Impulsive neutral stochastic functional integro-differential equations with infinite delay driven by fBm

Applied Mathematics and Computation 247 (2014) 205–212

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Impulsive neutral stochastic functional integro-differential equations with infinite delay driven by fBm q Yong Ren a,⇑, Xing Cheng a, R. Sakthivel b a b

Department of Mathematics, Anhui Normal University, Wuhu 241000, China Department of Mathematics, Sungkyunkwan University, Suwon 440-746, South Korea

a r t i c l e

i n f o

Keywords: Stochastic evolution equation Evolution operator Fractional Brownian motion Mild solution

a b s t r a c t In this paper, we study a class of impulsive neutral stochastic functional integro-differential equations with infinite delay driven by a standard cylindrical Wiener process and an independent cylindrical fractional Brownian motion (fBm) with Hurst parameter H 2 ð1=2; 1Þ in the Hilbert space. We prove the existence and uniqueness of the mild solution for this kind of equations with the coefficients satisfying some non-Lipschitz conditions, which include the classical Lipschitz conditions as special case. An example is provided to illustrate the theory. Some well-known results are generalized and extended. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction In this paper, we are concerned with the existence and uniqueness of the mild solution for the following impulsive neutral stochastic functional integro-differential equations with infinite delay driven by a standard Wiener process and an independent fractional Brownian motion hR i 8 t > d½xðtÞ þ gðt; xt Þ ¼ A½xðtÞ þ gðt; xt Þdt þ 0 Bðt  sÞ½xðsÞ þ gðs;xt Þds þ f ðt; xt Þ dt þ hðt; xt ÞdwðtÞ þ rðtÞdBHQ ðtÞ; t 2 J :¼ ½0; T; t – tk ; > <     Dxðtk Þ ¼ x t þk  x t k ¼ Ik ðxðt k ÞÞ; k ¼ 1; 2; .. . ; m; > > : xðtÞ ¼ uðtÞ 2 DBF 0 ðð1; 0;XÞ; t 2 J 0 :¼ ð1; 0;

ð1:1Þ

where xðÞ takes value in a real separable Hilbert space X with inner product ð; Þ and norm jj  jj; A is the infinitesimal generator of a strongly continuous semigroup ðSðtÞÞtP0 on X with domain DðAÞ; BðtÞ is a closed linear operator on X with domain DðBÞ  DðAÞ which is independent of t; BH is a fractional Brownian motion with Hurst parameter H 2 ð1=2; 1Þ and fwðtÞ : t 2 Jg is a standard Wiener process on a real and separable Hilbert space Y; T P 0 is a fixed real number. In the sequel, let ðX; F ; PÞ be a complete probability space and for t P 0; F t denote the r-field generated by fBHQ ðsÞ; wðsÞ; s 2 ½0; tg and the P-null sets. Further, we assume that w and BHQ are independent. Let LðY; XÞ be the space of all bounded, continuous and linear b ! X; h : ½0; þ1Þ  D b ! L0 ðY; XÞ; r : ½0; þ1Þ ! L0 ðY; XÞ are appropriate operators from Y into X. Assume that g; f : ½0; þ1Þ  D 2 Q b functions. Here, D ¼ Dðð1; 0; XÞ denotes the family of all right piecewise continuous functions with left-hand limit u from ð1; 0 to X. For equations with infinite delay, the segment xt : ð1; 0 ! X is defined by xt ðhÞ ¼ xðt þ hÞ for t P 0 belongs to b The space L0 and L0 will be defined in the next section. Here, Ik 2 CðX; XÞ ðk ¼ 1; 2; . . . ; mÞ are bounded the phase space D. 2 Q q

The work of Yong Ren is supported by the National Natural Science Foundation of China (Nos. 11371029 and 11201004).

⇑ Corresponding author.

