R E S E A R C H
Open Access
Second-order neutral impulsive stochastic
evolution equations with infinite delay
Chaohui Yue
**Correspondence:
[email protected] School of Sciences, Anhui Agricultural University, Hefei, 230036, China
Abstract
In this paper, we study a class of second-order neutral impulsive stochastic evolution equations with infinite delay (SNISEEIs in short), in which the initial value belongs to the abstract spaceBh. Sufficient conditions for the existence of the mild solutions for
SNISEEIs are derived by means of the Krasnoselskii-Schaefer fixed point theorem. Two examples are given to illustrate the obtained results.
Keywords: second-order stochastic evolution equation; impulsive effect; cosine family; fixed point theorem
1 Introduction
In this paper, we consider the second-order neutral impulsive stochastic evolution equa-tions with infinite delay (SNISEEIs in short) of the following form:
dx(t) –g(t,xt)=Ax(t) +f(t,xt)dt+σ(t,xt) dw(t), t∈J:= [,T], ()
x=φ∈Bh, ()
x() =ψ∈H, ()
x(tk) =Ik(xtk), x
(tk) =˜Ik(xt
k), k= , , . . . ,m. ()
Here, the statex(·) takes values in a separable real Hilbert spaceH with inner product (·,·) and norm · , whereA:D(A)⊂H→His the infinitesimal generator of a strongly continuous cosine familyC(t) onH. The historyxt : (–∞, ]→H,xt(θ) =x(t+θ), for
t≥, belongs to the phase spaceBh. Now, we present the abstract phase spaceBh. Assume
thath: (–∞, ]→(,∞) is a continuous function withl=–∞ h(t) dt<∞. For anya> , define
Bh=
ψ: (–∞, ]→H:Eψ(θ)/is a bounded and measurable
function on [–a, ] and
–∞h(s)s≤supθ≤
Eψ(θ)/ds<∞
.
We endowBhwith the norm
ψBh=
–∞h(s) sup
s≤θ≤
Eψ(θ)/ds, for allψ∈Bh,
then (Bh,·Bh) is a Banach space []. LetKbe another separable Hilbert space with inner
product (·,·)Kand norm · K. Suppose{w(t) :t≥}is a givenK-valued Wiener process
with a finite trace nuclear covariance operatorQ≥ defined on a complete probability space (,F,P) equipped with a normal filtration{Ft}t≥, which is generated by the Wiener
processw. We are also employing the same notation · for the operator normL(K;H), whereL(K;H) denotes the space of all bounded linear operators fromKintoH. Assume thatg,f :J×Bh→H(i= , ) andσ:J×Bh→LQ(K,H) are appropriate mappings
spec-ified later. Here,LQ(K,H) denotes the space of allQ-Hilbert-Schmidt operators fromK
intoH, which will be defined in the next section. The initial dataφ={φ(t) : –∞<t≤}is anF-adapted,Bh-valued stochastic process independent of the Wiener processwwith
finite second moment.ψis anF-adapted,H-valued random variable independent of the Wiener processwwith finite second moment.Ikand˜Ik:Bh→Hare appropriate
func-tions. Moreover, let =t<t<· · ·<tm<tm+=T, be given time points and the symbol ξ(t) represents the jump of the functionξatt, which is defined byξ(t) =ξ(t+) –ξ(t–). Stochastic partial differential equations (SPDEs in short) with delay have attracted great interest due to their applications in describing many sophisticated dynamical systems in physical, biological, medical and social sciences. One can see [–] and the references therein for details. Moreover, to describe the systems involving derivatives with delay, Hale and Lunel [] introduced the deterministic neutral functional differential equations, which are of great interest in theoretical and practical applications. Taking the environ-mental disturbances into account, Kolmanovskii and Myshkis [] introduced the neutral stochastic functional differential equations (NSFDEs in short) and gave its applications in chemical engineering and aero elasticity. The investigation of qualitative properties such as existence, uniqueness and stability for NSFDEs has received much attention. One can see [, , –] and the references therein. In addition, impulsive effects exist in many evo-lution processes in which states are changed abruptly at certain moments of time, involved in such fields as medicine and biology, economics, bioengineering, chemical technology
etc.(see [, ] and the references therein).
