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

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.


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 filtration {F } ≥0 satisfying the usual conditions; that is, it is right continuous and F 0 contains all -null sets. The initial data of the stochastic process is defined on (−∞, 0 ]. That 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 uniqueness of solutions to stochastic functional differential equations with infinite delay of type (1) in a fading memory phase space.

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 B. 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 : (−∞, ] → R is continuous (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. 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 B is a normed linear space of functions mapping (−∞, 0] into a Banach space ( , |⋅|), endowed with a norm |⋅| B and satisfying the following axioms.
For ∈ B, ≥ 0 and ≤ 0, we define the linear operator O( ) by (O( )) ≥0 is exactly the solution semigroup associated with the following trivial equation: We define Let C 00 be the set of continuous functions : (−∞, 0] → with compact support. We recall the following axiom.
where BC is the space of all bounded continuous functions mapping (−∞, 0] into provided with the uniform norm topology. (2) Let ∈ R and provided with the norm (3) For any continuous function : Consider the following conditions on : Properties of each phase space are summarized in Table 1.
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 (−∞, 0] with values in R , that is, Lemma 2 (page 22 in [3]). If the phase space B satisfies axiom
Lemma 4 (Borel-Cantelli, page 487 in [5]). If { } is a sequence of events and where . . is an abbreviation for "infinitively often." Definitions 1. R -value stochastic process ( ) defined on −∞ < ≤ is called the solution of (1) with initial data 0 , if ( ) has the following properties: Now, we establish existence and uniqueness of solutions for (1) with initial data 0 . We suppose a uniform Lipschitz condition and a weak linear growth condition.
By the Gronwall inequality, we infer That is, Consequently Letting → ∞, that implies the following inequality Now, to prove the second part or the lemma, suppose that ∈ M 2 ((−∞, 0]; R ). Then The demonstration of the lemma is complete.
Then from (43) and therefore Noting that the sequence { } → ( ) means that for any given > 0 there exists 0 such that ≥ 0 , for any ∈ (−∞, ], one then deduces that Finally, ( ) is the solution of (1), and the demonstration of existence is complete.