HU LING,WU ZHENG,WEI ZHANG-ZHI AND WANG LIANG-LONG

(School of Mathematical Sciences,Anhui University,Hefei,230601)

Abstract:In this paper we discuss stochastic differential equations with a kind of periodic boundary value conditions(in sense of mean value).Appealing to the decomposition of equations,the existence of solutions is obtained by using the contraction mapping principle and Leray-Schauder fixed point theorem,respectively.

Key words:stochastic differential equation,Leray-Schauder fixed point theorem,boundary value problem,contraction mapping principle

1 Introduction

As we all know,boundary value problems(BVPs)for differential equations are very important in applications.In these years,a lot of existence of solutions for BVPs of deterministic differential equations have been obtained(see[1]–[4]),but there is much less for stochastic differential equations.

In 1991,Nualart and Pardoux[5]–[6]discussed the first and second order stochastic differential equations under corresponding conditions:

and

Their method was similar to the method of variation of constants.

In 2009,by using the Green’s function and the fixed point theorem,Dhage and Badgire[7]got the existence of solution of the following equations:

They discussed the random equation for existence as well as for existence of extremal solutions under suitable conditions of the nonlinearity f which thereby generalizes several existence results.

Fedchenko and Prigarin[8]investigated stationary boundary value problem

The relative simplicity of linear problems studied allows one to obtain complete results on the existence and uniqueness of the solution,formulate the equivalent Cauchy problem, find a link with deterministic optimal control problems,and solve a number of other problems.

Cao et al.[9]in 2014 paid their attention to the numerical solutions as follows:

It was proved that under certain regularity conditions,the resulting numerical solution of the homotopy method converged at the same rate as the numerical algorithm used to solve the initial value problem.Their numerical experiments demonstrate that the homotopy continuation method may be less restrictive in selecting the initial iterative point than the shooting method.It should be noted that the convergence analysis of the shooting method was incomplete and the method developed in this paper can be used to obtain the convergence rate of the shooting method.

The above BVPs are deterministic,but in many models the boundary value conditions are indeterminate.Actually there is not much research in this area.Fortunately Wang and Han[10]discussed the periodic boundary value problem as follows:

Under the suitable conditions,the existence of solution was obtained by a new technique.The main idea is to decompose the stochastic process into a deterministic term and a new stochastic term with zero mean value.

It is easily known that the BVPs of stochastic differential equation(SDE)are of great sense in many fields.In this paper,we discuss the problem as follows:

Then by virtue of Leray-Schauder fixed point theorem and the contraction mapping principle,the solutions y(t)and z(t)are obtained respectively.

This paper is divided into three parts.In the next section,we give some notations and lemmas.The main results are given in Section 3.

2 Preliminaries

In this paper,we consider the boundary value problem for stochastic differential equation

where f:Rn×[0,T]→ Rn,g:Rn×[0,T]→ Rn×mand W=(w1,···,wm)is an mdimensional standard Brownian motion de fined on a complete probability space(?,F,P).We denote by

the filtration generated by W(s).The space of all mean square-integrable Ft-adapted and Rn-valued stochastic processes is denoted bywith the norm

where the notation|·|denotes the norm of the Euclidean space.

We decompose x:[0,T]→Rninto two parts

where

is a deterministic Rn-valued mapping,and

is an Rn-valued stochastic process with Ez(t)=0.Substituting(2.3)into(2.1),we have

and

because of Ez(t)=0,we have

Hence we obtain the equations and

We discuss(2.4) first.Adding y(t)on both sides of the first equation of(2.4)and multiplying et,we have

Solving this differential equation,we get

Considering the boundary value condition,we have

From simple calculation we know that for any z(t),(2.4)is equivalent to

Let S:=C([0,T],Rn)with the maximum norm

De fine the operator Ψ :S →S by

Lemma 2.1Given u(t),v(t)∈S,let z(u,t),z(v,t)be the corresponding solutions of(2.5)with any initial value.Then

Proof. For any initial value z(0)=η,(2.5)is equivalent to

Thus,by(H1),we have

Let

Then,by Gronwall’s inequality,we obtain

Lemma 2.2If the Lipschitz constants L and M satisfy

then for any z(t)the BVP(2.4)has a unique solution.

Proof.Obviously,Ψ:S→S is a continuous mapping.Note that

Thus,by the contraction mapping principle,Ψ has a fixed point.

Lemma 2.3[11]Consider the following SDE problem:

if there exist two constants L＞0 and M＞0 such that

for all x1,x2,x∈Rn,0≤t≤T,then there exists a unique solutionof the SDE(2.7).

Let y(t)be the solution of the problem(2.4).Consider

where η is any F0-adapted random variable satisfying Eη=0.For any y(t)∈ S,by Lemma 2.3,the SDE(2.8)has a unique solution z(t,y).

3 Main Results

Theorem 3.1Suppose that f,g are Lipschitz continuous and linear growth in x with Lipschitz constants L and M respectively.If the inequality

holds,then BVP(2.1)has at least one solution.

Proof. Note that y(t)and z(t)are solutions of(2.4)and(2.8)respectively,hence

satis fies the problem(2.1).

Remark 3.1From the above proof,we find that the solution of(2.1)is not unique.We will obtain some different x(t)depending on the choice of η in(2.8).However,the difference between two solutions of(2.1)is a stochastic process with zero mean value.

Theorem 3.2Suppose that f,g are Lipschitz continuous and linear growth in x.If f(x(t),t)is bounded in mean sense,i.e.,there is an N＞0 such that

then the BVP(2.1)has at least one solution.

Proof. For any y∈S,suppose that f(t,y)is bounded in mean sense.It is easy to proof that the operator Ψ de fined by(2.6)is completely continuous Let

where

For each y∈Br(0),we have

Thus

From the Leray-Schauder fixed point theorem,there exist a fixed point y?∈ Br(0),which is a solution of the problem(2.4).z?(t)is a solution of the SDE(2.8).By an argument similar to Theorem 2.1,we get that

is the solution of the SDE(2.1).