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

1. Introduction

Let [absolute value of (x)] denote the Euclidian norm in [R.sup.n]. If A is a vector or a matrix, its transpose is denoted by A' and its trace norm is represented by [absolute value of (A)] = [(Trace(A'A)).sup.1/2]. Let a [conjunction] b (a [disjunction] b) be the minimum (maximum) for a, b [member of] R.

Let ([OMEGA], F, P) be a complete probability space with a filtration [{[F.sub.t]}.sub.t[greater than or equal to]0] satisfying the usual conditions; that is, it is right continuous and [mathematical expression not reproducible] contains all P-null sets.

[M.sup.2]((-[infinity],T]; [R.sup.n]) denotes the family of all [F.sub.t]-measurable [R.sup.n] valued processes x(t), t [member of] (-[infinity], T] such that E([[integral].sup.T.sub.-[infinity]] [absolute value of (x(t).sup.2])] dt) < [infinity].

Assume that W(t) is an m-dimensional Brownian motion which is defined on ([OMEGA], F, P); that is, W(t) = ([W.sub.1](t), [W.sub.2](t), ..., [W.sub.m](t))'.

Let [C.sup.[mu]] = {[phi] [member of] C(-[infinity]; 0]; [R.sup.n]) : [lim.sub.[theta][right arrow]- [infinity]][e.sup.[mu][theta]][phi]([theta]) exists in [R.sup.n]} denote the family of continuous functions [phi] defined on (-[infinity], 0] with norm [[absolute value of ([[phi].sub.u])] = [sup.sub.[theta][less than or equal to]0][e.sup.[mu][theta]] [absolute value of ([phi]([theta]))].

Consider the n-dimensional stochastic functional differential equation

dx (t) = f ([x.sub.t], t)dt + g ([x.sub.t], t) dW (t), [t.sub.0] [less than or equal to] t [less than or equal to] T, (1)

where [x.sub.t] : (-[infinity], 0] [right arrow] [R.sup.n]; [theta] [??] [x.sub.t]([theta]) = x(t + [theta]);- [infinity] < [theta] [less than or equal to] 0 can be regarded as a [C.sup.[mu]]-value stochastic process, and f : [C.sup.[mu]] [[t.sub.0], T] [right arrow] [R.sup.n] and g : [C.sup.[mu]] x [[t.sub.0], T] [right arrow] [R.sup.nxm] are Borel measurable.

The initial data of the stochastic process is defined on (-[infinity],[t.sub.0]]. That is, the initial value [mathematical expression not reproducible]-measurable and [C.sup.[mu]]-value random variable such that [xi] [member of] [M.sup.2]([C.sup.[mu]]).

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 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 t [right arrow] [x.sub.t] is continuous if x : (-[infinity], a] [right arrow] [R.sup.n] is continuous (where a > 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 (-[infinity], 0] into a Banach space (X, [absolute value of (x)]), endowed with a norm [[absolute value of (x)].sub.B] and satisfying the following axioms.

([A.sub.1]) There exist a positive constant H and functions K(x), M(x) : [0, + [infinity]) [right arrow] [0, + [infinity]), with K being continuous and M being locally bounded, such that for any [sigma] [member of] R and a > 0, if x : (-[infinity], [sigma] + a] [right arrow] X, [x.sub.[sigma]] [member of] B, and x(*) is continuous on [[sigma], [sigma] + a], then for all t in [[sigma], [sigma] + a], the following conditions hold:

(i) [x.sub.t] [member of] B,

(ii) [absolute value of (x(t))] [less than or equal to] H [[absolute value of ([x.sub.t])].sub.B],

(iii) [[absolute value of ([x.sub.t])].sub.B] [less than or equal to] K(t - [sigma])[sup.sub.[sigma][less than or equal to]s[less than or equal to]t] [absolute value of (x(s))] + M(t - [sigma]) [[absolute value of ([x.sub.[sigma])].sub.B].

([A.sub.2]) For the function x(*) in ([A.sub.1]), t [right arrow] [x.sub.t] is a B-valued continuous function for t in [[sigma], [sigma] + a].

