Global Existence and Controllability to a Stochastic Integro-differential Equation

In this paper, we are focused upon the global uniqueness results for a stochastic integro-differential equation in Frechet spaces. The main results are proved by using the resolvent operators combined with a nonlinear alternative of Leray-Schauder type in Frechet spaces due to Frigon and Granas. As an application, a controllability result with one parameter is given to illustrate


Introduction
In this paper, we consider the uniqueness of mild solutions on a semi-infinite positive real interval J := [0, +∞) for a class of stochastic integro-differential equations in the abstract form dx(t) = Ax(t) + t 0 B(t − s)x(s)ds dt + f (t, x(t))dw(t), t ∈ J, (1.1) where A : D(A) ⊂ H → H, B(t) : D(B(t)) ⊂ H → H, t ≥ 0, are linear, closed, and densely defined operators in a Hilbert space H, f : J × H → L Q (K, H) is an appropriate function specified later and w(t), t ≥ 0 is a given K-valued Brownian motion, which will be defined in Section 2. The initial data x 0 is an F 0 -adapted, H-valued random variable independent of the Wiener process w.Stochastic differential and integro-differential equations have attracted great interest due to their applications in characterizing many problems in physics, biology, mechanics and so on.Qualitative properties such as existence, uniqueness and stability for various stochastic differential and integro-differential systems have been extensively studied by many researchers, see for instance [1,2,3,4,5,6,7,8,9] and the references therein.The theory of nonlinear functional integro-differential equations with resolvent operators serves as an abstract formulation of partial integrodifferential equations which arises in many physical phenomena [10,11,12,13,14].Just as pointed out by Ouahab in [15], the investigation of many properties of solutions for a given equation, such as stability, oscillation, often needs to guarantee its global existence.Thus it is of great importance to establish sufficient conditions for global existence results for functional differential equations.The existence of unique global solutions for deterministic functional differential evolution equations with infinite delay in Fréchet spaces were studied by Baghli et al. [16,17] and Benchohra et al. [18].Motivated by the works [16,17,19,18], the main purpose of this paper is to establish the global uniqueness of solutions for the problem (1.1)-(1.2).Our approach here is based on a recent Frigon and Granas nonlinear alternative of Leray-Schauder type in Fréchet spaces [20] combined with the resolvent operators theory.The obtained result can be seen as a contribution to this emerging field.
The rest of this paper is organized as follows: In section 2, we recall some basic definitions, notations, lemmas and theorems which will be needed in the sequel.In section 3, we prove the existence of the unique mild solutions for the problem (1.1)-(1.2).Section 4 is reserved for an application.

Preliminaries
This section is concerned with some basic concepts, notations, definitions, lemmas and preliminary facts which are used throughout this paper.For more details on this section, we refer the reader to [5,21].
Throughout the paper, (H, • , •, • ) and (K, • K , •, • K ) denote two real separable Hilbert spaces.Let (Ω, F , P) be a complete probability space equipped with some filtration {F t } t≥0 satisfying the usual conditions (i.e., it is right continuous and F 0 contains all P-null sets).Let {e i } ∞ i=1 be a complete orthonormal basis of K. We denote by {w(t), t ≥ 0} a cylindrical K-valued Wiener process with a finite trace nuclear covariance operator where {w i (t)} ∞ i=1 are mutually independent one-dimensional standard Wiener processes.We then let F t = σ{w(s) : 0 ≤ s ≤ t} be the σ-algebra generated by w.
Let L(K, H) denote the space of all linear bounded operators from K into H, equipped with the usual operator norm • L(K,H) .For φ ∈ L(K, H), we define H) with respect to the topology induced by the norm • Q where φ 2 Q = φ, φ is a Hilbert space with the above norm topology.
The collection of all strongly measurable, square integrable, H-valued random variables, denoted by L 2 (Ω, H), is a Banach space equipped with norm , where the expectation E is defined E[x] = Ω x(w)dP(w).An important subspace is given by L 0 2 (Ω, H) = {f ∈ L 2 (Ω, H) : f is F 0 − measurable}.Let C Ft (J, H) denote the space of all continuous and F t -adapted measurable processes from J into H.

Consider the space
Throughout the rest of the paper, A : D(A) ⊂ H → H is the infinitesimal generator of a resolvent operator R(t), t ≥ 0 in the Hilbert space H and B(t) : D(B(t)) ⊂ H → H, t ≥ 0 is a bounded linear operator.To obtain our results, we assume that the abstract Cauchy problem EJQTDE, 2010 No. 47, p. 3 has an associated resolvent operator of bounded linear operators R(t), t ≥ 0 on H.
Let X be a Fréchet space with a family of semi-norms With X, we associate a sequence of Banach spaces {(X n , • n )} as follows: For every n ∈ N, we consider the equivalence relation x ∼ n y if and only if x − y n = 0 for all x, y ∈ X.We denote X n = (X| ∼n , • n ) the quotient space, the completion of X n with respect to • n .To every Y ⊂ X, we associate a sequence the {Y n } of subsets Y n ⊂ X n as follows: For every x ∈ X, we denote [x] n the equivalence class of x of subset X n and we defined respectively, the closure, the interior and the boundary of Y n with respect to • n in X n .We assume that the family of semi-norms { • n } verifies: and for almost all t ∈ J.
EJQTDE, 2010 No. 47, p. 4 Definition 2.3 [20] A function G : X → X is said to be a contraction if for each n ∈ N there exists k n ∈ (0, 1) such that: Theorem 2.1 (Nonlinear Alternative of Granas-Frigon, [20]).Let X be a Fréchet space and Y ⊂ X a closed subset and N : Y → X be a contraction such that N(Y ) is bounded.Then one of the following statements hold: (C1) N has a unique fixed point; (C2) There exists λ ∈ [0, 1), n ∈ N, and

