤N1EZtpCαc1t, t≥0,
EGt, ZtpL0 2 ≤
N2EZtpCαc1t, t≥0
for any mild solution Zt of 1.1. If the constants N1 and N2 are small enough such that ω − K3/1−4p−1N0 > 0 withK3 determined by4.13below, then the solution is (the pth moment) exponentially stable. In other words, there exist positive constants >0andKp, , φ>0such that, for eacht≥0,
EZtpCα ≤K
p, , φe−et. 4.4
Proof. LetZtbe a mild solution of1.1, then
EZtpCα E
sup −r≤θ≤0
Ztθpα
≤4p−1E
sup −r≤θ≤0
Utθ,0φ0−k0, φ pα
4p−1E
sup −r≤θ≤0
ktθ, Ztθpα
4p−1E
sup −r≤θ≤0
tθ
0
Utθ, sFs, Zsds
p
α
4p−1E
sup −r≤θ≤0
tθ
0
Utθ, sGs, ZsdWs
p
α
:I12I13I14I15,
I12≤4p−1E
sup −r≤θ≤0
Mp1e−pωtθtθ−pα φ0−k0, φ p
≤4p−1Mp
1e−pωt−rt−r−
pαE φ0−k0, φ p
≤4p−1Mp1epωrr−pαE φ0−k0, φ p·e−ωt
:K0·e−ωt,
I13≤4p−1N0EZtpCαc0t
,
I14≤4p−1Mp1E
sup −r≤θ≤0
tθ
0
tθ−s−αe−ωtθ−sFs, Zsds
p
4p−1Mp1E
sup −r≤θ≤0
tθ
0
tθ−s−αe−p−1ω/ptθ−se−ω/ptθ−sFs, Zsds
≤4p−1Mp1E
⎛
⎝sup
−r≤θ≤0
tθ
0
tθ−s−p/p−1αe−ωtθ−sds
p−1
×
tθ
0
e−ωtθ−sFs, Zspds
⎞ ⎠
≤4p−1Mp1eωr
∞
0
tθ−s−p/p−1αe−ωtθ−sds
p−1 E
t
0
e−ωt−sFs, Zspds
≤4p−1Mp1eωr
Γ
1− p p−1α
ωp/p−1α−1
p−1
·
t
0
e−ωt−sN1EZspCαc1s
ds
:K1·
t
0
e−ωt−sN1EZspCαc1s
ds.
4.5
ForI15we have that
I154p−1E
sup −r≤θ≤0
tθ
0
Aαt0Utθ, stθ−sγ−1R2sds
p,
4.6
where
R2u
s
0
s−σ−γUs, σGs, ZsdWσ 4.7
withγsatisfyingα1/p < γ <1/2. Then,
I15≤4p−1Mp1sinπγ π
pE
⎛
⎝ sup
−r≤θ≤0
tθ
0
tθ−sqγ−α−1e−ωtθ−sds
p−1
·
tθ
0
e−ωtθ−sR2spds
⎞ ⎠
≤4p−1eωrMp1
Γ
1− p
p−11α−γ
ωp/p−11α−γ−1
p−1
·
t
0
e−ωt−sER2spds
.
Since, byLemma 2.2,
t
0
e−ωt−sER2spds
t
0
e−ωt−sE
s
0
s−σ−γUs, σGs, ZsdWσ
pds
≤CpMp
t
0
e−ωt−sE
s
0
s−σ−2γe−2ωs−σGs, Zs2L0 2
dσ
p/2 ds
≤CpMpE
t
0
s
0
s−σ−2γe−2ω1/qs−σe−2ω1/ps−σGs, Zs2L0 2
dσ
p/2 ds,
4.9
and the Young inequality enables us to get immediately that
E
t
0
e−ωt−sR2spds
≤CpMp
t
0
s−σ−2γe−2ω1/qs−σds
p/2t
0
e−ωt−σEGs, ZspL0 2
dσ
,
≤CpMp
Γ1−2γ
2ω q
2γ−1p/2t
0
e−ωt−σEGs, ZspL0 2
dσ
,
≤CpMP
Γ1−2γ
2ω q
2γ−1p/2t
0
e−ωt−σN2EZspCαc2s
dσ . 4.10 Hence,
I15≤4p−1eωrMp1
sinππγpΓ
1− p
p−11α−γ
ωp/p−11α−γ−1
p−1
·CpMP
Γ1−2γ
2ω q
2γ−1p/2t
0
e−ωt−sN2EZspCαc2s
ds
:K2·
t
0
e−ωt−sN2EZspCαc2s
ds.