On the other hand, the study of abstract deterministic second-order evolutions equa-tions governed by the generator of a strongly continuous cosine family was initiated by [] and subsequently studied by [, ]. The second-order stochastic differential equa-tions are the right model in continuous time to account for integrated processes that can be made stationary. For instance, it is useful for engineers to model mechanical vibra-tions or charge on a capacitor or condenser subjected to white noise excitation through a second-order stochastic differential equations. There are some interesting works that have been done on the second-order stochastic differential equations. For example, McKibben [] investigated the second-order damped functional stochastic evolution equations. For further work on this topic, one can see Mahmudov and McKibben []. Moreover, McKibben [] established the existence and uniqueness of mild solutions for a class of second-order neutral stochastic evolution equations with finite delay. Balasubramaniam and Muthukumar [] gave the sufficient conditions for the approximate controllability of the second-order neutral stochastic evolution equations with infinite delay. For more details of second-order stochastic differential equations, we refer the reader to Da Prato [] and the references therein.
solutions for SNISEEIs by means of the Krasnoselskii-Schaefer fixed point theorem. Two types of stochastic nonlinear wave equations with infinite delay and impulsive effects are provided to illustrate the obtained results.
The paper is organized as follows. In Section , we introduce some preliminaries. In Section , we prove the existence of the mild solutions for SNISEEIs by means of the Krasnoselskii-Schaefer fixed point theorem. In Section , we study the continuous de-pendence of solutions on the initial values. Two examples are provided in the last section to illustrate the theory.
2 Preliminaries
In this section, we mention some preliminaries needed to establish our results. For details as regards this section, the reader may refer to Da Prato and Zabczyk [], Fattorini [] and the references therein.
Let (,F,P;F) (F={Ft}t≥) be a complete filtered probability space satisfying thatF contains allP-null sets ofF. AnH-valued random variable is anF-measurable function
x(t) :→Hand the collection of random variablesS={x(t,ω) :→H|t∈J}is called a stochastic process. Generally, we just writex(t) instead ofx(t,ω) andx(t) :J→Hin the space ofS. Let{ei}∞i=be a complete orthonormal basis ofK. Suppose that{w(t) :t≥}is a cylindricalK-valued Wiener process with a finite trace nuclear covariance operatorQ≥, denoteTr(Q) =∞i=λi=λ<∞, which satisfiesQei=λiei, witheibeing a CONS of
eigen-vectors, and then, w.r.t. this spectral representation ofQthe drivingQ-Wiener process can be represented asw(t) =∞i=√λiwi(t)ei, where{wi(t)}∞i=are mutually independent one-dimensional standard Wiener processes. We assume thatFt=σ{w(s) : ≤s≤t}, which
is aσ-algebra generated bywandFT=F. Letψ∈L(K,H) and define
ψQ=TrψQψ∗= ∞
n=
λnψen.
IfψQ<∞, thenψis called aQ-Hilbert-Schmidt operator. LetLQ(K,H) denote the space
of allQ-Hilbert-Schmidt operatorsψ:K→H. The completionLQ(K,H) ofL(K,H) with
respect to the topology induced by the norm · QwithψQ= (ψ,ψ) is a Hilbert space
with the above norm topology.
The collection of all strongly measurable, square-integrable, H-valued random vari-ables, denoted by L(,F,P;H)≡ L(,H), is a Banach space equipped with norm
x(·)L = (Ex(·,ω)H)/, where the expectationE is defined by Ex=
x(ω) dP. Let C(J,L(,H)) be the Banach space of all continuous maps fromJ intoL(,H) satisfy-ing the conditionsupt∈JEx(t)<∞. An important subspace is given byL
(,H) ={f ∈
L(,H) :f isF-measurable}.
We say that a functionx: [ν,τ]→His a normalized piecewise continuous function on [ν,τ] ifxis piecewise continuous and left continuous on (ν,τ]. We denote byPC([ν,τ];H) the space formed by the normalized piecewise continuous stochastic processes from{x(t) :
t∈[ν,τ]}. In particular, we introduce the spacePC formed by allH-valued stochastic processes{x(t) :t∈[,T]}such thatxis continuous att=tk,x(t–k) =x(tk) andx(t+k) exists, for allk= , . . . ,m. In the sequel, we always assume thatPC is endowed with the norm
xPC= (sups∈JEx(s))/. It is clear that (PC, ·
To simplify the notations, we put t = , tn+ = T. For x ∈ PC, we denote xk˜ ∈
C([tk,tk+];L(,H)),k= , , . . . ,n, given by
˜
xk(t) =
⎧ ⎨ ⎩
x(t), fort∈(tk,tk+],
x(t+
k), fort=tk.
