ZHAO Jiaxin,LI Zhi and XU Liping

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

Abstract. Using a novel approach,we present explicit criteria for the quasi contraction of stochastic functional differential equations.As an application,some sufficient conditions ensuring the contraction property of the solution to the considered equations are obtained.Finally,some examples are investigated to illustrate the theory.

Key Words: Quasi contraction;contraction;stochastic functional differential equations.

1 Introduction

Recently,the contraction problem of stochastic differential equations has attracted lots of attention and become one of the most active areas in biology [1],control theory [2],observer design [3],synchronization of coupled oscillators [4],traffic networks [5],and so on.For example,Dahlquist[6]employed logarithmic norms to demonstrate the contractivity of differential equations.Aminzare et al.[7]investigated nonlinear system contraction methods.The contraction analysis for hybrid systems was studied by Burden et al.[8].Margaliot et al.[9]proposed three generalizations of contraction based on a norm that allows contraction to take place after small transients in time or amplitude.

The Lyapunov function method is a well-known method for determining contractibility in stochastic differential equations.Contractibility of stochastic differential equations has been achieved using Lyapunov functions and functionals (see[10-12]).For stochastic differential equations,finding a Lyapunov function is difficult,and the contractibility criteria produced by the Lyapunov function approach are frequently expressed in terms of differential inequalities,matrix inequalities,and so on.The Lyapunov function’s stated requirements are not only a little bit strong,but also broadly implicit and difficult to investigate.Furthermore,when studying the contraction of stochastic differential equations by integral inequalities,we find that there are two flaws:the coefficients must typically satisfy the Lipschitz condition,and the Lipschitz constants must usually be sufficiently small.

In this paper,we will investigate quasi contraction of stochastic differential equations using a novel approach that does not require any integral inequality.Furthermore,We establish the explicit exponential contraction condition for stochastic differential equations.Usingformulae,we will get some new sufficient conditions ensuring the contraction of stochastic differential equations based on a comparison principle and proof by reductio ad absurdum.

The following is how the rest of the paper is structured:We introduce some necessary notations and preliminaries in Section 2.The quasi contraction and exponential contraction of stochastic differential equations are discussed in Section 3.In Section 4,we give some instances to show how our findings are beneficial.

2 Preliminary

Let (Ω,F,P) be a complete probability space equipped with some filtration{Ft}t≥0satisfying the usual conditions,i.e.,the filtration is right continuous andF0contains all P-null sets.LetH,Kbe two real separable Hilbert spaces and we denote by 〈·,·〉H,〈·,·〉Ktheir inner products and by‖·‖H,‖·‖Ktheir vector norms,respectively.We denote byL(K,H) the set of all linear bounded operators fromKintoH,equipped with the usual operator norm‖·‖.Letτ＞0 andC:=C([-τ,0];H) denote the family of all continuous functions from [-τ,0] toH.The spaceC([-τ,0];H) is assumed to be equipped with the norm‖φ‖C=sup-τ≤θ≤0‖φ(θ)‖H.We also denotebe the family of all almost surely bounded,F0-measurable,C([-τ,0];H)-valued random variables.

Let{W(t),t≥0}denote aK-valued{Ft}t≥0-Wiener process defined on (Ω,F,P) with covariance operatorQ,i.e.,

whereQis a positive,self-adjoint,trace class operator onK.In particular,we shall call suchW(t),t≥0,aK-valuedQ-Wiener process with respect to{Ft}t≥0.

In order to define stochastic integrals with respect to theQ-Wiener processW(t),we introduce the subspaceK0=Q1/2(K) ofKwhich,endowed with the inner product

Clearly,for any bounded operators,this norm reduces to.

For arbitrarily givenT≥0,letJ(t,ω),t ∈[0,T],be anFt-adapted,-valued process,and we define the following norm for arbitraryt∈[0,T],

In particular,we denote all-valued predictable processesJsatisfying‖J‖T ＜∞by.The stochastic integral,t≥0,may be defined for allby

Consider the following semilinear stochastic partial functional differential equation

whereAis the infinitesimal generator ofC0-semigroup{S(t)}t≥0of bounded linear operators overHwith domainD(A)?H,the mappingf:R+×C([-τ,0];H)→Handg:R+×C([-τ,0];H)→L(K,H).

Definition 2.1.A stochastic process{x(t),t ∈[0,T]},0≤T ＜∞,is called a strong solution of(2.1)if

(i) x(t)is adapted to Ft,t≥0;

(ii) x(t)∈D(A)on[0,T]×Ωwithalmost surely and for arbitrary0≤t≤T,

for any.

Definition 2.2.A stochastic process{x(t),t∈[0,T]},0≤T＜∞,is called a mild solution of(2.1)if

(i) x(t)is adapted to Ft,t≥0;

(ii) x(t)∈H has the continuous paths on t∈[0,T]almost surely,and for arbitrary0≤t≤T,

and.