E-mail addresses: [email protected], [email protected] (Y. Ren). http://dx.doi.org/10.1016/j.amc.2014.08.095 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

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Y. Ren et al. / Applied Mathematics and Computation 247 (2014) 205–212

 functions and the fixed times tk satisfies 0 ¼ t0 < t 1 < t 2 <    < tm < T; xðtþ k Þ and xðt k Þ denote the right and left limits of xðtÞ  at time t k . And Dxðtk Þ ¼ xðtþ Þ  xðt Þ represents the jimp in the state x at time t , where Ik determines the size of the jump. k k k b stochastic process independent of the wiener proThe initial data u ¼ fuðtÞ : 1 < t 6 0g is an F 0 -measurable, D-valued cess w and the fBm BHQ with finite second moment. In the past decades, the theory of impulsive integro-differential equations has became an active area of investigation due to their applications in the fields such as mechanics, electrical engineering, medicine biology, ecology and so on. One can see [7,17] and the references therein. Several authors have established the existence results of mild solutions for these equations (see [2,3,15,18] and the references therein). For the potential applications in telecommunications networks, finance markets, biology and other fields ([8,10,13]), stochastic differential equations driven by fractional Brownian motion (fBm) have attracted researchers’ great interest. Especially, Duncan et al. [9] proved the existence and uniqueness of a mild solution for a class of stochastic differential equations in a Hilbert space with a standard, cylindrical fBm with the Hurst parameter in the interval (1/2, 1). Moreover, Maslowski and Nualart [14] established the existence and uniqueness of a mild solution for nonlinear stochastic evolution equations in a Hilbert space driven by a cylindrical fBm under some regularity and boundedness conditions on the coefficients. very recently, Caraballo et al. [6] investigated the existence and uniqueness of mild solutions to stochastic delay evolution equations driven by a fBm with Hurst parameter H 2 ð1=2; 1Þ. An existence and uniqueness result of mild solutions for a class of neutral stochastic differential equation with finite delay, driven by a fBm in a Hilbert space has been recently established in Boufoussi and Hajji [5]. To the best of our knowledge, there is no work on the impulsive neutral stochastic functional integro-differential equations driven by a standard cylindrical Wiener process and an independent cylindrical fBm with Hurst parameter H 2 ð1=2; 1Þ in the Hilbert space. Motivated by the previously mentioned paper, in this work, we aim to study this interesting problem. We prove the existence and uniqueness of the mild solution for this kind of equations with the coefficients satisfying some non-Lipschitz conditions, which include the classical Lipschitz condition as special case. An example is provided to illustrate the theory. We would like to mention we have extended and generalized the results appeared in Anguraj and Vinodkumar [1], Boufoussi and Hajji [5], Caraballo et al. [6], Ren and Xia [19]. The paper is organized as follows. In Section 2, we introduce some preliminaries. Section 3 proves the existence and uniqueness of a mild solution for the system (1.1). An example is provided in the last section to illustrate the theory.

2. Preliminaries In this section, we provide some preliminaries needed to establish our main results. For details of this section, we refer the reader to [5,6,11] and the references therein. Throughout this paper ðX; k  k; h; iÞ and ðY; k  kY ; h; iY Þ are two real separable R Hilbert spaces. The notation L2 ðX; XÞ stands for the space of all X-valued random variables x such that Ekxk2 ¼ X kxk2 dP R 12 < 1. For x 2 L2 ðX; XÞ, let kxk2 ¼ X kxk2 dP . It is easy to check that L2 ðX; XÞ is a Hilbert space equipped with the norm k  k2 . Let LðY; XÞ denotes the space of all bounded linear operators form Y to X, we abbreviate this notation to LðYÞ whenever X ¼ Y and Q 2 LðYÞ represents a non-negative self-adjoint operator. Let Y 0 be an arbitrary separable Hilbert space and L02 ¼ L2 ðY 0 ; XÞ be a separable Hilbert space with respect to the Hilbert– 1

