BLAISE PASCAL

By using successive approximation, we prove existence and uniqueness result for a class of neutral functional stochastic diﬀerential equations in Hilbert spaces with non-Lipschitzian coeﬃcients


Introduction
The purpose of this paper is to prove the existence and uniqueness of mild solutions for a class of neutral functional stochastic differential equations (FSDEs) described in the form where A is the infinitesimal generator of an analytic semigroup of bounded linear operators, (T (t)) t≥0 , in a Hilbert space H; x t ∈ C r = C([−r, 0], H) and f : [0, T ]×C r → H, g : [0, T ]×C r → H, σ : [0, T ]×C r → L 2 (Q 1/2 E, H are appropriate functions.Here L 2 (Q 1/2 E, H) denotes the space of all Q-Hilbert-Schmidt operators from Q 1/2 E into H (see section 2 below).Neutral FSDEs arises in many areas of applied mathematics and such equations have received much attention in recent years.The theory of neutral FSDEs in finite dimensional spaces has been extensively studied in the literature; see Kolmanovskii and Nosov [8], Mao [13]- [12], Kolmanovskii et al. [7], and Liu and Xia [10].However, in the infinite-dimensional Hilbert space, only a few results have been obtained in this field despite the importance and interest of the model (1.1).In this respect, it is worth mentioning that this kind of neutral equation arises from problems related to coupled oscillators in a noisy environment, or in problems of viscoeslastic materials under random or stochastic influences (see [15] for a description of these problems in the deterministic case).To the best of our knowledge, there exist only few papers already published in this field.To be more precise, a version of (1.1), in the particular case where the delays are constant, is considered in [9] and some stability properties of the mild solutions are analyzed in a similar way as Dakto proved in [4] in the deterministic case, while in [6] the existence and uniqueness of mild solutions to model (1.1) is studied, as well as some results on the stability of the null solution.In [2] Caraballo et al studied the problem in a variational point of view.So far little is known about the neutral FSDEs in Hilbert spaces.Our idea is inspired by a paper of Mahmudov [11] in which the author study the existence and uniqueness of equation (1.1) without delay.
The paper is organized as follows, In Section 2 we give a brief review and preliminaries needed to establish our results.Section 3 is devoted to the study of existence and uniqueness by using a Picard type iteration.

Preliminaries
In this section, we introduce notations, definitions and preliminary results which we require to establish the existence and uniqueness of a solution of equation (1.1).Let E and H be two real separable Hilbert spaces.Denote by L(E, H) the family of bounded linear operators from E to H. Fix a non-negative and symmetric operator Q ∈ L(E, E).Let W be a cylindrical Q-Wiener process in E (cf.[3]) defined on some complete filtered probability space (Ω, F, P; Let A : D(A) → H be the infinitesimal generator of an analytic semigroup, (T (t)) t≥0 , of bounded linear operators on H.For the theory of strongly continuous semigroup, we refer to Pazy [14] and Goldstein [5].We will point out here some notations and properties that will be used in this work.It is well known that there exist M ≥ 1 and λ ∈ R such that T (t) ≤ M e λt for every t ≥ 0. If (T (t)) t≥0 is a uniformly bounded and analytic semigroup such that 0 ∈ ρ(A), where ρ(A) is the resolvent set of A, then it is possible to define the fractional power (−A) α for 0 < α ≤ 1, as a closed linear operator on its domain D(−A) α .Furthermore, the subspace D(−A) α is dense in H, and the expression If H α represents the space D(−A) α endowed with the norm .α , then the following properties are well known (cf.[14], p. 74).

