MildSolutionofStochasticEquationswithL`evyJumps: Existence,Uniqueness,Regularity and Stability

ZHOU Guoli,GUO Bolingand HOU ZhentingCollege ofMathematicsandStatistics,ChongQingUniversity,ChongQing4013,

China.

2Institute of Applied Physics and Computational Mathematics,P.O.Box 8009,

Beijing 100088,China.

3College of Mathematicsand Statistics,Central SouthUniversity,Changsha410075,

China.

Received 29 March 2013;Accepted 10 June 2013

MildSolutionofStochasticEquationswithL`evyJumps: Existence,Uniqueness,Regularity and Stability

ZHOU Guoli1,2,?,GUO Boling2and HOU Zhenting31College ofMathematicsandStatistics,ChongQingUniversity,ChongQing401331,

China.

2Institute of Applied Physics and Computational Mathematics,P.O.Box 8009,

Beijing 100088,China.

3College of Mathematicsand Statistics,Central SouthUniversity,Changsha410075,

China.

Received 29 March 2013;Accepted 10 June 2013

.The existence and uniqueness of mild solution to stochastic equations with jumps are established,a stochastic Fubini theorem and a type of Burkholder-Davis-Gundy inequality are proved,and the two formulas are used to study the regularity property of the mild solution of a general stochastic evolution equation perturbed by L′evy process.Then the authors prove the moment exponential stability,almost sure exponential stability and comparison principles of the mild solution.As applications, the stability and comparison principles of stochastic heat equation with L′evy jump are given.

Stochastic evolution equation;L′evy processes;mild solution;stability.

1 Introduction

In recent years,there are so many monographs concerning stochastic partial differential equations with L′evy jump and it’s applications in physics,economics,statistical mechanics,fluid dynamics and finance etc.For these theory and applications,one can see[1–3] and references therein.In this article,the existence,uniqueness,regularity and stability for the mild solution of the stochastic partial differential equations with L′evy jumpare studied.There are a lot of works dealing with existence and uniqueness for stochastic partial differential equations with jump process.In[4],existence and uniqueness for solutions of stochastic reaction diffusion equations driven by Poisson random measures are obtained.In[5],Malliavin calculus is applied to study the absolute continuity of the law of the solutions of stochastic reaction diffusion equations driven by Poisson random measures.In[6],a minimal solutionis obtained for the stochastic heat equation drivenby non-negative L′evy noise with coefficients of polynomial growth.In[7],a weak solution is established for the stochastic heat equation driven by stable noise with coefficients of polynomial growth.In[8],existence and uniqueness for solutions of stochastic generalized porous media equations with L′evy jump are obtained.

For stability of stochastic partial differential equations,there are also lots of works. One can find the related topics in[9–11]and reference therein.The stability of a linear equation with jump coefficient is studied in[11].In[12],the stability of a semilinear stochastic differential equation with Wiener process is investigated.The exponential stability of general nonlinear stochastic differential equations with Wiener processes is studied in[13],and the asymptotic and exponential stability of the nonlinear stochastic delay differential equations driven by Wiener processes are considered in[14]and[15] respectively.In[16],stability of infinite dimensional stochastic evolution equations with memory and L′evy jumps is studied.

The main aim of this paper is to study existence,uniqueness,regularity and stability of stochastic equation:

In[16],the intensity measure λ ofeN(dt,dx)is finite,while the intensity measure in this article is σ-finite,and also the classical Lipschitz condition(2.5)in[16]is relaxed to condition A.5 in this article.The authors of this article prove the existence and uniqueness of mild solution of(1.1),the continuity of the solution with respect to initial data.And then they prove the stochastic Fubini theorem for compensated Poisson random measure whose intensity measure is σ-finite compared to finite case in[16].Furthermore a new typeofBurkholder-Davis-Gundyinequalitywhichis moreprecisethanLemma2.2in[16] is got.Using the two basic tools,the authors get the regularity property of mild solution of(1.1)without conditions(2.15)and(2.16)in[16]which are critical there.Then the authors also prove the almost sure exponential stability without condition(2.28)in[16].