([A.sub.3]) The space B is complete.

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

For [phi] [member of] B, t [greater than or equal to] 0 and [theta] [less than or equal to] 0, we define the linear operator O(t) by

[mathematical expression not reproducible]. (2)

[(O(t)).sub.t[greater than or equal to]0] is exactly the solution semigroup associated with the following trivial equation:

d/dt u(t) = 0

[u.sub.0] = [phi]. (3)

We define

[O.sub.0](t) = O(t)/[B.sub.0], where [B.sub.0] [??] {[phi] [member of] B: [phi](0) = 0}. (4)

Let [C.sub.00] be the set of continuous functions [phi] : (-[infinity], 0] [right arrow] X with compact support. We recall the following axiom.

([A.sub.4]) If a uniformly bounded sequence [([[phi].sub.n]).sub.n[greater than or equal to]0] in [C.sub.00] converges to a function [phi] compactly on (-[infinity], 0], then [phi] [member of] B and [[absolute value of ([[phi].sub.n] - [phi])].sub.B] [right arrow] 0.

Definition 1.

(1) B is called a fading memory space if it satisfies the axioms ([A.sub.1]),([A.sub.2]),([A.sub.3]),([A.sub.4]) and [absolute value of ([O.sub.0](t)[phi])] [right arrow] 0 as

(2) B is called a uniform fading memory space if it satisfies the axioms ([A.sub.1]),([A.sub.2]),([A.sub.3]),([A.sub.4]) and [absolute value of ([O.sub.0](t))] [right arrow] 0 as t [right arrow] +[infinity].

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

We start first with the phase space of X-valued bounded continuous functions [phi] defined on (-[infinity], 0], that is, BC((-[infinity],0];X) with norm [[absolute value of ([phi])].sub.BC] = [sup.sub.-[infinity][theta][less than or equal to]0] [absolute value of ([phi]([theta]))].

(1) Let

BU = {[phi] [member of] BC ((-[infinity], 0]; X)

: [phi] is bounded uniformly continuous}, (5)

where BC is the space of all bounded continuous functions mapping (-[infinity], 0] into X provided with the uniform norm topology.

(2) Let [mu] [member of] R and

[mathematical expression not reproducible], (6)

provided with the norm

[mathematical expression not reproducible]. (7)

(3) For any continuous function g : (-[infinity], 0] [right arrow] [0, +[infinity]), we define

[mathematical expression not reproducible], (8)

endowed with the norm

[mathematical expression not reproducible]. (9)

Consider the following conditions on g:

([g.sub.1]) [sup.sub.-[infinity]<[theta][less than or equal to]-t](g(t + [theta])/g([theta])) is locally bounded for t [greater than or equal to] 0,

([g.sub.2]) [lim.sub.[theta][right arrow]-[infinity]]g(d) = +[infinity],

([g.sub.3]) [lim.sub.t[right arrow]+[infinity][sup.sub.-[infinity][theta][less than or equal to]-t](g(t + [theta])/g([theta])) = 0.

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 [phi] defined on (-[infinity], 0] with values in [R.sup.n], that is, BC((-[infinity], 0]; [R.sup.n]) with norm [[absolute value of ([phi])].sub.BC] = [sup.sub.-[infinity]<[theta][less than or equal to]0][absolute value of ([phi]([theta]))].

Lemma2 (page 22 in [3]). If the phase space B satisfies axiom ([A.sub.4]), then BC((-[infinity], 0]; [R.sup.n]) is included in B.

3. Existence and Uniqueness

Lemma 3 (see [4]). If p [greater than or equal to] 2, g [member of] [L.sup.2] ([[t.sub.0], T]; [R.sup.nxm]) such that [mathematical expression not reproducible], then

[mathematical expression not reproducible]. (10)

Lemma 4 (Borel-Cantelli, page 487 in [5]). If {[E.sub.n]} is a sequence of events and

[[infinity].summation over (n=1)] P([E.sub.n]) < [infinity], (11)

then

P({[E.sub.n] i.o.}) = 0, (12)

where i.o. is an abbreviation for "infinitively often."