()
Moreover, forB⊆PC, we denote byBk,k= , , . . . ,n, the setBk={˜xk:x∈B}.
Lemma A set B⊆PCis relatively compact inPC,if and only if,the setBkis relatively compact in C([tk,tk+];L(,H)),for every k= , , . . . ,m.
Now, we consider the space
Bb=
x: (–∞,T]→H,xk∈PC(Jk,H) and there existxt–kandxtk+
withx(tk) =x
t–k,x=ϕ∈Bh,k= , , , . . . ,m
,
wherexkis the restriction ofxtoJk= (tk,tk+]. Set · bbe a semi-norm inBbdefined by
xb=xBh+ sup
≤s≤T
Ex(s)/, x∈Bb.
Then we have the following useful lemma appearing in [].
Lemma Assume that x∈Bb,then for t∈J,xt∈Bh.Moreover,we have lEx(t)/≤ xtBh≤l sup
≤s≤t
Ex(s)/+xBh,
where l=–∞ h(s) ds<∞.
Now, let us recall some facts about cosine families of operatorsC(t) andS(t) appeared in [, ].
Definition A one parameter family{C(t) :t∈R} ⊂L(H,H) satisfying that (i) C() =I,
(ii) C(t)xis continuous intonR, for allx∈H, (iii) C(t+s) +C(t–s) = C(t)C(s), for allt,s∈R,
is called a strongly continuous cosine family.
The corresponding strongly continuous sine family{S(t) :t∈R} ⊂L(H,H) is defined by
S(t)x=tC(s)xds,t∈R,x∈H.
The generatorA:H→Hof{C(t) :t∈R}is given byAx= d
dtC(t)x|t=for allx∈D(A) =
{x∈H:C(·)x∈C(R;H)}.
It is well known that the infinitesimal generatorAis a closed, densely defined operator onH. Such cosine and corresponding sine families and their generators satisfy the follow-ing properties appearfollow-ing in Fattorini []:
Proposition Suppose that A is the infinitesimal generator of a cosine family of operators
(i) there existM∗≥andα≥such thatC(t) ≤M∗eα|t|and henceS(t) ≤M∗eα|t|,
(ii) AsˆrS(u)xdu= [C(rˆ) –C(s)]xfor all≤s≤ ˆr<∞,
(iii) there existsN∗≥such thatS(s) –S(ˆr) ≤N∗sˆreα|s|ds,for all≤s≤ ˆr<∞.
The uniform boundedness principle, together with Proposition (i), implies that both
{C(t) :t∈[,T]}and{S(t) :t∈[,T]}are uniformly bounded.
To prove our results, we need the following Krasnoselskii-Schaefer type fixed point the-orem appearing in [].
Theorem Letandbe two operators of H such that (i) is a strict contraction,and
(ii) is completely continuous.
Then either
() the operator equationx+x=xhas a solution,or
() the setG={x∈H:λ(λx) +λx=x}is unbounded forλ∈(, ). 3 Existence result
In this section, we aim to give the existence of mild solutions for SNISEEIs ()-(). Firstly, let us propose the definition of the mild solution of SNISEEIs ()-().
Definition AnFt-adapted stochastic processx: (–∞,T]→His called a mild solution
of SNISEEIs ()-() if
(i) {xt:t∈J}isBh-valued andx(·)|J∈PC;
(ii) x(t)∈Hhas càdlàg paths ont∈Ja.s. and for eacht∈J,x(t)satisfies the following integral equation:
x(t) =C(t)φ() +S(t)ψ–g(,φ)+
t
C(t–s)g(s,xs) ds
+
t
S(t–s)f(s,xs) ds+
t
S(t–s)σ(s,xs) dw(s)
+
<tk<t
C(t–tk)Ik(xtk) + <tk<t
S(t–tk)˜Ik(xtk);
(iii) x=φ,x() =ψ.
In this paper, we need the following assumptions:
(H) The cosine family of operators{C(t) :t∈[,T]}onHand the corresponding sine family{S(t) :t∈[,T]}satisfyC(t)≤M,S(t)≤M,t≥for a positive constantM.
