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. Tis 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.
Tis paper is devoted to existence and uniqueness of solutions for some stochastic functional diferential equations with infnite
delay in a fading memory phase space.
1. Introduction
Let |⋅| denote the Euclidian norm in R
. If is a vector or
a matrix, its transpose is denoted by
and its trace norm is
represented by || = (Trace(
))
1/2
. Let ∧(∨) be the
minimum (maximum) for ,∈ R.
Let (Ω, F,) be a complete probability space with a
fltration {F
}
≥0
satisfying the usual conditions; that is, it is
right continuous and F
0
contains all -null sets.
M
2
((−∞, ]; R
) denotes the family of all F
-meas-
urable R
valued processes (), ∈ (−∞,] such that
(∫
−∞
|()|
2
) < ∞.
Assume that () is an -dimensional Brownian motion
which is defned on (Ω, F,); that is, () = (
1
(),
2
(),
...,
())
.
Let
= { ∈ (−∞; 0]; R
): lim
→−∞
() exists in
R
} denote the family of continuous functions defned on
(−∞,0] with norm ||
= sup
≤0
|()|.
Consider the -dimensional stochastic functional difer-
ential equation
() = (
,)+(
,)(),
0
≤ ≤ , (1)
where
: (−∞,0] → R
; →
() = ( + );−∞ <
≤0 can be regarded as a
-value stochastic process, and
:
× [
0
,]→ R
and :
× [
0
,]→ R
×
are Borel
measurable.
Te initial data of the stochastic process is defned on
(−∞,
0
]. Tat is, the initial value
0
= = {() : −∞ <
≤ 0} is a F
0
-measurable and
-value random variable
such that ∈ M
2
(
).
Our aim, in this paper, is to study existence and unique-
ness of solutions to stochastic functional diferential equa-
tions with infnite delay of type (1) in a fading memory phase
space.
2. Preliminary
Te theory of partial functional diferential equations with
delay has attracted widespread attention. However, when the
delay is infnite, one of the fundamental tasks is the choice
of a suitable phase space B. A large variety of phase spaces
could be utilized to build an appropriate theory for any class
of functional diferential equations with infnite delay. One of
the reasons for a best choice is to guarantee that the history
function →
is continuous if : (−∞, ] → R
is con-
tinuous (where >0). 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. Te
frst axiomatic approach was introduced by Coleman and
Mizel in [1]. Afer this paper, many contributions have been
published by various authors until 1978 when Hale and Kato
Hindawi
Chinese Journal of Mathematics
Volume 2017, Article ID 8219175, 9 pages
https://doi.org/10.1155/2017/8219175