Lemma 2.1. Suppose that the preceeding conditions are satisfied. (1) Let
We now recall the following Bihari's inequality [1] .
Lemma 2.2.Let ρ : R + → R + be a continuous and non-decreasing function and let g, h, λ be non-negative functions on R + such that dy is well defined for some x 0 > 0, G −1 is the inverse function of G and h * (t) := sup s≤t h(s).In particular, we have the Gronwall-Bellman lemma: If Finally, we remark that for the proof of our theorem we shall make use of the following Lemma (see [3], p. 184) Then, for any arbitrary p > 2 there exists a constant c(p, T ) > 0 such that:

then there exists a continuous version of the process {W φ
A ; t ≥ 0}

The main result
In this section we study the existence and uniqueness of mild solution of equation (1.1).Henceforth we will assume that A is the infinitesimal generator of an analytic semigroup, (T (t)) t≥0 , of bounded linear operators on H. Further, to avoid unnecessary notations, we suppose that 0 ∈ ρ(A) and that, see Lemma 2.1, In order to show the existence and the uniqueness of the equation (1.1), we are going to make the following hypotheses 2 is measurable, continuous in ξ for each fixed t ∈ [0, T ] and there exists a function ) is continuous non-decreasing and for each fixed has a global solution on [0, T ].
The main result of this paper is given in the next theorem.For the proof, we will need the following lemmas.
, and consider the equation Under condition (H.3), Equation (3.2) has a unique mild solution x ∈ B T .
Proof.Let us consider the set S T is a closed subset of B T provided with the norme .B T .
Let ψ be the function defined on S T by We will first prove that the function ψ is well defined.Let x ∈ S T and t ∈ [0, T ], we have We are going to show that each function t → I i (t) is continuous on [0, T ].
The continuity of I 1 follows directly from the continuity of t → T (t)h.By (H.3), the function (−A) β g is continuous and since the operator For the third term I 3 (t) = t 0 AT (t − s)g(s, x s )ds, we have By the strong continuity of T (t), we have for each s ∈ [0, T ], we conclude by the Lebesgue dominated theorem that lim h→0 I 31 (h) = 0. On the other hand, Similar computations can be used to show the continuity of I 4 .The continuity of the last term follows from the Lemma 2.3.Hence, we conclude that the function t → ψ(x)(t) is continuous on[0, T ] a.s.Next, to see that ψ(S T ) ⊂ S T , let x ∈ S T and t ∈ [0, T ].By using condition (3a) and Hölder's inequality, with 1 p + 1 q = 1, we have By using Lemma 2.1 and Lemma 2.3, we obtain The F t measurability is easily verified, so we conclude that ψ is well defined.
Now, we are going to show that ψ is a contraction mapping in S T 1 with some T 1 ≤ T to be specified later.Let x, y ∈ S T and t ∈ [0, T ], we have By condition (3b), Lemma 2.1 and Hölder's inequality, we have x(s) − y(s) p . where Then there exists 0 < T 1 ≤ T such that 0 < γ(T 1 ) < 1 and ψ is a contraction mapping on S T 1 and therefore has a unique fixed point, which is a mild solution of equation (3.2) on [0, T 1 ].This procedure can be repeated in order to extend the solution to the entire interval [−r, T ] in finitely many steps.
We now construct a successive approximation sequence using a Picard type iteration with the help of Lemma 3.3.Let x 0 be a solution of equation (3.2) with f = 0, σ = 0 .For n ≥ 0, let x n+1 be the solution of equation (2) x n (θ) p )ds. (3.5) Proof.• 1: For m, n ∈ N and t ∈ [0, T ] we have where By using condition (3b) for the terms I 1 and I 2 , we obtain By using Lemma 2.3 and condition (2b) for the term I 3 , we obtain Using the fact that 4 p−1 (−A) −β p M p g < 1 and the above inequalities, we obtain that: By Lemma 2.2, we obtain • 2: By the same method as in the proof of assertion (1), we obtain that x n (t) p ≤ u(t).Then, by (3.5), we obtain

By Lemma 2.2, we obtain
This completes the proof.
H) the Hilbert space of all square integrable and F t adapted processes with values in H. with F t = F 0 for −r ≤ t ≤ 0, let us denote by B T the Banach space of all H -valued F t adapted process x(t, ω) : [−r, T ] × Ω → H, which are continuous in t for a.e.fixed ω ∈ Ω and satisfy 0}is well defined and there exist positive constants M, D 0 , D 1 such that for all m, n ∈ N and t ∈ [0, T ] ) Lemma 3.4.Under conditions (H.1) − (H.3), the sequence {x n , n ≥