Definitions 1. [R.sup.n]-value stochastic process x(t) defined on -[infinity] < t [less than or equal to] T is called the solution of (1) with initial data [mathematical expression not reproducible], if x(t) has the following properties:

(i) x(t) is continuous and [mathematical expression not reproducible] is [F.sub.t]-adapted,

(ii) {f([x.sub.t], t)} [member of] [L.sup.1] ([[t.sub.0], T]; [R.sup.n]) and [mathematical expression not reproducible],

(iii) [mathematical expression not reproducible],

[mathematical expression not reproducible]. (13)

x(t) is called unique solution, if any other solution [bar.x](t) is distinguishable with x(t); that is,

P[x(t) = [bar.x](t), for any [t.sub.0] [less than or equal to] t [less than or equal to] T} = 1. (14)

Now, we establish existence and uniqueness of solutions for (1) with initial data [mathematical expression not reproducible]. We suppose a uniform Lipschitz condition and a weak linear growth condition.

Theorem 5. Assume that there exist two positive number K and [bar.K] such that,

(i) for any [phi], [psi] [member of] [C.sup.[mu]] and t [member of] [[t.sub.0], T], it follows that

[mathematical expression not reproducible], (15)

(ii) for any t [member of] [[t.sub.0],T], it follows that f(0,t),g(0,t) [member of] [L.sup.2]([C.sup.[mu]]) such that

[[absolute value of f(0,t)].sup.2] [disjunction] [[absolute value of g(0,t)].sup.2] [less than or equal to] K. (16)

Then, problem (1), with initial data [mathematical expression not reproducible], has a unique solution x(t). Moreover, x(t) [member of] [M.sup.2]((-[infinity], T]; [R.sup.n]).

Lemma 6. Let (15) and (16) hold. If x(t) is the solution of (1) with initial data [mathematical expression not reproducible], then

[mathematical expression not reproducible], (17)

where C = 3E[[absolute value of ([xi])].sup.2.sub.[mu]] + 6K(T - [t.sub.0] + 1)(T - [t.sub.0]) + 6[bar.K](T - [t.sub.0] + 1)(T - [t.sub.0])E[[absolute value of ([xi])].sup.2.sub.[mu]].

Moreover, if [xi] [member of] [M.sup.2]((-[infinity],0]; [R.sup.n]), then x(t) [member of] [M.sup.2]((-[infinity],t]; [R.sup.n]).

Proof. For each number q [greater than or equal to] l, define the stopping time

[[tau].sub.q] = T [conjunction] inf {t [member of] [[t.sub.0], T]: [[absolute value of ([x.sub.t])].sub.[mu]] [greater than or equal to] q}. (18)

Obviously, as q [right arrow] [infinity], [[tau].sub.q] [right arrow] T a.s. Let [x.sup.q](t) = x(t [conjunction] [[tau].sub.q]), t [member of] [[t.sub.0], T], and then [x.sup.q](t) satisfy the following equation:

[mathematical expression not reproducible]. (19)

Using the elementary inequality [(a + b + c).sup.2] [less than or equal to] 3([a.sup.2] + [b.sup.2] + [c.sup.2]), we get

[mathematical expression not reproducible]. (20)

Taking the expectation on both sides and using the Holder inequality, Lemma 3, and (15) and (16), we get for all t in [[t.sub.0], T]

[mathematical expression not reproducible]. (21)

We have also for each t in [[t.sub.0], T]

[mathematical expression not reproducible]. (22)

Letting t = T, we get

[mathematical expression not reproducible]. (23)

where C = 3E[[absolute value of ([xi])].sup.2.sub.[mu]] + 6K(T - [t.sub.0] + 1)(T - [t.sub.0]) + 6[bar.K](T - [t.sub.0] + 1) (T - [t.sub.0])E[[absolute value of ([xi])].sup.2.sub.[mu]].