For the stability of stochastic differential equations with Wiener noise,it has been deeplystudied,see[9–14,16],and referencetherein.Inthis article,thestability of stochastic equations with L′evy noise is considered.This will bring various difficulties from calculus and probability compared to studying stochastic equations with Wiener noise.By using method from[16]and[17],the authors derive some sufficient conditions to ensure stability of infinite dimensional stochastic systems in sense of both moment exponential stability and almost sure exponential stability,and in the following examples are given to illustrate the two kinds of stabilities.Finally,the authors compare the stability betweendeterministic and stochastic systems with two examples to illustrate our theorems.

The article is organized as follows.In Section 2,we present the framework.Existence, uniqueness and regularity are proved in Section 3.In Section 4,we get moment exponential stability.In Section 5,almost sure exponential stability is obtained.In Section 6, comparison principles are proved.In Section 7,two examples including stochastic heat equations with L′evy jump are given.

2 Some preliminaries

with initial condition X(0)=η,and the coefficients in Eq.(2.1)satisfy the following conditions:

(A.1)A:D(A)→H is the infinitesimal generator of a C0-semigroup S(t),t≥0.

(A.2)B:H→H isB(H)/B(H)-measurable,and there exists a positive constant C satisfies

(A.3)Q:H→L(U,H)is strongly continuous,i.e.that the mapping

is continuous from H to H for each u∈U.

(A.4)For all t∈]0,T]and x∈H we have

(A.5)There is a square integrable mapping M:[0,+∞[→[0,∞[such that

and

for all t∈]0,T]and x,y∈H.

(A.6)There is a positive constant C like(A.2)such that

and

where F:H×Z→H isB(H)×B(Z)/B(H)measurable. For fixed T＞0,we define:

Then for X∈H2(T,H)and λ≥0 we have

For the reader’s convenience,before giving our main results we cite some theorems here which will be needed later:

Theorem 2.1(Contraction Theorem).(i)Let(E,‖‖E)and(Λ,‖‖Λ)be two Banach space.Let mapping G:Λ×E→E satisfy

for all λ∈Λ and x,y∈E.Then there exists exactly one mapping ?:Λ→E such that

for all λ∈Λ.

(ii)If we assume in addition that the mapping λ→G(λ,x)is continuous from Λ to E for all x∈E we get that ?:Λ→E is continuous.

(iii)If the mapping λ→G(λ,x)are not only continuous from Λ to E for all x∈E but there even exists an L≥0 such that

for all x∈E then the mapping ?:Λ→E is Lipschitz continuous.

Theorem 2.2.[18]If a process Ψ is adapted toFt,t∈[0,T],and stochastically continuous with values in a Banach space E then there exists a predictable version of Ψ.

Definition 2.1.An H-valued predictable process X(t),t∈[0,T]is called a mild solution of(2.1) if

for each t∈[0,T].

3 Existence,uniqueness and regularity

Theorem 3.1.Assume conditions from(A.1)to(A.6)hold,then there exists a unique mild solution X(η)∈H2(T,H)of(2.1)with initial condition

In addition we get that the mapping

is Lipschitz continuous.

In the following two steps,we will prove that

The first step:We prove that the mappingFis well defined.

1.Let h∈H,as

2.Let{ei}and{hi}be orthonormal basis respectively of U and H.Because

3.Similarly to 1,it is easy to see that 1[0,t](s)S(t?s)F(X(s?),x)is predictable,and by (A.6)we have

1.Obviously S(t)η,t∈[0,T],is an element ofH2(T,H).

Letting t→t0,and then r→1,by(A.2),strong continuous of S(t)and dominated convergence theorem,we know for all most w∈? we have

As we have

Let’s define

The last step follows by dominated convergence theorem,since

has predictable version for all t∈[0,T].Furthermore by(A.5),we have

4.There is a version of

which is inH2(T,H).It is easy to show 1[0,t](s)S(t?s)F(X(s?),x)is predictable.By(A.6), we have

Letting t＜t0,we have

Since

letting t→t0,and r↓1,by dominated convergence theorem we have the following result:

If t＞t0,similarly we can get

has a predictable version which is an element inH2(T,H).

The third step:We are going to show

Dividing by eλtboth sides of(3.2),we have

For the second term of the(3.1),by Burkholder-Davis-Gundy inequality and(A.5)we have

Dividing by eλtboth sides of(3.3),we have

Obviously,

By Burkholder-Davis-Gundy inequality and(A.6),we have the following estimate for the third term of the right hand side of(3.1):

Thus we have

Obviously

Therefore we have finally proved that there exists an a(λ)＜1 with

So there exists a unique

satisfying

which is the unique solution of(2.1).

are Lipschitz continuous for all Y∈H2(T,H)where the Lipschitz constant does not depend on Y.