(H) The functionf :J×Bh→Hsatisfies the following properties:
. f(·,φ) :J→His strongly measurable for everyφ∈Bh;
. f(t,·) :Bh→His continuous for eacht∈J;
. there exist an integrable functionm:J→[,∞)and a continuous
nondecreasing function: [,∞)→(,∞)such that for every(t,φ)∈J×Bh,
we have
Ef(t,φ)≤m(t)φB
h
, lim inf
ζ→∞ (ζ)
(H) The functionσ:J×Bh→LQ(K,H)satisfies the following properties:
. σ(t,·) :Bh→LQ(K,H)is continuous for almost allt∈J;
. σ(·,x) :J→LQ(K,H)is stronglyFt-measurable for eachx∈Bh;
. there exists a positive constantLσ such that
Eσ(t,x) –σ(t,x)≤Lσx–xBh, (t,xi)∈J×Bh,i= , ; Eσ(t,x)≤LσxBh+ , (t,x)∈J×Bh.
(H) The functiong:J×Bh→His continuous and there exists a positive constantLg
such that
Eg(t,x) –g(t,x)≤Lgx–xBh, (t,xi)∈J×Bh,i= , ; Eg(t,x)≤LgxBh+ , (t,x)∈J×Bh.
(H) The functionsIkand˜Ik:Bh→Hare continuous and there are positive constants LIk,LI˜k,k= , , . . . ,msuch that
EIk(x) –Ik(y)≤LIkx–yBh, x,y∈Bh,k= , , . . . ,m, EIk˜(x) –Ik˜(y)≤L˜Ikx–y
Bh, x,y∈Bh,k= , , . . . ,m.
The main result of this section is the following theorem.
Theorem Assume the conditions(H)-(H)hold and assume that S(t)is compact.Then there exists a mild solution of SNISEEIs()-()provided that
Ml
T
T
m(s) ds+
m
k=
(LIk+LI˜k) + T
TLg+Tr(Q)Lσ
< ()
and
L= Ml
TTLg+Tr(Q)Lσ+
m
k=
(LIk+L˜Ik)
< . ()
Proof In the sequel, the notationBr(x,Z) stands for the closed ball with center atxand
radiusr> inZ, where (Z, · Z) is a Banach space. Lety: (–∞,T]→Hbe defined by
y(t) =
⎧ ⎨ ⎩
φ(t), t∈(–∞, ],
C(t)φ() +S(t)ψ, t∈J.
On the spaceY={x∈PC:x() =φ()}endowed with the uniform convergence topology, we define the operator:Y→Yby
x(t) =C(t)φ() +S(t)ψ–g(,φ)+
t
C(t–s)g(s,xs) ds
+
t
S(t–s)f(s,xs) ds+
t
S(t–s)σ(s,xs) dw(s)
+
<tk<t
C(t–tk)Ik(xtk) + <tk<t
wherexis such thatx=φandx=xonJ. From Lemma and the assumption onφ, we infer thatx∈PC. Our proof will be split into the following three steps.
Step .is completely continuous onBr(y|J,Y).
which shows the desired result of the claim.
SinceS(t) is a compact operator, the setVε(t) ={ε
x(t) :x∈(Br(y|J,Y))}is relatively
com-pact inHfor everyε, <ε<t. Moreover, for eachx∈(Br(y|J,Y)), we have
E(x)(t) –
εx(t)≤ε
t
t–ε
S(t–s)Ef(s,xs)ds
≤Mε
t
t–ε
m(t)xsBhds
≤Mr∗ε
t
t–ε m(s) ds.
Therefore, we have
E(x)(t) –
εx(t)→, asε→+,
and there are precompact sets arbitrary close to the setV(t) ={x(t) :x∈(Br(y|J,Y))}.
Thus, the setV(t) ={x(t) :x∈(Br(y|J,Y))}is precompact in (Br(y|J,Y)). Therefore, from
the Arzela-Ascoli theorem, the operatoris completely continuous. From Theorem , we infer that there exists a mild solution for the system ()-().
4 Examples
In this section, two types of stochastic nonlinear wave equations with infinite delay and impulsive effects are provided to illustrate the theory obtained.