4.11
Therefore, combining the above estimates yields that
1−4p−1N0
EZtpCα ≤K0e−ωt4p−1c0t K3e−ωt
t
0
eωsEZspCαds
e−ωt
t
0 eωsK
1c1s K2c2sds,
where
K3 K1N1K2N2. 4.13
Now, Taking arbitrarily∈Rwith 0< < ωandT >0 large enough, we obtain that
1−4p−1N0
T
0
etEZtpCαdt
≤K0
T
0
e−ω−tdt
T
0
4p−1c0tetdtK3
T
0 et−ωt
t
0
eωsEZspCαds dt
T
0 e−ω−t
t
0
eωsK1c1s K2c2sds dt.
4.14
While,
T
0 et−ωt
t
0
eωsEZspCαds dt
T
0
T
s
eωsEZspCαet−ωtdt ds
≤ 1
ω−
T
0
esEZspCαds.
4.15
Substituting4.15into4.14gives that
1−4p−1N0
T
0
etEZtpCαdt
≤K0
T
0
e−ω−tdt
T
0
4p−1c0tetdt K3 ω−
T
0
esEZspCαds
T
0 e−ω−t
t
0
eωsK1c1s K2c2sds dt.
4.16
SinceK3 can be small enough by assumption, it is possible to choose a suitable∈Rwith 0< < ω−K3/1−4p−1N0such that
1−4p−1N0− K3
Hence lettingT → ∞in4.16yields that
∞
0
etEZtpCαdt≤
1 1−4p−1N
0−K3/ω−
K 0
∞
0
e−ω−tdt
∞
0
4p−1c0tetdt
∞
0
e−ω−t
t
0
eωsK1c1s K2c2sds dt
:Kp, , φ<∞.
4.18
On the other hand, from the deduction of4.12it is not difficult to see that it also holds withωsubstituted by, that is,
1−4p−1N 0
EZtpCα ≤K0e−t4p−1c0t K3e−t
t
0 esEZ
spCαds
e−t
t
0
esK1c
1s K2c2sds,
4.19
then it follows from4.18and4.19thatnote the conditions forci·
EZtpCα ≤
K04p−1c0tetK3Kp, , φ
∞
0
esK1c1s K2c2sds
·e−t
:Kp, , φe−t,
4.20
which is our desired inequality. Then the proof is completed.
We also have the following result for almost surely exponential stability.
Theorem 4.2. Suppose that all the conditions ofTheorem 4.1are satisfied. Then the solution is almost surely exponentially stable. Moreover, there exists a positive constant >0such that
lim sup
t→∞
logZtα
t ≤ − ε
2p, a.s. 4.21
Proof. The proof is similar to that of5, Theorem 3.3and we omit it.
5. Examples
Example 5.1. We have
dZt, x−bZt−r, x
at, x ∂ 2
∂2xZt, x−bZt−r, x ft, Ztr1t, x
dt
gt, Ztr2t, xdβt,
Zt,0 Zt, π 0, t≥0,
Zθ, x φx, θ∈−r,0, 0≤x≤π,
5.1
where−r < rit<0,at, xis a continuous function and is uniformly H ¨older continuous in
twith exponent 0 < δ ≤1and satisfies that supx|at, x−a∞, x| → 0 ast → ∞. Let HL20, πandKR,βtdenote a one-dimensional standard Brownian motion.
We define the operatorsAtby
Atu−at, xu 5.2
with the domain
DA u·∈H:u, uabsolutely continuous, u∈H, u0 uπ 0 . 5.3
Then At generates an evolution operator Ut, s satisfying assumptions B1–B4 see
23. SetHαDAαt0for somet0>0 andCαC−r,0, Hα.
In order to discuss the system 5.1, we also need the following assumptions on functionsfandg.
A1The functionsf :0,∞×R → R,g:0,∞×R → Rare continuous and global Lipschitz continuous in the second variable.
A2There exist real numbers N3, N4 > 0 and continuous functions c3·, c4· :
0,∞ → Rsuch that
Ft, y|p≤N3y|P c3t, fort≥0, y∈R,
G
t, y|p≤N4y|P c4t, fort≥0, y∈R,
5.4
Now we can definek:0,∞×C−r,0;Hα → Hα,F:0,∞×C−r,0;Hα → H
andG:0,∞×C−r,0;Hα → L02R, Has
kt, φx:bφ−rx,
Ft, φx:ft, φr1tx,
Gt, φx:gt, φr1tx
.
