XIE Qiaoqiao ,YANG Bin,? and LI Zhi

1 College of Education and Sports Sciences,Yangtze University,Jingzhou 434023,China.

2 School of Information and Mathematics,Yangtze University,Jingzhou 434023,China.

Abstract. By means of the Banach fixed point principle,we establish some sufficient conditions ensuring the existence of the global attracting sets and the exponential decay in the mean square of mild solutions for a class of neutral stochastic functional differential equations by Poisson jumps.An example is presented to illustrate the effectiveness of the obtained result.

Key Words:Global attracting set;mild solution;Banach fixed point principle;Poisson jumps.

1 Introduction

Attracting sets of stochastic dynamical systems have attracted the increasing attention over the last a few decades due to weaken the stability conditions of stochastic system.Many different arguments have been developed to establish some sufficient conditions ensuring the existence of the global attracting sets.Among others,the (delay) integral inequalities introduced in [1] has been efficiently applied,see,e.g.,Wang and Li [2]obtained the global attracting sets of impulsive stochastic partial differential equations with infinite delays by establishing some impulsive-integral inequalities;Long et al.[3]considered the global attracting set and stability of stochastic neutral partial functional differential equations with impulses;Liu and Li [4] gave the global attracting sets and the conditions of exponential stability of neutral stochastic partial functional differential equations driven byα-stable processes;Li[5] and Xu and Luo[6] studied the global attracting sets and the conditions of exponential stability of mild solutions to a class of neutral stochastic functional differential equations driven by fBm with Hurst parameterrespectively.

On the other hand,stochastic differential equations driven by Poisson jump processes have attracted a great deal of attention.For example,Knoche [7] proved the existence and uniqueness of mild solution for stochastic evolution equations with Poisson jump processes;R?ckner and Zhang[8]studied the existence,uniqueness and large deviation principle for stochastic evolution equations driven by Poisson jump processes;Luo and Taniguchi[9]investigated the existence and uniqueness for non-Lipschitz stochastic neutral delay evolution equations driven by Poisson jumps;Hou et al.[10] considered the stability of energy solutions for SPDEs with variable delay and Poisson jump processes;Cui at al.[11]studied the exponential stability for neutral stochastic evolution equations driven by Poisson jumps;Budhiraja et al.[12]obtained the large deviations for stochastic partial differential equations driven by a Poisson random measure.

But,as far as we know that there is no paper which investigates the global attracting sets for stochastic differential equations driven by Poisson jump processes.In order to fill this gap,in this paper,we will investigate by employing the Banach fixed point principle,without requiring to any integral inequality,the existence of the global attracting sets of the mild solution for a class of neutral stochastic functional differential equation driven by Poisson jump processes.

The rest of this paper is organized as follows.In Section 2,we introduce some necessary notations and preliminaries.In Section 3,we state and prove our main results.

2 Preliminary

Let (?,F,{Ft}t≥0,P) be a filtered complete probability space satisfying the usual condition,which means that the filtration is a right continuous increasing family and F0contains allP-null sets.Let(H,〈·,〉,‖·‖)and(K,〈·,〉K,‖·‖K)be two real,separable Hilbert spaces and L(K,H)be the space of bounded linear operator fromKtoH.

Suppose{p(t),t≥0} is aσ-finite stationary Ft-adapted Poisson point process which takes values in a measurable space (U,B(U)).The random measureNpdefined byNp((0,t]×A):=∑s∈(0,t]IA(p(s)) forA∈B(U)is called the Poisson random measure induced byp(·).Then,we can define the compensated Poisson random measureby=Np(dt,dy)?v(dy)dt,wherevis the characteristic measure ofNp.LetW=W(t)t≥0be aK-valued Wiener process which is independent of the Poisson point process,defined on(?,F,{Ft}t≥0,P)with covariance operatorQ,that is

whereQis a positive,self-adjoint,trace class operator onK.Let(K,H)denote the space of allQ-Hilbert-Schmidt operators fromKintoHwith the norm for?∈L(K,H)

The reader is referred to Da Prato and Zabczyk[13]for a systematic theory about stochastic integrals in Hilbert space.

LetA:D(A)→Hbe the infinitesimal generator of an analytic semigroup,{S(t)}t≥0,of bounded linear operators onH.It is well known that there existM≥1 andλ∈R such that‖S(t)‖≤Meλtfor everyt≥0.If{S(t)}t≥0is a uniformly bounded and analytic semigroup such that 0∈ρ(A),whereρ(A)is the resolvent set ofA,then it is possible to define the fractional power(?A)αfor 0<α≤1,as a closed linear operator on its domainD(?A)α.Furthermore,the subspaceD(?A)αis dense inH,and the expression

defines a norm inD(?A)α.IfHαrepresents the spaceD(?A)αendowed with the norm‖·‖α,then the following properties are well known(cf.[14],Theorem 6.13 p.74).

Lemma 2.1.Suppose that the preceding conditions are satisfied.

(1)Let0<α≤1.Then Hα is a Banach space.

(2)If0<β≤α then the injection HαHβ is continuous.

(3)For every0<β≤1,there exist Mβ>0and λ>0such that

(4)For every α∈(0,1],there exists a positive constant Cα,such that

For a Borel setZ∈B(U?{0}),we consider the following neutral stochastic functional differential equations driven by Poisson jump process

whereD:=D([?r,0];H)is the space of all F0-measurable c`adl`ag functions from[?r,0]toHequipped with the norm‖φ‖D=supt∈[?r,0]‖φ‖,G,f:[0,+∞)×H→H,g:[0,+∞)×H→,h[0,+∞)×H×U→Hare some given functions to be specified later,andρ(t),τ(t),δ(t),θ(t):[0,+∞)→[0,r],r>0 are some continuous functions.

