Tong Ouyang,　Weiguo Liu

(School of Math.and Information Science,Guangzhou University,Guangzhou 510006)

FIXED POINTS AND EXPONENTIAL STABILITY OF ALMOST PERIODIC MILD SOLUTIONS TO STOCHASTIC VOLTERRA-LEVIN EQUATIONS??

Tong Ouyang?,Weiguo Liu

(School of Math.and Information Science,Guangzhou University,Guangzhou 510006)

In this paper,we consider stochastic Volterra-Levin equations.Based on semigroup of operators and fixed point method,under some suitable assumptions to ensure the existence and stability of pth-mean almost periodic mild solutions to the system.

stochastic differential equation;fixed points theory,almost periodic solutions

2000 Mathematics Subject Classification 65C30;37C25;70H12

Ann.of Applied Math.

31:2(2015),190-199

1　Introduction

Stochastic differential equations have attracted much attention since stochastic modeling plays an important role in physics,engineering,finance,social science and so on.Qualitative properties such as the existence,uniqueness and stability of stochastic differential systems have been extensively studied by many researchers,see for instance[5,9,12-14].Recently, the concept of quadratic mean almost periodicity was introduced by Bezandry and Diagana [2].In[2],the authors proved the existence and uniqueness of a quadratic mean almost periodic solution to the stochastic evolution equations.Bezandry[4]considered the existence of quadratic mean almost periodic solutions to semi-linear functional stochastic integrodifferential equations.For more results on this topic,we refer the reader to the papers [1,3,6,7,11]and references therein.

On the other hand,Volterra equations have been used to model the circulating fuel nuclear reactor,the neutron density and the neural networks and so on.In[15],Luo used the fixed point theory to study the exponential stability in mean square and the exponential stability for Volterra-Levin equations.Zhao,Yuan and Zhang[18]improved some wellknown results in Luo[15].We refer the reader to the papers[8,17,19]and the references therein.

As far as we know so far no one has studied the almost periodic mild solutions to stochastic Volterra-Levin equations.Motivated by the above works,we investigate the existence and stability of pth-mean almost periodic mild solutions to stochastic Volterra-Levin equations in the abstract form

where B(t)is a Brownian motion.Some sufficient conditions ensure the existence and stability of p-mean almost periodic mild solutions.

The rest of this paper is organized as follows.In Section 2 some necessary preliminaries on some notations and lemmas are established.In Section 3 the existence and stability of pth-mean almost periodic mild solutions are proved.

2　Preliminaries

In this section,in order to prove the existence and stability of the pth-mean almost periodic mild solutions of equation(1.1),we need some notations,definitions and lemmas.

Let{?,F,P}be a complete probability space equipped with some filtration{Ft}t≥0satisfying the usual conditions,that is,the filtration is right continuous and F0contains all P-null sets.Let(B,‖·‖)be a Banach space and p≥2,denote by Lp(P,B)the Banach space of all B-value random variables y satisfying

Next we introduce the following useful definitions[2].

Definition 2.1 A continuous stochastic process X:R→Lp(P,B)is said to be p-mean almost periodic if for each ε＞0 there exists an l(ε)＞0 such that any interval of length l(ε)contains at least a number κ for which

Consider the Banach space CUB(R;Lp(P,B))=CUB(R;Lp(?,F,P,B))of all continuous and uniformly bounded process from R into Lp(P,B)equipped with the sup norm

Denote by AP(R,Lp(P,B))the collection of all p-mean almost periodic stochastic processes.

Lemma 2.1 If X belongs to AP(R,Lp(P,B)),then:

(i)The mapping t→E‖X(t)‖pis uniformly continuous;

(ii)there exists a constant M＞0 such that E‖X(t)‖p≤M,for all t∈R.

Lemma 2.2 AP(R,Lp(P,B))?CUB(R,Lp(P,B))is a closed subspace.

Let(B1,‖·‖1)and(B2,‖·‖2)be Banach spaces and Lp(P,B1),Lp(P,B2)be their corresponding Lp-spaces respectively.

Definition 2.2 A function f:R×Lp(P,B1)→Lp(P,B2),which is jointly continuous, is said to be p-mean almost periodic in t∈R uniformly in Y∈K,where K ?Lp(P,B1) is a compact,if for any ε＞0,there exists an l(ε,K)＞0 such that any interval of length l(ε,K)contains at least a number κ for whichfor each stochastic process Y:R→K.

Denote the set of such functions by AP(R×Lp(P,B1),Lp(P,B2)).

Let(U,‖·‖U,〈·,·〉U)and(V,‖·‖V,〈·,·〉V)be separable Hilbert spaces.Denote by L(V,U) the space of all bounded linear operators from V to U.Let Q∈L(V,V)be a non-negative self-adjoint operator anddenotes the space of all ξ∈L(V,U)such thatis a Hilbert-Schmidt operator.The norm is given by

Let{Bn(t)}n∈Nbe a sequence of real-valued one-dimensional standard Brownian motions mutually independent of(?,F,P),and{en}n∈Nbe a complete orthonormal basis in V.We call the V-valued stochastic process

is a Q-Wiener process,where λn,n∈N are nonnegative real numbers and Q is a nonnegative self-adjoint operator such that Qen=λnenwith

Let A:Dom(A)?U→ U be the infinitesimal generator of an analytic semigroup {S(t)}t≥0in U.Then(A-βI)is an invertible and bounded analytic semigroup for β＞0 large enough.Suppose that 0∈ρ(A),where ρ(A)is the resolvent set of A.Then,for β∈(0,1],it is possible to define the fraction power(-A)βas a closed linear operator on its domain Dom((-A)β).Furthermore,the subspace Dom((-A)β)is dense in U,and the expression

defines a norm in Dom((-A)β).If Uβrepresents the space Dom((-A)β)endowed with the norm‖·‖β,then the following properties are well known(cf.Pazy[16,Theorem 6.13 p.74]).

