LI Zhiand LUO JiaowanSchool of Information and Mathematics,Yangtze University,Jingzhou 434023, China.

2SchoolofMathematicsandInformationSciences,GuangzhouUniversity,Guangzhou 510006,China.

Received 23 April 2013;Accepted 21 December 2013

Neutral Functional Partial Differential Equations Driven by Fractional Brownian Motion with
Non-Lipschitz Coefficients

LI Zhi1,?and LUO Jiaowan21School of Information and Mathematics,Yangtze University,Jingzhou 434023, China.

2SchoolofMathematicsandInformationSciences,GuangzhouUniversity,Guangzhou 510006,China.

Received 23 April 2013;Accepted 21 December 2013

.Under a non-Lipschitz condition being considered as a generalized case of Lipschitz condition,the existence and uniqueness of mild solutions to neutral stochastic functional differential equations driven by fractional Brownian motion with Hurst parameter 1/2＜H＜1 are investigated.Some known results are generalized and improved.

Neutral functional partial differential equation;mild solution;Wiener integral;fractional Brownian motion.

1 Introduction

Recently,the theory for stochastic differential equations(without delay)driven by a fractional Brownian motion(fBm)has been studied intensively(see e.g.[1-6]and the references therein).

As for the stochastic functional differential equations driven by a fBm,even much less has been done,as far as we know,there exists only a few papers published in this field.In[7],the authors studied the existence and regularity of the density by using the Skorohodintegral based on Malliavin calculus.In[8],Neuenkirch et al.studied the problem by using rough path analysis.In[9],Ferrante and Rovira studied the existence and convergence when the delay goes to zero by using the Riemann-Stieltjes integral.Usingalso the Riemann-Stieltjes integral,[10]proved the existence and uniqueness of mild solution in infinite dimensional space.In infinite dimensional space,[11]have discussed the existence,uniqueness and exponential asymptotic behavior of mild solutions by using Wiener integral.Very recently,[12]first investigated the following neutral stochastic functional differential equations drivenby a fractional Brownian motionundertheglobal Lipschitz and linear growth condition

Where A is the infinitesimal generator of an analytic semigroup of bounded linear operators,(S(t))t≥0,in a Hilbert space X,BHis a Q-fractional Brownian motion on a real and separable Hilbert space Y,r,ρ:[0,T]→[0,τ](τ＞0)are continuous,f,g:[0,T]×X→X, σ:[0,T]→L02(Y,X)are appropriate functions and ?∈C([?τ,0];L2(?,X)).HereL02(Y,X) denotes the space of all Q-Hilbert-Schmidt operators from Y into X(see Section 2).

Unfortunately,for many practical situations,the nonlinear terms do not obey the global Lipschitz and linear growth condition,even the local Lipschitz condition.Motivated by the above papers,in this paper,we aim to extend the existence and uniqueness of mild solutions to cover a class of more general neutral stochastic functional differential equations driven by a fractional Brownian motion with Hurst parameter 1/2＜H＜1 undera non-Lipschitz condition,with the Lipschitz condition being regarded as a special case,and a weakened linear growth condition.

The rest of this paper is organized as follows.In Section 2,we introduce some notations,concepts,and basic results about fractional Brownian motion,Wiener integral over Hilbert spaces and we recall some preliminary results about analytic semigroups and fractional power associated to its generator.In Section 3,the existence and uniqueness of mild solutions are proved.

2 Preliminaries

In this sectionwe collect some notions,conceptionsand lemmas on Wienerintegrals with respect to an infinite dimensional fractional Brownian motion.In addition,we also recall some basic results about analytical semi-groups and fractional powers of their infinitesimal generators which will be used throughout the whole of this paper.

Let(?,F,P)be a complete probability space.Consider a time interval[0,T]with arbitrary fixed horizon T and let{βH(t),t∈[0,T]}be the one-dimensional fractional Brownian motion with Hurst parameter H∈(1/2,1).This means by definition that βHis a centered Gaussian process with covariance function:

Moreover βHhas the following Wiener integral representation:

where β={β(t):t∈[0,T]}is a Wiener process,and KH(t,s)is the kernel given by

We put KH(t,s)=0 if t≤s.