Corollary 2.1.Let γ be a constant and non-decreasing functions η(·):[-τ,0]→R+.Then,

(I) Assume that there exists a constant β≥0such that for any t∈R+,

Then,the mild solution of(2.1)is quasi constractive in mean square,if

(II) If for any t∈R+,

Then,the mild solution of(2.1)is exponentially constractive in mean square,if

From the Corollary 3.1 and Corollary 3.2,we immediate obtain the following Corollary 3.3.

Corollary 2.2.Let hi(·):R+→R,i=0,1,···,n with0:=h0(t)≤h1(t)≤h2(t)≤···≤hn(t)≤τ,t∈R+,be locally bounded Borel measurable functions.

(I) Suppose that there exist constants γi,i=0,1,···,n,β≥0and the Borel measurable function θ:[-τ,0]→R+,such that for any t∈R+,

Then,the mild solution of(2.1)is quasi constractive in mean square,if

(II) Suppose that there exist constants γi,i=0,1,···,n and the Borel measurable function θ:[-τ,0]→R+,such that for any t∈R+,

Then,the mild solution of(2.1)is exponentially constractive in mean square,if

In the section,we will work under the following hypotheses:

(H1)Ais the infinitesimal generator of a contractionC0-semigroupS(t),t≥0.

(H2) For anyb∈[0,∞),there exists a constantM＞0 such that for anyt∈[0,b]andx,y∈C([-τ,0];H),

(H3) For anyb∈[0,∞),f(t,0)∈L2([0,b];H) and,),where the twoL2spaces are defined in[14].

It is well known that (2.1) has an unique mild solution under the hypotheses (H1),(H2) and (H3).e.g.,see[13-15].

3 Quasi contraction

Since the mild solutions do not have stochastic differentials,by theformula,we cannot deal with mild solutions directly in most arguments.To this end,we introduce the following approximating system:

wheren∈ρ(A),the resolvent set ofAandR(n)=nR(n,A),R(n,A) is the resolvent ofA.The following lemma is important to prove our result in this section,we can refer to[13,18].

Lemma 3.1.Letbe an arbitrarily given initial datum and assume that conditions (H1) to (H3) hold.Then(3.1)has a unique strong solution xn ∈D(A),which lies in C([0,T];L2(Ω,F,P;H))for all T＞0.Moreover,xn converges to the mild solution x(t)of(2.1)almost surely in C([0,T];L2(Ω,F,P;H))as n→∞.

To state the main result of this section,let us define some functions.Letη(t,θ):R+×[-τ,0]→H,be non-decreasing inθfor eacht∈R+.Furthermore,η(t,θ) is normalized to be continuous from the left inθon[-τ,0].Assume that

is a locally bounded Borel-measurable function intfor each? ∈C([-τ,0];H).Here,the integral in (3.2) is the Riemann-Stieltjes integral.

Definition 3.1.The mild solution of(2.1)is said to be quasi constractive in mean square if there exists a pair of positive constants δ,K and a constant β such that

for any t∈R+and for any ξ,.

Definition 3.2.The mild solution of(2.1)is said to be exponentially constractive in mean square if there exists a pair of positive constants δ,K and a constant β such that

for any t∈R+and for any ξ,.

Theorem 3.1.Let γ(·):R+→Rbe a locally bounded Borel-measurable function.Assume that there exists a constant β≥0such that for any t∈R+,

Then the mild solution of(2.1)is said to be quasi constractive in mean square,if there exists δ＞0such that for any t∈R+,

Proof.FixK＞1 and letsuch that.For the sake of simplicity,we denotexn(t):=xn(t,ξ),yn(t):=xn(t,ζ) andx(t):=x(t,ξ),y(t):=x(t,ζ),t≥-τ,wherexn(t,ξ) andx(t,ξ) are the strong solution to (3.1) and the mild solution to (2.1),respectively.LetandZ(t):=Ke-δtE‖ξ-,t≥0.We will show

Assume on the contrary that there existst1＞0 such thatXn(t1)＞Z(t1).Lett*:=inf{t＞0:Xn(t)＞Z(t)}.By continuity ofXn(t) andZ(t),

From (3.5) and using‖R(n)‖≤2,the dominated convergence theorem and the Fubini’s theorem,it follows that

which conflicts with (3.7).Therefore

The proof is complete.

Theorem 3.2.Let γ(·):R+→Rbe a locally bounded Borel-measurable function,such that for any t∈R+,

Then,the mild solution of(2.1)is exponentially constractive in mean square if there exists δ＞0such that for any t∈R+,

By view of[16]and the Theorem 3.1,we easily obtain the following corollaries.