The proof is complete.

Before giving the regularity of mild solution of(2.1),we first need a Stochastic Fubini theoremwhich is a fundamentaltool.Set ?T×Z=[0,T]×?×Z,Zbe σ-algebra generated by open subset in Z,PTbe the predictable σ-algebra of[0,T]×?,ds?P?λ(dz)be the product of the Lebesgue measure,P and λ(dz)on[0,T]×?×Z.L2(?T×Z):=L2(([0,T]×?×Z,ds?P?λ(dz));H).

Proposition 3.1.Let(Y,Y,μ)be a finite measure space and ψ:?T×Y×Z→H aPT?Y?Z-measurable mapping such that

Then:

(1)the process indexed by t∈[0,T]

is progressively measurable and belongs to L2(?T×Z);

(2)the process indexed by y∈Y

has anFT?Y-measurable version m:?×Y→H such that

(3)we have

Proof.(1)Follows from the following inequality:Let f be a nonnegativePT?Y?Z-measurable function.Then

because

and we get(3.4)by the Schwarz inequality.Now suppose that m in(2)exists.Then by taking ? instead of ?T×Z in(3.4)we get

by the Burkholder inequality.Hence

is defined P-almost everywhere.Now take ψnsatisfying the assumption of the proposition such that the sequence of the integrals

converges to zero.Then there exists a subsequence(nk:k∈N)such that

Let us introduce the setDof allPT?Y?Z-measurable processes ψ with

such that there exists an m satisfying(2)and(3).It is easy to see thatDis a linear space and if we can find ψn∈Dsuch that

then we would finish the proof.Indeed,take the corresponding functions mn.Then the sequence(mn:n∈N)is Cauchy in L1(?×Y;H)due to(3.5)and ψ belongs toDdue to (a)and(b).Now we will show how to construct the approximating sequence ψn.We assume λ(dz)is finite.By[18,Lemma A.1.4.]we can find mappings Fnon H.The simple functions Fnψ take values in finite dimensional subspace of H.Moreover

and if Fnψ∈D,n∈N then ψ∈D.Now to show that Fnψ∈D,we will take advantage of the fact that each Fnψ is bounded in H and

if and only if

for φnuniformly bounded in H.So as Fnis of the form

where(Ck:k≤m)is aPT?Y?Z-decomposition of ?T×Y×Z and Bk,k≤m are elements in finite dimensional subspace of H.We conclude that Fnψ∈Dprovided ICkBk∈Ddue to linearity ofD.Another reduction shows that this is true if

for s＜t,Cs∈Fsbut this is obvious.When λ(dz)is σ-finite,there exists a sequence Ansatisfying λ(An)＜+∞and An↑Z.Obviously

therefore ψ∈D.

In the following we give a type of Burkholder-Davis-Gundy inequality which will play an important role in proving the regularity property of the mild solution of(2.1).

Lemma 3.1.If A is the infinitesimal generator of pseudo-contraction C0-semigroup S(t),t≤T,

and g(t,z)is H-valued progressively measurable mappings with respect to{Ft}t≥0such that

then for any t∈[0,T],

for some number r≥0.

Proof.Let’s define

Assume g(t,z)takes values inD(A)and

Since A is the infinitesimal generator of a C0-semigroup,A is closed.And by(3.6)we can check that

for all t≥0 and

By(3.6)and Proposition 3.1 with Y replaced with[0,T],we have

By It?o’s formula,we get

Set

So we have

Since A is pseudo-contraction,there exists r≥0 such that

Define Tn:=inf{t|‖Y(t)‖＞n}.Then by(3.7),(3.8)and B-D-G inequality,we can get the following estimate

So,by Gronwall inequality we have

By Fatou lemma,we have

Without assuming g(t,z)takes values inD(A),we define gn(t,z)=nR(n,A)g(t,z)where R(n,A),n∈N,is the resolvent of A,then we know gn(t,z)takes values inD(A)and satisfies(3.6)under the following condition

Define

Then it is easy to check

So under condition(3.10),(3.9)also follows without assumption that g(t,z)take values inD(A).In order to relax(3.10),we define stopping times

Then

Therefore by Fatou lemma,the assertion follows.