5.5
Then it is not difficult to verify thatk, F,andGsatisfy the conditionsH1,H2,andH3, respectively, due to AssumptionsA1,A2, since, by the embody property ofHαalso see
25, Corollary 2.6.11, x ≤ CAαt
0x for some constant C > 0. Hence we have, by Theorems3.3and4.1, the following.
Theorem 5.2. Let0< α < min{p−2/2p, δ}andp >2/δ. Suppose that all the above assumptions are satisfied. Then for the stochastic system5.1there exists a global mild solutionZt Zt,0, φ∈ Hα, and it is exponentially stable provided thatN3andN4are small enough.
We present another system for which the linear evolution is given explicitly, and so all the coefficients for the conditions of the obtained results can be estimated properly.
Example 5.3. We have
d
Zt, x
t
−r
π
0
bs−t, y, xZs, ydy ds
∂2 ∂x2 at
Zt, x
t
−r
π
0
bs−t, y, xZs, ydy ds
ft, Zt−r, x
dt
gt, Zt−r, xdβt, t≥0,
Zt,0 Zt, π 0,
Zθ, x φθ, x,−r ≤θ≤0, 0≤x≤π,
5.6
whereatis a positive function and is H ¨older continuous intwith parameter 0< δ <1.f,g andβare as inExample 5.1,b∈C1andbs, y, π bs, y,0 0.
LetHL20, π,KR.Atis defined by
Atf−f−atf 5.7
with the domain
Then it is not difficult to verify thatAtgenerates an evolution operatorUt, ssatisfying assumptionsB1–B4and
Ut, s Tt−se−tsaτdτ, 5.9
whereTtis the compact analytic semigroup generated by the operator−Awith−Af−f forf ∈DA. It is easy to compute thatAhas a discrete spectrum, and the eigenvalues are n2, n∈N, with the corresponding normalized eigenvectorsz
nx
2/πsinnx. Thus for f∈DA, there holds
−Atf ∞
n1
−n2−atf, znzn, 5.10
and clearly the common domain coincides with that of the operatorA. Furthermore, We may defineAαt0 t0∈0, afor self-adjoint operatorAt0by the classical spectral theorem, and
it is easy to deduce that
Aαt0f ∞
n1
n2at0
α
f, znzn 5.11
on the domainDAα {f·∈H,∞
n1n2at0αf, znzn∈H}. Particularly,
A1/2t0f ∞
n1
!
n2at
0f, znzn. 5.12
Therefore, we have that, for eachf∈H,
Ut, sf ∞
n1
e−n2t−s−t
saτdτf, znzn,
Aαt0A−βt0f ∞
n1
n2at0
α−β
f, znzn,
Aαt0Ut, sf ∞
n1
n2at0
α
e−n2t−s−tsaτdτf, znzn.
5.13
Then,
fort, s∈0, T, 0< α < β. And
AβtUt, sf 2
∞
n1
n2at2βe−2n2t−s−2tsaτdτf, zn2
t−s−2β ∞
n1
n2att−s2βe−2n2att−satt−s−2tsaτdτf, zn2
t−s−2β ∞
n1
e2βlogn2att−s−2n2att−satt−s−2staτdτf, zn2
≤t−s−2β ∞
n1
β2βeatt−s−2tsaτdτ"f, zn#2, notec logx−x≤clogc−c,
5.15
which shows that AβtUt, s ≤ C
β/t−sβ for Cβ ββmax{eatt−s−
t
saτdτ : t, s ∈ 0, T}>0.
Now we defineF, Gas above and
kt, ψx
0
−r
π
0
bs, y, xψs, ydy ds 5.16
for allψ ∈ Cα C−r,0, Hαand anyx ∈ 0, π. Thus5.6has the form 1.1. Thus we
can easily obtain its existence and stability of mild solutions for5.6by Theorems3.3and4.1
under some proper conditions.
Acknowledgments
The author would like to thank the referees very much for their valuable suggestions to this paper. This work is supported by the NNSF of Chinano. 10671069, NSF of Shanghaino. 09ZR1408900, and Shanghai Leading Academic Discipline Projectno. B407.