Schmidt norm k  kL0 . Let L0Q ðY; XÞ be the space of all w 2 LðY; XÞ such that wQ 2 is a Hilbert–Schmidt operator. The norm is 2  1 2  2 2 given by kwkL0 ¼ wQ  ¼ trðwQ w Þ. Then w is called a Q-Hilbert–Schmidt operator from Y to X. In the sequel, L02 ðX; XÞ Q

denotes the space of F 0 -measurable, X-valued and square integrable stochastic processes. 2.1. fractional Brownian motion and the phase space Now, we recall some basic knowledge on the fBm as well as the Wiener integral with respect to it. For more details, one can see Caraballo et al. [6] and Nualart [16]. Consider a time interval ½0; T with arbitrary fixed horizon T and let fbH ðtÞ; t 2 ½0; Tg be a one-dimensional fBm with Hurst parameter H 2 ð1=2; 1Þ. By definition, it means that bH is a continuous centered Gaussian process with the covariance function:

RH ðr; sÞ ¼

 1  2H s þ r 2H  js  rj2H : 2

Further, bH has the following Wiener integral representation;

bH ðtÞ ¼

Z

t

K H ðt; sÞ dbðsÞ;

0

where b ¼ fbðtÞ : t 2 ½0; Tg is a Wiener process and K H ðt; sÞ is the kernel given by 1

K H ðt; sÞ ¼ cH s2H here cH ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Hð2H1Þ Bð22H;H12Þ

Z

t

ðy  sÞH3=2 yH1=2 du for t > s;

s

with BðÞ represents the Beta function. We take K H ðt; sÞ ¼ 0 if t 6 s.

Y. Ren et al. / Applied Mathematics and Computation 247 (2014) 205–212

207

Let H be reproducing kernel Hilbert space of the fBm. Also, H is the closure of set of indicator functions fv½0;t ; t 2 ½0; Tg D E v½0;t ; v½0;s ¼ RH ðt; sÞ. The mapping v½0;t ! bH ðtÞ can be extended to an isometry between

with respect to the scalar product

H

H and first Wiener chaos. Denote by bH ðuÞ the image of u by the previous isometry. Let us define the operator K H from H to L2 ð½0; TÞ by

ðK H uÞðsÞ ¼

Z

T

uðuÞ

s

@K ðu; sÞ du: @u

RT Then K H is an isometry between H and L2 ð½0; TÞ (see [16]). Further, for any u 2 H, we have bH ðuÞ ¼ 0 K H ðuÞdbðtÞ. Let H fbn ðtÞgn2N be a sequence of two-sided one dimensional standard fBm mutually independent on ðX; F ; PÞ. Consider the following series 1 X bHn ðtÞen ;

t P 0;

n¼1

where fen gn2N is a complete orthonormal basis in Y, the series does not necessarily converge in the space Y. Therefore, we consider a Y-valued stochastic process BHQ ðtÞ given by the following series:

BHQ ðtÞ ¼

1 X 1 bHn ðtÞQ 2 en ;

t P 0:

n¼1

Moreover, if Q is a non-negative self-adjoint trace class operator, then this series converges in the space Y, that is, it holds that BHQ ðtÞ 2 L2 ðX; YÞ. Then, we say that the above BHQ ðtÞ is a Y-valued Q-cylindrical fBm with covariance operator Q. For example, if frn gn2N is a bounded sequence of non-negative real numbers such that Qen ¼ rn en , assuming that Q is a nuclear operator in Y, then the stochastic process

BHQ ðtÞ ¼

1 1 X X pffiffiffiffiffiffi H 1 bHn ðtÞQ 2 en ¼ rn bn ðtÞen ; n¼1

t P 0;

n¼1

is well-defined as a Y-valued Q-cylindrical fBm. P1 1  2 Definition 2.1 [6]. Let u : ½0; T ! L0Q ðY; XÞ such that n¼1 kK H ðuQ en ÞkL2 ð½0;T;HÞ < 1. Then, its stochastic integral with H respect to the fBm BQ ðtÞ is defined, for t P 0, as follows:

Z 0

t

uðsÞ dBHQ ðsÞ :¼

1 Z X

t

0

n¼1

1

uðsÞQ 2 en dbHn ¼

1 Z X n¼1

t

0

1

ðK H ðuQ 2 en ÞÞðsÞ dbðsÞ:

Lemma 2.2. [6] For any u : ½0; T ! L0Q ðY; XÞ satisfies that we have

Z  E 

b

a

2 

uðsÞ dBHQ ðsÞ 6 cHð2H  1Þða  bÞ2H1

where c ¼ cðHÞ. In addition, if

Z  E 

a

b

P1

b

1

uQ 2 en ÞkL1=H ð½0;T;XÞ < 1 holds, and for any a; b 2 ½0; T with a > b,

n¼1 k

1

kuQ 2 en k2 ds;

a

n¼1

1

uðtÞQ 2 en k is uniformly convergent for t 2 ½0; T, then

n¼1 k

2 

1 Z X

P1

uðsÞ dBHQ ðsÞ 6 cHð2H  1Þða  bÞ2H1

Z a

b

kuk2L0 ds: Q

b ¼ Dðð1; 0; XÞ denotes the family of all right piecewise continuous functions with left-hand limit u from In this paper, D B b is assumed to be equipped with the norm jjujj ¼ sup ð1; 0 to X. The phase space D 1
jjujjBT ¼



sup Ejjujj2t

1=2 :

06t6T

2.2. partial integro-differential equations in Banach space In this subsection, we recall some knowledge on partial integro-differential equations and the related resolvent operators. Let X and Y be two Banach spaces such that jjyjj :¼ jjAyjj þ jjyjj; y 2 Y. A and BðtÞ are closed linear operators on X. Let Cð½0; þ1Þ; YÞ; BðY; XÞ stand for the space of all continuous functions from ½0; þ1Þ into Y, the set of all bounded linear operators from Y into X, respectively. In what follows, we suppose the following assumptions.

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Y. Ren et al. / Applied Mathematics and Computation 247 (2014) 205–212

(H1) A is the infinitesimal generator of a strongly continuous semigroup ðSðtÞÞtP0 on X. (H2) For all t P 0; BðtÞ is a closed linear operator from DðAÞ to X, and BðtÞ 2 BðY; XÞ. For any y 2 Y, the map t ! BðtÞy is bounded, differentiable and the derivative t ! B0 ðtÞy is bounded and uniformly continuous on Rþ . By Grimmer [11], under the assumptions (H1) and (H2), the following Cauchy problem

(

v 0 ðtÞ ¼ Av ðtÞ þ v ð0Þ ¼ v 0 2 X

Rt 0

Bðt  sÞv ðsÞds;

t P 0;

ð2:1Þ

has an associated resolvent operator of bounded linear operator valued function RðtÞ 2 LðXÞ for t P 0. Definition 2.3 [12]. A resolvent operator associated with (2.1) is a bounded linear operator valued function RðtÞ 2 LðXÞ for t P 0, satisfying the following properties: (i) Rð0Þ ¼ I and jRðtÞj 6 Nebt for some constants N and b. (ii) For each x 2 X; RðtÞx is strongly continuous for t P 0. (iii) RðtÞ 2 LðYÞ for t P 0. For x 2 Y; RðÞx 2 C1 ð½0; þ1Þ; XÞ \ Cð½0; þ1Þ; YÞ and

R0 ðtÞx ¼ ARðtÞx þ

Z

t

Bðt  sÞRðsÞxds ¼ RðtÞAx þ

0

Z

t

Bðt  sÞBðsÞxds;

t P 0:

0

By Grimmer [11], we can establish the existence and uniqueness of the mild solution to the following integro-differential equation