We will denote byHthe reproducing kernel Hilbert space of the fBm.In factHis the closure of set of indicator functions{I[0,t],t∈[0,T]}with respect to the scalar product

Themapping I[0,t]→βH(t)can be extendedto an isometrybetweenHand thefirst Wiener chaos and we will denote by βH(?)the image of ? by the previous isometry.

We recall that for ψ,?∈Htheir scalar product inHis given by

We refer to[13]for the proof of the fact that K?His an isometry betweenHand L2([0,T]). Moreover for any ?∈H,we have

It follows from[13]that the elements ofHmay be not functions but distributions of negative order.In order to obtain a space of functions contained inH,we consider the linear space|H|generated by the measurable functions ψ such that

where αH=H(2H?1).The space|H|is a Banach space with the norm‖ψ‖|H|and we have the following conclusions[13].

Lemma 2.1.

and for any ψ∈L2([0,T]),we have

Let X and Y be two real,separable Hilbert spaces and letL(Y,X)be the space of bounded linear operator from Y to X.For the sake of convenience,we shall use the same notation to denote the norms in Y,X andL(Y,X).Let Q∈L(Y,X)be an operator defined by Qen=λnenwithfinite trace trare non-negative real numbers and{en}(n=1,2···)is a complete orthonormal basis in Y.We define the infinite dimensional fBm on Y with covariance Q as

separable Hilbert space.

Now,let φ(s),s∈[0,T]be a function with values in.The Wiener integral of φ with respect to BHis defined by where βnis the standard Brownian motion used to present.

Now we end this subsection by stating the following result in[12].

Now we turn to state some notations and basic facts about the theory of semi-groups and fractional power operators.Let A:D(A)→X be the infinitesimal generator of an analytic semigroup,(S(t))t≥0,of bounded linear operators on X.For the theory of strongly continuous semigroup,we refer to Pazy[14].We will point out here some notations and properties that will be used in this work.It is well known that there exist M≥1 and λ∈R such that‖S(t)‖≤Meλtfor every t≥0.If(S(t))t≥0is a uniformly bounded and analytic semigroup such that 0∈ρ(A),where ρ(A)is the resolvent set of A,then it is possible to define the fractional power(?A)αfor 0＜α≤1,as a closed linear operator on its domain D(?A)α.Furthermore,the subspace D(?A)αis dense in X,and the expression

defines a norm in D(?A)α.If Xαrepresents the space D(?A)αendowed with the norm‖·‖α,then the following properties are well known(cf.Pazy[14,Theorem 6.13]).

Lemma 2.3.Suppose that the preceding conditions are satisfied.

(1)Let 0＜α≤1.Then Xαis a Banach space.

(2)If 0＜β≤α then the injection Xα■→Xβis continuous.

(3)For every 0＜β≤1 there exists Mβ＞0 such that

We also need the following Lemma 2.4.

Lemma 2.4.(Caraballo[15],Lemma 1)For u,v∈X,and 0＜c＜1,

3 Existence and uniqueness

In this section we study the existence and uniqueness of mild solution for Eq.(1.1).For this equation we assume that the following conditions hold.

(H1)A is the infinitesimal generatorof an analytic semigroup,S(t)t≥0,of boundedlinear operators on X.Further,to avoid unnecessary notations,we suppose that 0∈ρ(A), and that,see Lemma 2.3,

for some constants M,Mβand every t∈[0,T].

(H2)The function f satisfies the following non-Lipschitz condition:for any x,y∈X and t≥0,

where κ is a concave nondecreasing function from R+to R+such that κ(0)=0, κ(u)＞0 andR0+du/κ(u)=∞,e.g.,κ～uα,1/2＜α＜1.We further assume that there is an M′＞0 such that sup0≤t≤T‖f(t,0)‖≤M′.

(H3)There exist constants 1/2＜β≤1,K1≥0 such that the function g is Xβ-valued and satisfies for any x,y∈X and t≥0,

and

We further assume that g(t,0)≡0 for t≥0 and the function(?A)βis continuous in the quadratic mean sense:

Definition 3.1.A X-valued process x(t)is called a mild solution of(1.1)if

satisfies

Lemma 3.1.([16,Theorem 1.8.2])Let T＞0 and c＞0.Let κ:R+→R+be a continuous nondecreasing function such that κ(t)＞0 for all t＞0.Let u(·)be a Borel measurable bounded nonnegative function on[0,T],and let υ(·)be a nonnegative integrable function on[0,T].If

Then

holds for all such t∈[0,T]that

where

and J?1is the inverse function of J.In particular,if,moreover,c=0 andR0+ds/κ(s)=∞,then u(t)=0 for all t∈[0,T].