References
1 G. Da Prato and J. Zabczyk,Stochastic Equations in Infinite Dimensions, vol. 44 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1992.
2 D. A. Dawson, “Stochastic evolution equations and related measure processes,”Journal of Multivariate Analysis, vol. 5, pp. 1–52, 1975.
3 A. Ichikawa, “Stability of semilinear stochastic evolution equations,”Journal of Mathematical Analysis and Applications, vol. 90, no. 1, pp. 12–44, 1982.
4 P. Kotelenez, “On the semigroup approach to stochastic evolution equations,” inStochastic Space Time Models and Limit Theorems, Reidel, Dordrecht, The Netherlands, 1985.
6 P. Balasubramaniam and D. Vinayagam, “Existence of solutions of nonlinear neutral stochastic differential inclusions in a Hilbert space,”Stochastic Analysis and Applications, vol. 23, no. 1, pp. 137– 151, 2005.
7 T. E. Govindan, “Stability of mild solutions of stochastic evolution equations with variable delay,”
Stochastic Analysis and Applications, vol. 21, no. 5, pp. 1059–1077, 2003.
8 T. E. Govindan, “Almost sure exponential stability for stochastic neutral partial functional differential equations,”Stochastics, vol. 77, no. 2, pp. 139–154, 2005.
9 N. I. Mahmudov, “Existence and uniqueness results for neutral SDEs in Hilbert spaces,”Stochastic Analysis and Applications, vol. 24, no. 1, pp. 79–95, 2006.
10 T. Caraballo, “Existence and uniqueness of solutions for nonlinear stochastic partial differential equations,”Collectanea Mathematica, vol. 42, no. 1, pp. 51–74, 1991.
11 K. Liu and X. Xia, “On the exponential stability in mean square of neutral stochastic functional differential equations,”Systems & Control Letters, vol. 37, no. 4, pp. 207–215, 1999.
12 T. Taniguchi, “Almost sure exponential stability for stochastic partial functional-differential equations,”Stochastic Analysis and Applications, vol. 16, no. 5, pp. 965–975, 1998.
13 T. Taniguchi, “Successive approximations to solutions of stochastic differential equations,”Journal of Differential Equations, vol. 96, no. 1, pp. 152–169, 1992.
14 E. Mohammed,Stochastic Functional Differential Equations, Research Notes in Mathematics, no. 99, Pitman, Boston, Mass, USA, 1984.
15 D. N. Keck and M. A. McKibben, “Abstract stochastic integrodifferential delay equations,”Journal of Applied Mathematics and Stochastic Analysis, vol. 2005, no. 3, pp. 275–305, 2005.
16 M. A. McKibben, “Second-order neutral stochastic evolution equations with heredity,” Journal of Applied Mathematics and Stochastic Analysis, no. 2, pp. 177–192, 2004.
17 W. E. Fitzgibbon, “Semilinear functional differential equations in Banach space,”Journal of Differential Equations, vol. 29, no. 1, pp. 1–14, 1978.
18 S. M. Rankin III, “Existence and asymptotic behavior of a functional-differential equation in Banach space,”Journal of Mathematical Analysis and Applications, vol. 88, no. 2, pp. 531–542, 1982.
19 X. Fu and X. Liu, “Existence of periodic solutions for abstract neutral non-autonomous equations with infinite delay,”Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 249–267, 2007. 20 R. K. George, “Approximate controllability of nonautonomous semilinear systems,” Nonlinear
Analysis: Theory, Methods & Applications, vol. 24, no. 9, pp. 1377–1393, 1995.
21 T. Caraballo, J. Real, and T. Taniguchi, “The exponential stability of neutral stochastic delay partial differential equations,”Discrete and Continuous Dynamical Systems. Series A, vol. 18, no. 2-3, pp. 295– 313, 2007.
22 J. Wu,Theory and Applications of Partial Functional-Differential Equations, vol. 119 ofApplied Mathematical Sciences, Springer, New York, NY, USA, 1996.
23 A. Friedman,Partial Differential Equations, Holt, Rinehart and Winston, New York, NY, USA, 1969. 24 P. Sobolevskii, “On equations of parabolic type in Banach space,”American Mathematical Society
Translations, vol. 49, pp. 1–62, 1965.
25 A. Pazy,Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 ofApplied Mathematical Sciences, Springer, New York, NY, USA, 1983.