In order to study the regularity property of the mild solution of(2.1),we introduce an approximation system of(2.1)in the following:

where l∈ρ(A)which is the resolvent set of A,and R(l):=lR(l,A),R(l,A)is the resolvent of A.We say a stochastic process is RCLL if each of it’s sample path is right continuous with left limit.So a stochastic process X(t)is called a strongsolutionof(2.1),if it is RCLL, Ft-adapted,X(t)∈D(A)and satisfies(2.1).

Theorem 3.2.Let η∈L2(?,H)andF0-measurable.In addition to assumptions in Theorem 3.1 we assume A is dissipative with x∈H,Then the mild solution of(2.1)is RCLL.

Proof.Let l∈ρ(A),obviously

are bounded operators.So(3.11)has unique strong solution.In fact by Theorem 3.1 we know(3.11)has a unique mild solution denoted by Xl(t)and the following hold:

Thus by Fubini theorem,we have

On the other hand,by the stochastic Fubini theorem for Q-Wiener processes in[18],we have

And by Proposition 3.1

Hence,AXl(t)is integrable almost surely and

So far,we prove Xl(t)∈D(A),t∈[0,T],is the unique strong solution of(3.11).Let X(t)be the mild solution of(2.1).Then we consider

for any t≥0.We have that for any T≥0,

We consider

When l is big enough,there exists constant M such that‖R(l)‖≤M,so we have

Then we consider

where C(l)→0 as l→∞.

By(A.6)and Lemma 3.1 we have

Finally we consider

It is easy to check

By dominated convergence theorem,we have

By Burkholder-Davis-Gundy inequality,we have

for h∈H,we get

By Lemma 3.1,we get

After discussing about Ii,i=1,2,3,4,we can get the following estimate

where g(l)→0 as l→∞.By Gronwall theorem,we have

Therefore we know X(t)is RCLL on[0,T]a.s..

4 Moment exponential stability

In the following,we will study the moment exponential stability of mild solution of(2.1) by methods of the Razumikhin-Lyapunov function techniques.In order to obtain the stability,we assume that

Obviously(2.1)has a trivial solution when η=0 in Theorem 3.1.Let C2,0(H;R+)be the family ofallnon-negativefunctionV(x)on H whichare continuouslytwicedifferentiable with respect to x.Denote D([0,T];H)the collection of RCLL processes defined on[0,T]×? taking values in H.For any φ∈D([0,T];H)with φ∈D(A),we set:

where〈,〉Hdenote the inner product in H.

Theorem 4.1.Let A be infinitesimal generator of pseudo-contraction C0-semigroup and the conditions in Theorem 3.1 and(4.1)hold.Assume that there exist functions V(x)∈C2,0(H;R+)

satisfying:

where x∈H,c＞0 is independent of l,|·|denote the absolute value of real number.Further more, we assume

where γ∈(0,σ).Especially,when V=‖·‖2,

Letting l→+∞,by(4.2)-(4.5)and dominated convergence theorem,we have

Example 4.1.Consider the following stochastic heat equation

and

where Bt,t≥0,is a real standard Brownian motion,eN(dt,dz)is a poisson compensated martingale measure with character measure λ(dz)on R,b∈L2(0,1):=H and f is a real Lipschitz continuous function on H satisfying|f(u)|≤c‖u‖for some c＞0 and u∈H.Let

then we have

For the sake of simplicity,we assume‖b‖=1.Let V(u),u∈H,be a twice Fr′echet differentiable function on H and define L by

Then

If

then(4.6)is satisfied.(4.2)is easy to check,we omit the proof.The inequality(4.5) follows from

Finally,we check(4.4).

where we denote

So far,all the conditions(4.2)-(4.4)are checked,by Theorem 4.1,the null solution of (4.11)is exponentially stable in mean square.

5 Almost sure exponential stability

In this section,we intend to investigate the almost sure stability,which is in most situations the kind of stability one usually wants to have in practical applications,of the trivial solution of(2.1).