By the Gronwall inequality, we infer

[mathematical expression not reproducible]. (24)

That is,

[mathematical expression not reproducible]. (25)

Consequently

[mathematical expression not reproducible]. (26)

Letting q [right arrow] [infinity], that implies the following inequality

[mathematical expression not reproducible]. (27)

Now, to prove the second part or the lemma, suppose that [xi] [member of] [M.sup.2]((-[infinity],0]; [R.sup.n]). Then

[mathematical expression not reproducible]. (28)

The demonstration of the lemma is complete.

Proof of Theorem 5. We begin by checking uniqueness of solution. Let x(t) and [bar.x](t) be two solutions of (1), by Lemma 6 x(t) and [bar.x](t) [member of] [M.sup.2]((-[infinity], T]; [R.sup.n]). Note that

[mathematical expression not reproducible]. (29)

By the elementary inequality, [(a + b).sup.2] [less than or equal to] 2([a.sup.2] + [b.sup.2]), one then gets

[mathematical expression not reproducible]. (30)

By Holder inequality, Lemma 3, and (15) and (16), we have

[mathematical expression not reproducible]. (31)

From the fact [mathematical expression not reproducible], and

[mathematical expression not reproducible]. (32)

We have

[mathematical expression not reproducible]. (33)

Applying the Gronwall inequality yields

E([[absolute value x(t)-[bar.x](t)].sup.2]) = 0, [t.sub.0] [less than or equal to] t [less than or equal to] T. (34)

The above expression means that x(t) = [bar.x](t) a.s. for [t.sub.0] [less than or equal to] t [less than or equal to] T. Therefore, for all -[infinity] < t [less than or equal to] T, x(t) = [bar.x](t) a.s., the proof of uniqueness is complete.

Next, to check the existence, define [mathematical expression not reproducible], and [x.sup.0](t) = [xi](0) [t.sub.0] [less than or equal to] t [less than or equal to] T. Let [mathematical expression not reproducible] ..., and define Picard sequence

[mathematical expression not reproducible]. (35)

Obviously [x.sup.0](t) [member of] [M.sup.2]((-[infinity],T]; [R.sup.n]). By induction, we can see that [x.sup.k](t) [member of] [M.sup.2]((-[infinity],T]; [R.sup.n]).

In fact, by elementary in equality [(a+b+c).sup.2] [less than or equal to] 3([a.sup.2]+[b.sup.2]+[c.sup.2])

[mathematical expression not reproducible]. (36)

From the Holder inequality and Lemma 3, we have

[mathematical expression not reproducible]. (37)

Again the elementary inequality [(a + b).sup.2] [less than or equal to] 2[a.sup.2] + 2[b.sup.2], (22), (15), and (16) imply that

[mathematical expression not reproducible], (38)

where [C.sub.1] = 3E[[absolute value of ([xi])].sup.2.sub.[mu]] + 6K(t - [t.sub.0] + 1) (t - [t.sub.0]) and [C.sub.2] = 6[bar.K](t - [t.sub.0] + 1)(t - [t.sub.0]) E[[absolute value of ([xi])].sup.2.sub.[mu]].

Hence, for any l [greater than or equal to] 1, one can derive that

[mathematical expression not reproducible]. (39)

Note that

[mathematical expression not reproducible], (40)

and then

[mathematical expression not reproducible], (41)

where [C.sub.3] = [C.sub.1] + 2[C.sub.2].

From the Gronwall inequality, we have

[mathematical expression not reproducible]. (42)

Since k is arbitrary

[mathematical expression not reproducible]. (43)

From the Holder inequality, Lemma 3, and (15) and (16), as in a similar earlier inequality, one then has

[mathematical expression not reproducible]. (44)

That is,

[mathematical expression not reproducible]. (45)

By similar arguments as above, we also have

[mathematical expression not reproducible]. (46)

Then

[mathematical expression not reproducible], (47)

where M = 2[bar.K](t - [t.sub.0] + 1). Similarly

[mathematical expression not reproducible]. (48)

Continue this process to find that

[mathematical expression not reproducible]. (49)

Now we claim that for any k [greater than or equal to] 0

[mathematical expression not reproducible]. (50)

So, for k = 0,1,2,3, inequality (50) holds. We suppose that (50) holds for some k, and check (50) for k + 1. In fact

[mathematical expression not reproducible]. (51)

From (50)

[mathematical expression not reproducible], (52)

which means that (50) holds for k + 1. Therefore, by induction (50) holds for any k [greater than or equal to] 0.