To complete our main results,we need to prepare several lemmas which will be utilize in the sequel.

Note that g(t,0)≡0 and

Then we easily get that‖(?A)βg(t,x)‖2≤K21‖x‖2.Thus,by[12,Theorem 5],we can introduce the following successive approximating procedure:for each integer n=1,2,3,···,

and for n=0,

While for n=1,2,···,

Lemma 3.2.Let the hypothesis(H1)-(H4)hold.Then there is a positive constant C1,which is independent of n≥1,such that for any t∈[0,T]

Proof.For 0≤t≤T,it follows easily from(3.1)that

Note from[14]that(?A)?βfor 0＜β≤1 is a bounded operator.Employing the assumption(H3),it follows that

Applying the H¨older’sinequality and taking into account Lemma 2.3 as well as(H3),and the fact that 1/2＜β＜1,we obtain

On the other hand,in view of(H2),we obtain that

Next,by Lemma 2.2,we have

Since κ(u)is concave on u≥0,there is a pair of positive constants a,b such that

Putting(3.4)-(3.7)into(3.3)yields that,for some positive constants C2and C3,

While,for‖(?A)?β‖K1＜1,by Lemma 2.4,

which further implies that

Thus,by(3.8)we have

Observing that

we then derive that,for some positive constants C4,C5,

Now,an application of the well-known Gronwall’s inequality yields that

The required assertion(3.2)is obtained since k is arbitrary.

Lemma 3.3.Let the condition(H1)-(H4)be satisfied.For 1/2＜β≤1,we further assume that

where Γ(·)is the Gamma function and M1?βis a constant in Lemma 2.3.Then there exists a positive constant C such that,for all 0≤t≤T and n,m≥1,

Proof.From(3.1),it is easy to see that for any 0≤t≤T

Following from the proof of Lemma 3.2,there exists a positive C6satisfying

thelast inequality holdsfromthe Jensen’sinequality.Now,by thecondition(H3),Lemma

2.3 and H¨older’s inequality,

On the other hand,Lemma 2.4 and(H3)give that

So the desired assertion(3.10)follows from(3.11).

We can now state the main result of this paper.

Theorem 3.1.Under the conditions of Lemma 3.3,then Eq.(1.1)admits a unique mild solution. Proof.Uniqueness:Denote by x(t)and x(t)the mild solutions to(1.1).In the same way as Lemma 3.3 was done,we can show that for some K＞0

This,together with Lemma 3.1,leads to

which further implies x(t)=x(t)almost surely for any 0≤t≤T.

Existence:Following the proof of Lemma 3.3,there exists a positive C such that,for all 0≤t≤T and n,m≥1,

Integrating both sides and applying Jensen’s inequality gives that

Then

where

While by Lemma 3.2,it is easy to see that

Soletting h(t):=limsupn,m→∞hn,m(t)and takinginto account theFatou’slemma,we yield that

Now,applying the Lemma 3.1 immediately reveals h(t)=0 for any t∈[0,T].This further means{xn(t),n∈N is a Cauchy sequence in L2.So there is a x∈L2such that

In addition,by Lemma 3.2,it is easy to follow that E‖x(t)‖2≤C1.In what follows,we claim that x(t)is a mild solution to(1.1).On one hand,by(H3),

whenever n→∞.On the other hand,by(H3)and Lemma 2.3,compute for t∈[0,T]

While,applying(H2),the H¨older’s inequality and[17,Theorem 1.2.6]and letting n→∞, we can also claim that for t∈[0,T]

Hence,taking limits on both sides of(3.1),

This certainly demonstrates by the Definition 3.1 that x(t)is a mild solution to(1.1)on the interval[0,T].

Acknowledgments

This research is partially supported by the NNSF of China(No.11271093)and the cultivation project of Yangtze University for the NSF of China(2013cjp09).

We are grateful to anonymous referees for many helpful comments and valuable suggestions on this paper.

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?Corresponding author.Email addresses:lizhi csu@126.com(Z.Li),jluo@gzhu.edu.cn(J.Luo)

AMS Subject Classifications:60H15,60G15,60H05

Chinese Library Classifications:O211.63,O175.2