Chinese Journal of Mathematics
Volume 2017 (2017), Article ID 8219175, 9 pages
https://doi.org/10.1155/2017/8219175

Research Article

Some Stochastic Functional Differential Equations with Infinite Delay: A Result on Existence and Uniqueness of Solutions in a Concrete Fading Memory Space

Hassane Bouzahir, Brahim Benaid, and Chafai Imzegouan

LISTI, ENSA, Ibn Zohr University, P.O. Box 1136, Agadir, Morocco

Correspondence should be addressed to Hassane Bouzahir; hbouzahir@yahoo.fr

Received 4 February 2017; Accepted 2 April 2017; Published 16 April 2017

Academic Editor: Chuanzhi Bai

Copyright © 2017 Hassane Bouzahir et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper is devoted to existence and uniqueness of solutions for some stochastic functional differential equations with infinite delay in a fading memory phase space.

1. Introduction

Let denote the Euclidian norm in . If is a vector or a matrix, its transpose is denoted by and its trace norm is represented by . Let be the minimum (maximum) for .

Let be a complete probability space with a filtration satisfying the usual conditions; that is, it is right continuous and contains all -null sets.

denotes the family of all -measurable valued processes , such that

Assume that is an -dimensional Brownian motion which is defined on ; that is, .

Let denote the family of continuous functions defined on with norm .

Consider the -dimensional stochastic functional differential equationwhere can be regarded as a -value stochastic process, and and are Borel measurable.

The initial data of the stochastic process is defined on . That is, the initial value is a -measurable and -value random variable such that

Our aim, in this paper, is to study existence and uniqueness of solutions to stochastic functional differential equations with infinite delay of type (1) in a fading memory phase space.

2. Preliminary

The theory of partial functional differential equations with delay has attracted widespread attention. However, when the delay is infinite, one of the fundamental tasks is the choice of a suitable phase space . A large variety of phase spaces could be utilized to build an appropriate theory for any class of functional differential equations with infinite delay. One of the reasons for a best choice is to guarantee that the history function is continuous if is continuous (where ). In general, the selection of the phase space plays an important role in the study of both qualitative and quantitative analysis of solutions. Sometimes, it becomes desirable to approach the problem purely axiomatically. The first axiomatic approach was introduced by Coleman and Mizel in [1]. After this paper, many contributions have been published by various authors until 1978 when Hale and Kato organized the study of functional differential equations with infinite delay in [2]. They assumed that is a normed linear space of functions mapping into a Banach space , endowed with a norm and satisfying the following axioms.

There exist a positive constant and functions , with being continuous and being locally bounded, such that for any and , if , , and is continuous on , then for all in , the following conditions hold:(i),(ii),(iii).

For the function in , is a -valued continuous function for in .

The space is complete.

Later on, the concept of fading and uniform fading memory spaces has been adopted as the best choice.

For and , we define the linear operator by is exactly the solution semigroup associated with the following trivial equation: We define Let be the set of continuous functions with compact support. We recall the following axiom.

If a uniformly bounded sequence in converges to a function compactly on , then and

Definition 1. (1) is called a fading memory space if it satisfies the axioms as , for all (2) is called a uniform fading memory space if it satisfies the axioms as .

Examples. We recall the definitions of some standard examples of phase spaces .

We start first with the phase space of -valued bounded continuous functions defined on , that is, with norm .

(1) Let where is the space of all bounded continuous functions mapping into provided with the uniform norm topology.

(2) Let and provided with the norm

(3) For any continuous function , we define endowed with the norm

Consider the following conditions on : () is locally bounded for , () , ()

Properties of each phase space are summarized in Table 1.

Table 1

()
()
()
()
Uniform fading memory space

Yes
No
Yes
No
No

Yes
Yes
Yes
No
No

Under ()
Under ()
Yes
Under ()
Under ()

Under ()
Under ()
Yes
Under ()
Under ()

Yes
Yes
Yes
Under
Under

For other examples, properties, and details about phase spaces, we refer to the book by Hino et al. [3].

Fengying and Ke [4] discussed existence and uniqueness of solutions to stochastic functional differential equation with infinite delay in the phase space of bounded continuous functions defined on with values in , that is, with norm .