Next to verify {[x.sup.k](t)} converge to x(t) in [L.sup.2] with x(t) in [M.sup.2]((-[infinity],T]; [R.sup.n]) and x(t) is the solution of (1) with initial data [mathematical expression not reproducible]. For (50), taking t = T, then

[mathematical expression not reproducible]. (53)

By the Chebyshev inequality

[mathematical expression not reproducible]. (54)

By using Alembert's rule, we show that [[summation].sup.+[infinity].sub.k=0] (R[[4M(T - [t.sub.0])].sup.k]/k!) converge.

That is, [[summation].sup.+[infinity].sub.k=0] (R[[4M(T - [t.sub.0])].sup.k]/k!) < [infinity], and by Borel-Cantelli's lemma, for almost all [omega] [member of] [OMEGA], there exists a positive integer [k.sub.0] = [k.sub.0]([omega]) such that

[mathematical expression not reproducible], (55)

and then, {[x.sup.k](t)} is also a Cauchy sequence in [L.sup.2]. Hence, {[x.sup.k](t)} converges uniformly and let x(t) be its limit for any t [member of] (-[infinity],T]; since [x.sup.k](t) is continuous on t [member of] (-[infinity],T] and [F.sub.t] adapted, x(t) is also continuous and [F.sub.t] adapted.

So, as k [right arrow] +[infinity], [x.sup.k](t) [right arrow] x(t) in [L.sup.2]. That is, E[[absolute value of [x.sup.k](t) x(t)].sup.2] [right arrow] 0 as k [right arrow] [infinity].

Then from (43)

[mathematical expression not reproducible], (56)

and therefore

[mathematical expression not reproducible], (57)

That is, x(t) [member of] [M.sup.2]((-[infinity], T]; [R.sup.n]).

Now, to show that x(t) satisfies (1)

[mathematical expression not reproducible]. (58)

Noting that the sequence {[x.sup.k]} [right arrow] x(t) means that for any given [epsilon] > 0 there exists [k.sub.0] such that k [greater than or equal to] [k.sub.0], for any t [member of] (-[infinity], T], one then deduces that

[mathematical expression not reproducible], (59)

which means that, for any t [member of] [[t.sub.0], T], one has

[mathematical expression not reproducible]. (60)

For [t.sub.0] [less than or equal to] t [less than or equal to] T, taking limits on both sides of (35), we deduce that

[mathematical expression not reproducible], (61)

and consequently

[mathematical expression not reproducible]. (62)

Finally, x(t) is the solution of (1), and the demonstration of existence is complete.

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

Conflicts of Interest

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

References

[1] 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.

[2] 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.

[3] 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.

[4] 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.

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

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

```Table 1

([A.sub.1])         ([A.sub.2])

BC                       Yes                  No
BU                       Yes                 Yes
[C.sub.g]         Under {[g.sub.1])   Under ([g.sub.1])
[C.sup.0.sub.g]   Under {[g.sub.1])   Under ([g.sub.1])
[C.sup.[mu]]             Yes                 Yes

([A.sub.3])      ([A.sub.4])

BC                    Yes               No
BU                    Yes               No
[C.sub.g]             Yes       Under ([g.sub.2])
[C.sup.0.sub.g]       Yes       Under ([g.sub.2])
[C.sup.[mu]]          Yes         Under [mu] > 0

memory space

BC                        No
BU                        No
[C.sub.g]         Under ([g.sub.3])
[C.sup.0.sub.g]   Under ([g.sub.3])
[C.sup.[mu]]        Under [mu] > 0
```