(

R

v 0 ðtÞ ¼ Av ðtÞ þ 0t Bðt  sÞv ðsÞds þ uðtÞ; v ð0Þ ¼ v 0 2 X; R v ðtÞ ¼ RðtÞv 0 þ 0t Bðt  sÞuðsÞds; t P 0:

t P 0;

where u : ½0; þ1Þ ! X is a continuous function. 2.3. definition and assumptions Now, we present the definition of the mild solution for the system (1.1). Definition 2.4. A càdlàg stochastic process x : ð1; T ! X is said to be a mild solution of (1.1) if b (i) xðtÞ is measurable, F t -adapted and xt : t 2 ½0; T is D-valued; (ii) for each t 2 J; xðtÞ satisfies the following integral equation:

xðtÞ ¼ RðtÞ½uð0Þ  gð0; uÞ  gðt; xt Þ þ X Rðt  tk ÞIk ðxðtk ÞÞ: þ

Z

t

Rðt  sÞf ðs; xs Þds þ

0

Z

t

Rðt  sÞhðs; xs ÞdwðsÞ þ

0

Z 0

t

Rðt  sÞrðsÞdBHQ ðsÞ

0
In the sequel, we will work under the following assumptions. (H3) There is a positive constant M such that sup jjRðt  sÞjj 6 M. 06s;t6T

b ! X satisfy the following conditions: (H4) The functions f ; h : J  D

jjf ðt; xt Þ  f ðt; yt Þjj2 _ jjhðt; xt Þ  hðt; yt Þjj2 6 Hðjjx  yjj2t Þ;

b x; y 2 D;

t 2 J;

R ds where HðÞ is a concave non-decreasing function from Rþ to Rþ , such that Hð0Þ ¼ 0; HðsÞ > 0 for s > 0 and 0þ HðsÞ ¼ 1. b ! X and there exist a positive number K g such that K g < 1=12, for t 2 ½0; T, we have (H5) The functions g : J  D

jjgðt; xt Þ  gðs; yt Þjj2 6 K g jjx  yjj2t ;

b x; y 2 D;

t 2 J:

(H6) The function r : J ! L0Q ðY; XÞ satisfies that exists a positive constant L such that jjrðsÞjj2L0 6 L uniformly in J. 2 (H7) The function Ik 2 CðX; XÞ and there exist positive constants dk such that

jjIk ðxðt k ÞÞ  Ik ðyðt k ÞÞjj2 6 dk jjx  yjj2t ;

k ¼ 1; 2; . . . ; m:

(H8) For all t 2 ½0; T, it follows that f ðt; 0Þ; gðt; 0Þ; hðt; 0Þ; Ik ð0Þ 2 L2 , such that

jjf ðt; 0Þjj2 _ jjgðt; 0Þjj2 _ jjhðt; 0Þjj2 _ jjIk ð0Þjj2 6 c0 ; where c0 is a constant.

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Y. Ren et al. / Applied Mathematics and Computation 247 (2014) 205–212

In order to obtain the uniqueness of solution, we give Bihari inequality, which appeared in [4]. Lemma 2.5 (Bihari inequality). Let T > 0 and u0 P 0; uðtÞ; v ðtÞ be continuous functions on ½0; T. Let k : Rþ ! Rþ be a concave continuous and nondecreasing function such that kðrÞ P 0 for r > 0. If

uðtÞ 6 u0 þ

Z

t

v ðsÞkðuðsÞÞds

for all 0 6 t 6 T;

0

then

 Z uðtÞ 6 G1 Gðu0 Þ þ

t

v ðsÞds



0

for all such t 2 ½0; T that

Gðu0 Þ þ

Z

t

v ðsÞds 2 DomðG1 Þ;

0

where GðrÞ ¼ all 0 6 t 6 T.

Rr

ds ;r 1 kðsÞ

P 0 and G1 is the inverse function of G. In particular, if, moreover, u0 ¼ 0 and

R

ds 0þ kðsÞ

¼ 1, then uðtÞ ¼ 0 for

3. Existence and uniqueness of the mild solution In this section, we aim to establish the existence and uniqueness theorem of the mild solution for the system (1.1). We construct the sequence of successive approximations defined as follows:

8 0 x ðtÞ ¼ RðtÞuð0Þ; t 2 J; > > > > Rt Rt Rt > H n n n1 n1 > > < x ðtÞ ¼ RðtÞ½uð0Þ  gð0; uÞ  gðt; xt Þ þ 0 Rðt  sÞf ðs; xs Þds þ 0 Rðt  sÞhðs; xs ÞdwðsÞ þ 0 Rðt  sÞrðsÞdBQ ðsÞ X Rðt  t k ÞIk ðxðt k ÞÞ; t 2 J; n P 1; þ > > > > 0 > > : n x ðtÞ ¼ uðtÞ; 1 < t 6 0; n P 1: The main results of this paper is the following theorem. P Theorem 3.1. Let (H1)–(H8) hold, then there exists a unique mild solution of (1.1) in the space BT if 4ðK g þ M 2 m m k¼1 dk Þ < 1. Proof. The proof will be split into the following steps.

Step 1. For all t 2 ð1; T, the sequence xn ðtÞ; n P 1 2 BT is bounded. It is obvious that x0 ðtÞ 2 BT . Let x0 be a fixed initial approximation to (3.1). To begin with under assumption (H1)–(H8) and the Hölder inequality, we have

h i Ekxn ðtÞk2 6 6M2 Ekuð0Þ þ gð0; uÞk2 þ 12E kgðt; xnt Þ  gðt; 0Þk2 þ kgðt; 0Þk2 þ 12TM2 E Z th Z th i i 2 2 2 kf ðs; xn1 Þ  f ðs; 0Þk þ kf ðs; 0Þk E khðs; xn1 Þ  hðs; 0Þk2 þ khðs; 0Þk2 ds ds þ 12M  s s 0

2

þ 6M cHð2H  1ÞT

2H1

E

0

Z

t

kr

0

ðsÞk2L2 ds Q

2

þ 12M mE

Xm h k¼1

i kIk ðxn1 ðt k ÞÞ  Ik ð0Þk2 þ kIk ð0Þk2 :

Thus,

Ekxn k2t 6

B1 12M 2 ðT þ 1Þ þ E 1  12K g 1  12K g

Z 0

t

Hðkxn1 k2s Þds þ

P 12M 2 m m k¼1 dk Ekxn1 k2t ; 1  12K g

where

!  m X 1 2 2 2H 2 2 B1 ¼ 12M Ekuð0Þk þ K g Ekuk0 þ cHð2H  1ÞT L þ 12 1 þ M TðT þ 1Þ þ M m dk c0 : 2 k¼1 2

Given that HðÞ is concave and Hð0Þ ¼ 0, we can find a pair of positive constants a and b such that

HðtÞ 6 a þ bt;

for t P 0:

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Y. Ren et al. / Applied Mathematics and Computation 247 (2014) 205–212

Then we have,

Ekxn k2t 6 B2 þ

12M2 ðT þ 1Þb 1  12K g

Z

t 0

Ekxn1 k2s ds þ

P 12M 2 m m k¼1 dk Ekxn1 k2t ; 1  12K g

n P 1;

2

ðTþ1ÞTa B1 where B2 ¼ 112K þ 12M112K . Since Ekx0 ðtÞk 6 M 2 Ekuð0Þk2 :¼ B3 < 1. Thus Ekxn k2t < 1 for n P 1 and t 2 ½0; T, which shows g g

that the sequence xn ðtÞ; n P 1 is a bounded one in the space BT .

Step 2. The sequence xn ðtÞ; n P 1 is a Cauchy sequence. From (3.1), for all n P 1 and 0 6 t 6 T, we have

Ekxnþ1 ðtÞ  xn ðtÞk2 6 4K g Ekxnþ1  xn k2t þ 4M2 ðT þ 1Þ

Z

t 0

Xm HðEkxn  xn1 k2s Þds þ 4M 2 m k¼1 dk Ekxn  xn1 k2t :

Thus,

Ekxnþ1  xn k2t 6

4M 2 ðT þ 1Þ 1  4K g

Let Un ðtÞ ¼ supt2½0;T Ekx

nþ1

2

Un ðtÞ 6

4M ðT þ 1Þ 1  4K g



xn k2t .