Lemma 2 (page  22 in [3]). If the phase space satisfies axiom , then is included in .

3. Existence and Uniqueness

Lemma 3 (see [4]). If such that , then

Lemma 4 (Borel-Cantelli, page  487 in [5]). If is a sequence of events and then where is an abbreviation for “infinitively often.”

Definitions 1. -value stochastic process defined on is called the solution of (1) with initial data , if has the following properties:(i) is continuous and is -adapted,(ii) and ,(iii), for each , is called unique solution, if any other solution is distinguishable with ; that is, Now, we establish existence and uniqueness of solutions for (1) with initial data . We suppose a uniform Lipschitz condition and a weak linear growth condition.

Theorem 5. Assume that there exist two positive number and such that,
(i) for any and , it follows that(ii) for any , it follows that such thatThen, problem (1), with initial data , has a unique solution . Moreover, .

Lemma 6. Let (15) and (16) hold. If is the solution of (1) with initial data , then where
Moreover, if , then .

Proof. For each number , define the stopping time Obviously, as a.s. Let , and then satisfy the following equation: Using the elementary inequality , we get Taking the expectation on both sides and using the Hölder inequality, Lemma 3, and (15) and (16), we get for all in We have also for each in Letting , we get where
By the Gronwall inequality, we inferThat is, Consequently Letting , that implies the following inequalityNow, to prove the second part or the lemma, suppose that . ThenThe demonstration of the lemma is complete.

Proof of Theorem 5. We begin by checking uniqueness of solution. Let and be two solutions of (1), by Lemma 6   and . Note that By the elementary inequality, , one then gets By Hölder inequality, Lemma 3, and (15) and (16), we have From the fact , and We have Applying the Gronwall inequality yields The above expression means that a.s. for . Therefore, for all a.s., the proof of uniqueness is complete.

Next, to check the existence, define and for . Let , and define Picard sequenceObviously . By induction, we can see that .

In fact, by elementary inequality From the Hölder inequality and Lemma 3, we have Again the elementary inequality , (22), (15), and (16) imply that where and .

Hence, for any , one can derive that Note that and then where .

From the Gronwall inequality, we have Since is arbitraryFrom the Hölder inequality, Lemma 3, and (15) and (16), as in a similar earlier inequality, one then has That is,By similar arguments as above, we also have Then where . Similarly Continue this process to find that Now we claim that for any So, for , inequality (50) holds. We suppose that (50) holds for some , and check (50) for . In fact From (50) which means that (50) holds for . Therefore, by induction (50) holds for any .

Next to verify converge to in with in and is the solution of (1) with initial data . For (50), taking , then By the Chebyshev inequality By using Alembert’s rule, we show that .

That is, , and by Borel-Cantelli’s lemma, for almost all , there exists a positive integer such that and then, is also a Cauchy sequence in . Hence, converges uniformly and let be its limit for any ; since is continuous on and adapted, is also continuous and adapted.

So, as , in . That is, as .

Then from (43) and therefore That is, .

Now, to show that satisfies (1) Noting that the sequence means that for any given there exists such that , for any , one then deduces that which means that, for any , one has For , taking limits on both sides of (35), we deduce that and consequently Finally, is the solution of (1), and the demonstration of existence is complete.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

References

B. D. Coleman and V. J. Mizel, “On the general theory of fading memory,” Archive for Rational Mechanics and Analysis, vol. 29, pp. 18–31, 1968.

J. K. Hale and J. Kato, “Phase space for retarded equations with infinite delay,” Funkcialaj Ekvacioj. Serio Internacia, vol. 21, no. 1, pp. 11–41, 1978.

Y. Hino, T. Naito, N. Van, and J. S. Shin, Almost Periodic Solutions of Differential Equations in Banach Spaces, vol. 15 of Stability and Control: Theory, Methods and Applications, Taylor & Francis, London, UK, 2000.

W. Fengying and W. Ke, “The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 516–531, 2007.

S. Resnick, Adventures in Stochastic Processes, Springer Science+Business Media, New York, NY, USA, 3rd edition, 2002.