Example We consider the following second-order stochastic Volterra integro-differen-tial equations with iniintegro-differen-tial-boundary conditions and impulsive effects:
d
∂x(t,ξ) ∂t –
t
–∞K(t,s)F
x(s,ξ)ds
=∂ x(t,ξ)
∂ξ dt+
t
–∞K(t,s)F
x(s,ξ)ds
dt
+
t
–∞K(t,s)F
x(s,ξ)ds
dw(t),
<ξ<π, ≤t≤T,t=tk,
k= , , . . . ,m, ()
x(t,ξ) =φ(t,ξ), –∞<t≤, <ξ<π, ()
x(t, ) =x(t,π) = , ≤t≤T, () ∂x(,ξ)
∂t =x(ξ), <ξ<π, ()
xtk+–xtk–=Ikx(tk), xt+k–xtk–=˜Ikx(tk),
k= , , . . . ,m, ()
LetH=L[,π]. The operatorAis defined by
(Az)(ξ) =d z(ξ)
dξ , with domainD(A) =
z∈H:z() =z(π).
The spectrum ofAconsists of the eigenvalues –nforn∈N, with associated eigenvectors zn(ξ) = (π)/sin(nξ). Furthermore, the set {zn;n∈N}is an orthonormal basis ofH. In particular,
Ax= ∞
n=
–nx,znzn, x∈D(A).
The operatorsC(t) defined by
C(t)x= ∞
n=
cos(nt)x,znzn, t∈R,
form a cosine function onH, with associated sine function
S(t)x= ∞
n= sin(nt)
n x,znzn, t∈R.
From [], for allx∈H,t∈R,S(t) ≤ andC(t) ≤.
LetKi(t,s)∈C(R,R),i= , , and assume that there exists a positive continuous func-tionf(s) onR–such that
Ki(t,t+s)≤f(s), i= , , , l=
–∞f(s) ds<∞.
Now, we give the phase spaceBh. Assume thath: (–∞, ]→(,∞) is a continuous
func-tion withl=–∞ h(t) dt<∞. For anya> , define
Bh=
ψ: (–∞, ]→H:Eψ(θ)/is a bounded and measurable
function on [–a, ] and –∞h(s)
Eψ(s)/ds<∞
.
We endowBhwith the norm
ψBh=
–∞h(s)
Eψ(s)/ds, for allψ∈Bh.
Then (Bh, · Bh) is a Banach space. Let
φ(ξ) =
–∞h(s)φ(s,ξ) ds, g(t,φ)(ξ) =
–∞K(t,t+s)F
φ(s,ξ)ds,
f(t,φ)(ξ) =
–∞K(t,t+s)F
φ(s,ξ)ds, σ(t,φ)(ξ) =
–∞K(t,t+s)
Then ()-() can be rewritten in the abstract form ()-(). We can propose suitable condi-tions on the coefficients appeared in the above equation to guarantee ()-() has at least one mild solution by means of Theorem .
Example We consider the following stochastic nonlinear wave equation with impulsive effects and infinite delay:
d
∂x(t,ξ) ∂t –f
t,x(t–r,ξ)
=∂ x(t,ξ)
∂ξ dt+f
t,x(t–r,ξ)dt
+σt,x(t–r,ξ)dw(t),
≤ξ≤π, ≤t≤T,r> ,t=tk,k= , , . . . ,m, ()
x(t,ξ) =φ(t,ξ), –∞<t≤, <ξ<π, ()
x(t, ) =x(t,π) = , ≤t≤T, () ∂x(,ξ)
∂t =x(ξ), <ξ<π, ()
xtk+–xtk–=Ikx(tk), xt+k–xtk–=˜Ikx(tk),
k= , , . . . ,m, ()
where x ∈L(;H), φ ∈Bh, Bh is defined as Example , H=L([,π]), and w is an H-valued Wiener process.
LetA,C(t) andS(t) be defined as Example . Then the above system ()-() can be rewritten in the form of ()-(). Further, we assume that fi: [,T]×R→R(i= , ), σ : [,T]×R→BL(H) andIk,˜Iksatisfy (H)-(H). Then ()-() has at least one mild
solution.
Competing interests
The author declares that he has no competing interests.
Author’s contributions
The author is the only person who is responsible to this work.
Acknowledgements
The author sincerely thanks the reviewers for their valuable suggestions and useful comments. This work is partially supported by Mathematical discipline project of Anhui Agricultural University (Project number. XKXWD2013020; XK2013029).
Received: 23 October 2013 Accepted: 14 April 2014 Published:30 Apr 2014
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