Z

t

Z

t

0

HðEkxn  xn1 k2s Þds þ

P 4M 2 m m k¼1 dk Ekxn  xn1 k2t : 1  4K g

Then, we have in the above inequality,

HðUn1 ðsÞÞds þ

0

P 4M 2 m m k¼1 dk Un1 ðtÞ; 1  4K g

0 6 t 6 T:

ð3:2Þ

Choose T 1 2 ½0; T such that

C1

Z

t

HðUn1 ðsÞÞds 6 C 1

Z

0

t

Un1 ðsÞds;

n P 1;

0 6 t 6 T 1:

0

Moveover,

 Z t Z t  1 0 0 0 kx1 ðtÞ  x0 ðtÞk2 ¼  RðtÞgð0; u Þ  ½gðt; x Þ  gðt; x Þ  gðt; x Þ þ Rðt  sÞf ðs; x Þds þ Rðt  sÞhðs; x0s ÞdwðsÞ t t t s  0 0 2 Z t  H Rðt  sÞrðsÞdQ ðsÞ þ um0
Then, we get

Ekx1  x0 k2t 6

B4 14K g þ 14M 2 m þ 1  7K g 1  7K g

Pm

k¼1 dk

Ekx0 k2t þ

14M 2 ðT þ 1Þ 1  7K g

Z 0

t

HðEkx0 k2s Þds;

where

    Xm B4 ¼ 7M 2 K g Ekuk20 þ cHð2H  1ÞT 2H L þ 14 1 þ M2 TðT þ 1Þ þ M2 m k¼1 dk c0 ; We take the supermum over t, and use the Un , we have

U0 ðtÞ ¼ sup Ekx1  x0 k0t 6 B5 þ t2½0;T

14M2 ðT þ 1Þ 1  7K g

Z

t

HðB3 Þds :¼ B6 ;

0

where

B5 ¼

P B4 14K g þ 14M2 m m k¼1 dk þ B3 : 1  7K g 1  7K g

Now, for n ¼ 1 in (3.2), we have

U1 ðtÞ 6

4M 2 ðT þ 1Þ 1  4K g

Z

t

HðU0 ðsÞÞds þ

0

P Z t Z t 4M 2 m m k¼1 dk U0 ðtÞ ¼ C 1 HðU0 ðsÞÞds þ C 2 U0 ðtÞ 6 C 1 B6 ds þ C 2 B6 1  4K g 0 0

6 ðC 1 þ C 2 ÞT 1 B6 : And for n ¼ 2 in (3.2), we have

U2 ðtÞ 6 C 1

Z

t

HðU1 ðsÞÞds þ C 2 U1 ðtÞ 6 C 1 0

Z 0

t

ðC 1 þ C 2 ÞB6 ds þ C 2 ðC 1 þ C 2 ÞT 1 B6 6 ðC 1 þ C 2 Þ2

T 21 B6 : 2!

Y. Ren et al. / Applied Mathematics and Computation 247 (2014) 205–212

211

Thus, by applying mathematical induction in (3.2) and using the above work, we have

Un ðtÞ 6

ðC 1 þ C 2 Þn T n1 B6 ; n!

n P 1;

t 2 ½0; T 1 :

So, for any m P n P 0, we have

sup Ekxm ðtÞ  xn ðtÞk2 6 t2½0;T 1 

þ1 X

sup Ekxrþ1  xr k2t 6

r¼n t2½0;T 1 

þ1 X ðC 1 þ C 2 Þr T r

1

r!

r¼n

B6 ! 0; ðn ! 1Þ:

This is shows the assertion. Step3 . The existence and uniqueness of the solution for (1.1). The Borel–Cantelli Lemma shows that as n ! 1; xn ðtÞ ! xðtÞ uniformly for 0 6 t 6 T. So, taking limits on both side of (3.1), for all 1 < t 6 T, we obtain that xðtÞ is a solution to (1.1). This shows the existence. Now we prove the uniqueness. Let x1 ; x2 2 BT be two solutions on ð1; T. The uniqueness is obvious for t 2 ð1; 0. For 0 6 t 6 T, we have