Corollary 3.1.Under the hypothesis of Lemma3.1,letY (·,·):R+×[-τ,0]→R+,γi(·),hi(·):R+→R,i=0,1,···,n with0:=h0(t)≤h1(t)≤h2(t)≤···≤hn(t)≤τ,t∈R+,be locally bounded Borel measurable functions.Then

(I) Assume that there exists a constant β≥0such that for any t∈R+,

Then,the mild solution of(2.1)is said to be quasi constractive in mean square,if there exists δ＞0such that for any t∈R+,

(II) If for any t∈R+,

Then,the mild solution of(2.1)is exponentially constractive in mean square if there exists δ＞0such that for any t∈R+,

Corollary 3.2.Let γ be a constant and non-decreasing functions η(·):[-τ,0]→R+.Then,

(I) Assume that there exists a constant β≥0such that for any t∈R+,

Then,the mild solution of(2.1)is quasi constractive in mean square,if

(II) If for any t∈R+,

Then,the mild solution of(2.1)is exponentially constractive in mean square if

From the Corollary 3.1 and the Corollary 3.2,we immediate obtain the following Corollary 3.3.

Corollary 3.3.Let hi(·):R+→R,i=0,1,···,n with0:=h0(t)≤h1(t)≤h2(t)≤···≤hn(t)≤τ,t∈R+,be locally bounded Borel measurable functions.

(I) Suppose that there exist constants γi,i=0,1,···,n,β≥0and the Borel measurable function θ:[-τ,0]→R+,such that for any t∈R+,

Then,the mild solution of(2.1)is quasi constractive in mean square,if

(II) Suppose that there exist constants γi,i=0,1,···,n and the Borel measurable function θ:[-τ,0]→R+,such that for any t∈R+,

Then,the mild solution of(2.1)is exponentially constractive in mean square,if

Remark 3.1.As Xu,Wang and Yang pointed out in [15].Condition (H3) is necessary for the existence and uniqueness of the mild solution of (2.1) and the strong solution of (3.1).Condition (H2) can be reduced to the local condition [15] or the non-Lipschitz condition by the existence of mild solutions of stochastic partial functional differential equations with non-Lipschitz coefficients [17] and following the proof of Lemma 4.1 in[15].Especially,for the deterministic functional differential equations,condition (H2) may be weakened down to the requirement of Eq.(1) in[16].

4 Illustrate some examples

Example 4.1.Consider the following the semilinear stochastic functional differential equation with time-varying delay

where 0＜δ1(t),δ2(t)≤τ,t∈R+are locally bounded Borel measurable functions.

Assume that there exists a constantλ＞0 such that

Suppose that there exist locally bounded measurable functionsγi(·):R+→R,i=1,2,3,4 such that

for allt∈R+,x1,x2,y1,y2∈H.For anyβ＞0,by view of (4.2) and (4.3),we have

By the Corollary 3.3,we declare that for anyδ＞0 such that for anyt≥0,

Then,the solution of (4.1) is quasi constractive in mean square.

In the case thatH=K=Rn,A=0,consider the following stochastic functional differential equation

By the Corollary 3.1,the solution of (4.5) is exponentially constractive in mean square if

To illustrate further the effectiveness of the obtained result,we consider scalar stochastic functional differential equation

whereα＞0 stands for a parameter andw(t) is the 1-dimensional Brownian motion.

Clearly,(4.10) is the form of (4.5) withτ=1,

which means that (4.7) and (4.8) hold withγ1(s)=e-2sandγ2(s)=e-s.

So,by the Corollary 3.1 we deduce that the zero solution of (4.10) is exponentially constractive in mean square if

Example 4.2.Consider the following stochastic partial functional differential equation

whereh(·) is a function of bounded variation on[-τ,0].

Letgsatisfy the following non-Lipschitz condition:for anyu,v∈H,t≥0

whereρ(·) is a concave nondecreasing function from R+to R+such thatρ(0)=0,ρ(u)＞0 foru＞0 anddu=∞.By [17],we deduce that (4.11) has a unique mild solution.Clearly,(4.11) is of the form (2.1) where

fort ∈R+,φ ∈C([-τ,0];H).DefineV(s):=Var-τ,sh(·),s ∈[-h,0].Then,V(·) is nondecreasing on[-τ,0].By the properties of the Riemann-Stieltjes integral

By Theorem 3.2,the solutionx(t,t0,ξ) of (4.11) is said to be exponentially constractive in mean square,if there existsδ＞0 such that for anyt∈R+,

Definea(t):=2λ-2Var[-τ,0]h(·)-ρeδτ,t≥0.Let

Then,from the above equation we have for anyt∈R+,

For sufficiently smallδ∈(0,a0/2),we have

It means that for anyt∈R+,

SinceV(·) is non-decreasing,it follows that.Therefore,for anyt∈R+,

This means that solution of (4.11) is said to be exponentially constractive in mean square,ifa0=inft≥0a(t)＞0.