Theorem 5.1.Assume conditions in Theorem 3.1 and(4.1)hold.Then the mean square exponential stability of(2.1),i.e.,

implies the almost sure exponential stability

In the following,C is a positive constant changing from line to line.By(4.8),we have

where C does not depends on t.By H¨older inequality,(A.2),(4.1)and(4.8),we get

where C does not depends on t.By Burkholder-Davis-Gundy inequality,we have

By(A.5),(4.1)and(4.8),

where C does not depends on t.So,we have

By Lemma 3.1,we have

where C does not depends on t,the second inequality follows by(A.6)and(4.1),and the last inequality follows by(4.8).From(5.3)-(5.7),we get

where C does not depend on t.Define

For ?∈(0,(γ?2α)∧γ),we have

By the Borel-Cantelli lemma,for almost all w∈? there exists a random integer k0(w)≥2 such that if t∈[k?2,k?1]for any k＞k0,the following hold:

Consequently,

Since ? is an arbitrary positive number,we have

The proof is complete.

Example 5.1.In the following step we will apply the general Razumikhin type theorem established above to deal with the exponential stability of a class of stochastic differential equations with multiplicative L˙evy noise.To be precise,we intend to consider

Then the mild solution of(5.11)is mean square exponentially stable and its mean square Lyapunovexponentis lessthan?σ.Consequently,its trivial solutionis also almost surely exponentially stable and the sample Lyapunov exponent is less than?σ/2.

Proof.For any φ∈D([0,T];H)

Then(5.11)becomes(2.1).Moreover,by condition(5.12),we have

It is easy to check that equation(5.11)satisfies(4.2)-(4.6).So,by Theorem 4.1,the trivial solution of(5.11)is the mean square exponentially stable and its mean square Lyapunov exponent should be less than?σ.Furthermore,by Theorem 5.1,the trivial solution of(5.11)is also almost surely exponentially stable and its sample Lyapunov exponent should not be bigger than?σ/2.

6 Comparison principles and stability

The comparison principle is a useful tool to study qualitative properties of stochastic systems.In this part,we use Lyapunov functionals and differential inequality arguments to develop a comparison principle for the solutions of(2.1).By using this comparison principle,we obtain stability in probability,stability in the mean square and asymptotic stability in probability for the solutions of(2.1).

Consider the following one-dimension deterministic differential equation:

Under the Lipschitz conditions on F(·),it is not difficult to see there exists a solution of (6.1),we denote it by h(t,h0),t∈[0,T],T≥0,with h0as its initial value.In order to give the comparison principles,we need the following definition and lemma.

In the next,we are going to state the comparison principle for the solution of(2.1) which will play an important role in our stability analysis.

Theorem 6.1.Assume the following conditions hold.

(i)In addition to the conditions in Theorem 4.1 and[19,Lemma 6.1],let function F(t)be concave in t∈R+and satisfying:

(iii)Let X(t)=X(t,η),t≥0,be the mild solution of(2.1)with initial η∈L20,and

Then

Proof.Since

we get

So,we have

It is easy to calculate

where CTdenote a positive constant which depends on T,and in the following CTwill change from line to line,but still only depends on T.For I2,we have

where the second inequality follows by(A.2),(4.1)and by H¨older’s inequality.For I3, by Burkholder-Davis-Gundy inequality for stochastic convolutions driven by the Wiener process W(t)in[10],we have

where C＞0.For I4,by Lemma 3.1,we have

From(6.5)-(6.9),we get

By Gronwall inequality,we have

Let Xl(t)be thestrongsolutionof(3.11)suchthatby Theorem3.2,Xl(t)→X(t)uniformly with respect to t∈[0,T]almost surely as l→+∞.Define stopping times

Then for any l we have‖Xl(t∧Tn)‖≤n,t≥0.Applying It?o’s formula to the function V(Xl(t)),and by conditions(i),for any t≥0,h＞0,we have

Letting l→+∞in(6.11),we have

By(4.2),(6.10)and dominated convergence theorem,letting n→+∞in(6.12),we get

So,it follows that,

By(6.3)and[19,Theorem 8.1.4],we get

The proof is complete.

Theorem 6.2.Assume conditions in Theorem 6.1 hold.Let gi:[0,+∞)→[0,+∞),be strictly increasing,i=1,2.g2be right continuous and satisfy g2(x)≤cx2on R+for some c∈R+. Furthermore,assume

Then the stability of the trivial solution of(6.1)implies the stability in probability of the trivial solution of(2.1)in sense that for any ?,?′＞0,there exists δ=δ(?,?′)＞0 such that if‖x0‖＜δ, then

where x0independent of w∈? is initial value of X(t),t≥0. Proof.By(6.15),we have

Since the trivial solution of(6.1)is stable,for ?g1(?′)＞0,there exists a positive constant δ′such that if‖h0‖＜δ′,then h(t,h0)≤?g1(?′),for t≥0.Consequently,we have

By conditions on g2(t),for h0∈(0,δ′),there exists a constant δ＞0 such that if‖x0‖＜δ, then

So,by(6.17)and(6.19),we have

By Theorem 6.1 and(6.18),we have

The proof is complete.