! Z t m X Ekx1 ðtÞ  x2 ðtÞk 6 4 K g þ M m dk Ekx1  x2 k2t þ 4M2 ðT þ 1Þ HðEkx1  x2 k2s Þds: 2

2

0

k¼1

So,

Ekx1  x2 k2t 6

4M2 ðT þ 1Þ 1  B7

Z 0

t

HðEkx1  x2 k2s Þds;

P where B7 ¼ 4ðK g þ M 2 m m k¼1 dk Þ. Thus, the Bihari inequality yields that

sup Ekx1  x2 k2t ¼ 0:

t2½0;T

So, x1 ¼ x2 for all 1 < t 6 T. This complete the proof.

h

4. An example In this section, an example is provided to illustrate the theory obtained. Example 4.1. We consider the following stochastic partial integro-differential equations with infinite delay:

8@ Rt @2 @2 ½uðt; nÞ þ Gðt; uðt  h; nÞÞ ¼ @n ½uðs; nÞ þ Gðs; uðs  h; nÞÞds > 2 ½uðt; nÞ þ Gðt; uðt  h; nÞÞ þ 0 Bðt  sÞ > @n2 > @t > > > dBH dWðtÞþ > Q ðtÞ < þFðt; uðt  h; nÞÞ þ Hðt; uðt  h; nÞÞ dt H dt ; 0 6 n 6 p; h > 0; t 2 J :¼ ½0; T; uðt; 0Þ ¼ uð0; pÞ ¼ 0; t 2 J; t – tk ; k ¼ 1; 2; . . . ; m; > > > > > > Duðt k Þ ¼ ð1 þ bk Þuðnðtk ÞÞ; t ¼ tk ; k ¼ 1; 2; . . . ; m; > : uðt; nÞ ¼ Uðt; nÞ; 0 6 n 6 p; 1 < t 6 0: Let X ¼ L2 ð½0; pÞ; Y ¼ R; G; F; H; H be appropriate functions and U 2 M2 ðð1; 0Þ; XÞ; bk P 0 for k ¼ 1; 2; . . . ; m and Pm H k¼1 bk < 1; WðtÞ be a standard cylindrical wiener process, BQ ðtÞ be a fractional Brownian motion with Hurst parameter H 2 ð1=2; 1Þ which is independent of WðtÞ, defined on a complete probability space ðX; F ; PÞ. 2 Consider the operator Au ¼ @@nu2 with the domain

(

DðAÞ ¼

@u u 2 Xju and are absolutely continuous; @n

@2u @n2

)

2 X; uð0Þ ¼ uðnÞ ¼ 0 :

It is known that A is the infinitesimal of a strongly continuous semigroup SðtÞtP0 on X, which is given by qffiffiffi P1 n2 t hu; en ien ; u 2 DðAÞ, where en ðxÞ ¼ p2 sinðnxÞ; n ¼ 1; 2; . . ., is the orthonormal set of eigenvectors of A, and n¼1 e P n2 t hu; en ien ; u 2 X. Thus, assuming that F; G : J  B ! X; H : J  B ! L02 ðY; XÞ by f ðt; zÞðÞ ¼ Fðt; zðÞÞ; SðtÞu ¼ 1 n¼1 e gðt; zÞðÞ ¼ Gðt; zðÞÞ; hðt; zÞðÞ ¼ Hðt; zðÞÞ; rðtÞ ¼ HðtÞ and Ik ðzðtk ÞÞ ¼ ð1 þ bk Þuðzðt k ÞÞ; k ¼ 1; 2; . . . ; m, then, the system (4.1) can be rewriter as the abstract form as the system (1.1). Further, all the conditions of Theorem 3.1 have been fulfilled. So, we can conclude that the system (4.1) has a unique mild solution. SðtÞu ¼

212

Y. Ren et al. / Applied Mathematics and Computation 247 (2014) 205–212

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