We also impose the following assumptions:

(H1)Ais the infinitesimal generator of an analytic semigroupS(t)onH,i.e.,there exist constantsM≥1 andκ>0 such that for everyt≥0,it holds‖S(t)‖≤Me?κt.

(H2) There exist some constantsα∈(0,1]andK1>0 such that for anyx,y∈H,t≥0,G(t,x),G(t,y)∈D((?A)α)

(H3) There exists a positive constantK2such that for allx,y∈Handt≥0,

(H4) There exists a positive constantK3such that for allx,y∈Handt≥0,

(H5) There exists a positive constantK4such that for allx,y∈H,z∈Zandt≥0,

Definition 2.1.AnFt-adapted c`adl′ag stochastic process x(t),t≥0,is called a mild solution of(2.1)if it has the following properties:

(a) x0=?∈D([?r,0];H);

(b) For arbitrary t≥0,

Definition 2.2.A set S?H is called the global attracting set of(2.1)if for any initial value ?∈D([?r,0];H),the solution process x(t,?)of(2.1)converges to S as t→∞,i.e.,

wheredist(x,S)=infy∈SE‖x?y‖.

3 Main result

In this section we investigate the existence,uniqueness and the global attracting sets of the mild solution for (2.1).Our main method is the Banach fixed point theorem.The main result of this paper is given in the following theorem.

Theorem 3.1.Assume that(H1)-(H5)hold.Then Eq.(2.1)has a unique mild solution on[?r,∞)and the set S:=is a global attracting set of(2.1)provided that the following relations

whereΓ(·)is the standard Gamma function,

Proof.Denote by Λ the Banach space of all Ft-adapted c`adl′ag processx(t):?×[?r,∞)→H,endowed with a norm‖x‖Λ:=supt≥?rE‖x(t)‖2.Let?∈D([?r,0];H)and Λ?denote the set of allx∈Λ such thatx(t)=?(t),t∈[?r,0] and there exist some constantsN>0 andκ>κ′such that

Note that Λ?is a closed subset of Λ provided with the norm‖·‖Λ.Define the operator Ψ on Λ?for allt≥0 by:

and for allt≥0

We will show by using the Banach fixed point theorem that Ψ has a unique fixed point.

Step 1.We show that Ψ(Λ?)?Λ?.Letx∈Λ?.Then we have by using(3.2)

By the assumption(H2),we get that

By the assumption(H2)and the definition ofS?,we have

ForI3,using the Lemma 2.1,the assumption(H2)and the H?lder’s inequality,it follows that

ForI4,by the assumption(H3)and the H?lder’s inequality,we have

ForI5,by the assumption(H4)and theisometry property,we have

By view of the assumption(H5),we obtain

Then,combining(3.4)-(3.9)with(3.3)we know that

Thus,by using(3.1)we can obtain

It implies that Ψ(x)∈Λ?.

Step 2.Now,we will show that Ψ is a contraction mapping in Λ?.For anyx,y∈Λ?,we have

By the assumption(H2)and the fact thatx(s)=y(s)=?(s)fors∈[?r,0],we have

ForJ2,using Lemma 2.1,the assumption(H2)and the H?lder’s inequality,it follows that

ForJ3,by the assumption(H3)and the H?lder’s inequality,we have

ForJ4,by the assumption(H4)and theisometry property,we have

By view of the assumption(H5),we obtain

Then,combining(3.11)-(3.15)with(3.10)we have

Thus,by(3.1)we can deduce that Ψ is a contraction mapping on Λ?and therefore has a unique fixed pointx,which is a mild solution for(2.1)and satisfies for anyt>0

So,Sis a global attracting set of the mild solution for(2.1).The proof is complete.

Now,we shall focus on the exponential decay in mean square of the mild solution of the equation(2.1).

Definition 3.1.The mild solution of(2.1)with initial ?∈D([?r,0];H)is said to have exponential decay in mean square if there exists a pair of positive constants λ>0and M=M(?)≥0such that

Theorem 3.2.Assume that(H1)-(H5)hold for f(t,0)≡0,g(t,0)≡0and h(t,0,z)≡0.Then the mild solution of(2.1)has exponential decay in mean square provided

Proof.Bearing the estimates aboutI1,I2,I3of the last theorem in mind,we need only estimateI4,I5andI6here.To this end,we may obtain by usingf(t,0)≡0,g(t,0)≡0 andh(t,0,z)≡0 that

ForI5,by the assumption(H4)and theisometry property,we have

By view of the assumption(H5),we obtain

Thus,for anyJ>0 we have

Thus,by using(3.1)we can obtain for anyJ>0

LettingJ→0,we have E‖Ψ(x)(t)‖2≤N′e?κ′t.The proof is complete.

4 An example

In this section,we shall apply the previous results to an example to illustrate our theory.

Example 4.1.LetH=L2(0,π)andA=?is the standard Laplace operator onH.Define

which is an orthonormal basis ofHso that

Consider the following stochastic neutral partial differential equation driven by Poisson jump process,

whereαi>0,i=1,2,3,4 are constants andW(t) denote the standard cylindrical Wiener process andZ={z∈R:0≤|z|≤c,c>0}.

Note thatAgenerates an analytic and compact semigroupS(t),t≥0,inHand

and‖S(t)‖≤e?tfor allt≥0.Letp=3/2 and the linear operator(?A)3/4is given by

with domain

By using the same notions and notations as in the previous sections,we can thus get that

and further

Hence,by virtue of the Theorem 3.1,we know that

is a global attracting set of system(4.1)provided thatρ0<1.

In addition,if

andf(t,u,ξ)=α2u(t?r,ξ)satisfies condition(4.1),then by the Theorem 3.2,we know that the mild solution of system(4.1)is has exponential decay in the mean square.