Corollary 6.1.In case of Theorem 6.2,the asymptotic stability of the trivial solution of(6.1) implies the asymptotic stability in probability of the trivial solution of(2.1).

Proof.Let h(t,h0)be the trivial solution of(6.1)with initial value h0and X(t,x0)be the solutionof(2.1)withinitial value x0whichis independentof w∈?.Since h(t,h0)is stable, by Theorem 6.2,we know X(t,x0)is stable in probability.By(6.21),for arbitrary positive number ?′,there exists a positive number δ,when‖x0‖＜δ,we have

Letting t→+∞in(6.22),we prove the results of the theorem.Theorem 6.3.In addition to conditions in theorem 6.1,we assume g1(t),t≥0 is convex.Then the stability of the trivial solution of(6.1)implies the mean square stability of the trivial solution of(2.1)in sense that for arbitrary number ?＞0,there exists a number δ＞0,when the number x0satisfy‖x0‖＜δ,it comes to

Proof.Let h(t,h0)be the trivial solution of(6.1)with initial value h0.Since h(t,h0)is stable,for an arbitrary positive number ?,there exists a number δ＞0,when|h0|＜δ,we have|h(t,h0)|＜g1(?),for all t≥0.Denote the maximal solution of(6.1)byˉh(t,h0)with initial value h0,if|h0|＜δ,then we also have

By conditions on g2(t),for h0∈(0,δ),there exists δ′＞0,if‖x0‖＜δ′,then we have

So,by(6.15)we have

By Theorem 6.1 and(6.23),we get

Since g1(t)is convex,and by(6.24)we get

Finally,because g1is strictly increasing,we have

The proof is complete.

Example 6.1.Consider the following stochastic heat equation

where t≥0,a＞0,c＞0,b,σ,z,l∈R1,|·|denotestheabsolute value of thereal number,Btis a standardone-dimensionalBrownianmotion,eN(dt,dz)is acompensatedPoissonrandom measure on[?c,c]with character measure λ(dz),and X(t,x0)denotes the solutions of (6.25)with initial conditions X(0)=x0∈R1.

then it is easy to check

Letting the Razumkihin-Lyapunov function V(x)=‖x‖2,x∈H;for any g(t)∈D([0,π];H) with g(t)∈D(A),for t≥0,we have

By Theorem 4.1 and Theorem 5.1,we know that if

Example 6.2.Denote X(t,x)a process satisfying X(0,x)=x,then we consider the semilinear stochastic partial differential equation:

and for some M＞0,?(·):R+→R1is a bounded function with|?(t)|≤M for all t≥0, b(x):H→R1isnonlinearandLipschitzcontinuouswithb(0)=0,|b(x)|≤M‖x‖,W(t),t≥0, is an H-valued Q-wiener process with covariance operator Q,trQ＜∞.

So it is easy to deduce

For process Y(t)∈D([0,+∞);H),it is clear that

and

ThenlettingtheRazumikhin-LyapunovfunctionV(x)=‖x‖2,x∈H,wehavebyastraightforward computation that

By Theorem4.1 and Theorem5.1,we know if?2a+2M+M2trQ+C＜0,thenthe equation is mean square and almost surely exponentially stable.Finally if we let g1(t)=g2(t)=t2, t≥0 and V(x)=‖x‖2for x∈H,F(t)=?dt,t≥0,d＜2a?2M?M2trQ?C,thencomparison principles Theorem 6.2,Corollary 6.1 and Theorem 6.3 all hold.

Acknowledgments

The authors thank for Professor Dong Zhao for his valuable discussions.This work is partially supported by NNSF of China(Grant No.11071258/A0110,90820302),Tianyuan Foundation of NSF(Grant No.11126079)and CPSF(Grant No.2013M530559).

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10.4208/jpde.v26.n3.4 September 2013

?Corresponding author.Email addresses:zhouguoli736@126.com(G.Zhou),gbl@iapcm.ac.cn(B.Guo), zthou@csu.edu.cn(Z.Hou)

AMS Subject Classifications: 74H55,60H15

Chinese Library Classifications: O211