Chinese Journal of Mathematics

Volume 2017, Article ID 8219175, 9 pages

https://doi.org/10.1155/2017/8219175

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

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

Correspondence should be addressed to Hassane Bouzahir; rf.oohay@rihazuobh

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.