Lemma 2.3 Suppose that the preceding conditions are satisfied,then:

(1)For 0＜β≤1,Uβis a Banach space;

(2)if 0＜δ≤β then the injection Uβ■→Uδis continuous;

(3)for every 0＜δ≤1,there exists an Mδ＞0 such that

The following lemma was proved in[3,Theorem 4.4 p.125].

Lemma 2.4 Let F:R×Lp(P,B1)→Lp(P,B2),(t,Y)■→F(t,Y)be a p-mean almost periodic process in t∈R,uniformly for Y∈K,where K ?Lp(P,B1)is compact.Suppose that F is Lipschitzian in the following sense:

for t∈R and Y,Z∈Lp(P,B1),where G＞0;then for any p-mean almost periodic stochastic process Φ:R→Lp(P,B1),the stochastic process t→F(t,Φ(t))is p-mean almost periodic.

Definition 2.3 Equation(1.1)is said to be exponentially stable in pth-mean,if for any initial value φ,there exists a pair of constants α＞0 and C＞0 such that

3　Almost Periodic Mild Solutions

In this section,we consider the exponential stability in pth-mean of almost periodic mild solutions to stochastic Volterra-Levin functional differential equations

by means of the fixed-point theory,where B(t)is a Brownian motion,A:Dom(A)?U→U is the infinitesimal generator of an analytic semigroup S(·)on U,that is,for t≥0,‖S(t)‖U≤Me-λt,with M＞1,and we assume that λ≥M.Assume that f:R×Lp(P,U)→Lp(P,U) is an appropriate function satisfying f(t,0)=0,g∈C([-L,0];R),and σ:[0,∞)→The initial data{φ=φ(t):-L≤t≤0}is an F0-measurable U-valued random variable independent of B with finite second moment.

Definition 3.1An U-valued process x(t)is called a mild solution to(3.1)if x∈CUB([-L,∞);Lp(P,U)),x(t)=φ(t)for t∈[-L,0],and,for any t＞0,satisfies

In this paper,we always assume that the following assumptions hold:

(H1)For a constant β∈[0,1],the function f∈AP([0,T]×U,U),there exists a function Nf:R→R+such that

(H2)Nf(t)＜G,t∈R,where G is involved in Lemma 2.4;

(H3)there exists a constant Q＞0 such that

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Theorem 3.1Suppose that conditions(H1)-(H5)hold.Then equation(3.1)has a unique pth-mean almost periodic mild solution x(t),which is exponentially stable,if,for some constant α∈(0,1],

Proof Define by S the collection of all pth-mean almost periodic stochastic processes φ(t,ω):[-L,∞)×?→R,which is almost surely continuous in t for fixed ω∈?.Moreover, φ(s,ω)=φ(s)for s∈[-L,0]and eηtE‖φ(t,ω)‖pU→ 0 as t→ ∞,where η is a positive constant such that 0＜η＜λ.

Define an operator π:S→S by(πx)(t)=φ(t)for t∈[-L,0]and for t≥0,

For any constant α∈(0,1],(3.4)can be rewritten as

where

Firstly,we show that Φx(t)is p-mean almost periodic whenever x is p-mean almost periodic.Indeed,assuming that x is p-mean almost periodic,using condition(H1)and Lemma 2.4,one can see that s→f(s,x(s))is p-mean almost periodic.Therefore,for each ε＞0 there exists an l(ε)＞0 such that any interval of length l(ε)＞0 contains at least κ satisfying

for each s∈[0,t].Furthermore,

Secondly,we show that Φx(t)is p-mean almost periodic whenever x is p-mean almost periodic.We know that f(s,x(s))is p-mean almost periodic,therefore,for each ε＞0 there exists an l(ε)＞0 such that any interval of length l(ε)contains at least κ satisfying

Now using(H1),Lemma 2.4 and(3.7)we can obtain

Thirdly,by H?lder’s inequality and Lemma 7.7 in[10],for the chosen κ＞0 small enough,we have

where cp=(p(p-1))p/2.From the above discussion,it is clear that the operator π maps AP([0,∞),Lp(?,U))into itself.Thus,π is continuous in pth mean on[0,∞).Next,we show that π(S)?S.It follows from(3.4)that

Now we estimate the terms on the right-hand side of(3.8).Firstly,we obtain

Secondly,H?lder’s inequality and(H1)yield

For any x(t)∈S and any ε＞0,there exists a t1＞0 such that eη(u+s)E‖x(u+s)‖pU＜ε for t≥t1.Thus from(3.10)we can get

As e-(λ-η)t→ 0 as t→ ∞ and condition(3.3),there exists a t2≥t1such that for any t≥t2,we have

So from the above analysis and(3.11),we obtain for any t≥t2

That is,

As for the third term on the right-hand side of(3.8),by Lemma 7.7 in[10]we have

Thus,from(3.8),(3.9),(3.13)and(3.14),we know that eηtE‖(πx)(t)‖pU→0 as t→∞.So we conclude that π(S)?S.

Finally,we shall show that π is contractive.For x,y∈S,we can obtain

so π is a contraction mapping with contraction constant γ＜1.By the contraction mapping principle,π has a unique fixed point x(t)in S,which is the pth-mean almost periodic mild solution to equation(3.1)with x(t)=φ(t)on[-L,0]and eηtE‖x(t)‖pU→0 as t→∞.The proof is completed.

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(edited by Liangwei Huang)

?This research was partially supported by the NNSF of China(Grant No.11271093).

?Manuscript November 6,2014

?.E-mial:OMIyoung@yahoo.com