Existence Results
In this section, we prove that there is a unique global mild solution for the problem (1.1)-(1.2).We begin introducing the following concept of mild solutions.
is continuous and satisfies the following integral equation Let us list the following assumptions: (H1) A is the infinitesimal generator of a resolvent operator R(t), t ≥ 0 in the Hilbert space H and there exists a constant M > 0 such that

Carathéodory and satisfies the following conditions:
(i) There exist a function p ∈ L 1 loc (J, R + ) and a continuous nondecreasing function ψ : J → (0, +∞) such that for a.e.t ∈ J and each u ∈ H.
(ii) For all R > 0, there exists a function l R ∈ L 1 loc (J, R + ) such that Proof: Let us fix τ > 1.For every n ∈ N, we define in B +∞ the semi-norms where L * n (t) = t 0 l n (s)ds, and l n (t) = MT r(Q)l n (t) and l n is the function from (H2).Then B +∞ is a Fréchet space with the family of semi-norms • n∈N .
We transform (1.1)-(1.2) into a fixed point problem.Consider the operator Γ : Clearly fixed points of the operator Γ are mild solutions of the problem (1.1)-(1.2).For convenience, we set for n ∈ N Let x ∈ B +∞ be a possible fixed point of the operator Γ.By the hypotheses (H1) and (H2), we have for each We consider the function u defined by By the previous inequality, we have EJQTDE, 2010 No. 47, p. 6 Let us take the right-hand side of the above inequality as v(t).Then, we have Using the nondecreasing character of ψ, we get This implies that for each t ∈ [0, n], we have Thus by (3.1), for every t ∈ [0, n], there exists a constant Λ n , such that v(t) ≤ Λ n and hence u(t) ≤ Λ n .Since x n ≤ u(t), we have Clearly, X is a closed subset of B +∞ .We shall show that Γ : X → B +∞ , is a contraction operator.Indeed, consider x, x ∈ B +∞ , thus using (H1) and (H2) for each t ∈ [0, n] and n ∈ N So, the operator Γ is a contraction for all n ∈ N. From the choice of X there is no x ∈ ∂X n such that x = λΓ(x) for some λ ∈ (0, 1).
2) where w (t) denotes a K-valued Brownian motion, k : R Let H = L 2 ([0, π]) and define A : H → H by Az = z ′′ with domain Then A generates a strongly continuous semigroup and resolvent operator R(t) can be extracted from this semigroup [13].
).Then the system (3.2) takes the abstract form as (1.1)-(1.2).Further, we can impose some suitable conditions on the above-defined functions to verify the assumptions on Theorem 3.1, we can conclude that the system (3.2) admits a unique mild solution on J.

Controllability Results
As an application of Theorem 3.1, we consider the following controllability for stochastic functional integro-differential evolution equations of the form where the control function u(•) is given in L 2 (J, U), the Banach space of admissible control functions with U is real separable Hilbert space with the norm | • |, C is a bounded linear operator from U into H.And the functions A, B(t − s), f and x 0 are as in problem (1.1)-(1.2).For more results on the controllability defined on a compact interval, we refer to [25,26,27,28,29,30,31] and the references therein. with Then the system (4.1)-(4.2) is controllable on J.
Proof: Let us fix τ > 1.For every n ∈ N, we define in B +∞ the semi-norms x n := sup{e −τ L * n (t) E x(t) Using the condition (H3), for arbitrary function x(•), we define the control Noting that, we have Applying (H1)-(H3), we get We shall show that using this control the operator Ξ has a fixed point x(•).Then x(•) is a mild solution of the system (4.1)-(4.2).Let x ∈ B +∞ be a possible fixed point of the operator Ξ.By the conditions (H1)-(H3), we have for each We consider the function µ defined by Then, we have Consequently, Then by the condition (4.3), there exists Λ n such that µ(t) ≤ Λ n .Since x n ≤ µ(t), we have x n ≤ Λ n .Set X = {x ∈ B +∞ : sup{E x(t) 2 : 0 ≤ t ≤ n} ≤ Λ n + 1 for all n ∈ N}.
Clearly, X is a closed subset of B +∞ .We shall show that Ξ : X → B +∞ is a contraction operator.Indeed, consider x, x ∈ B +∞ .By (H1)-(H3) for each t ∈ [0, n] and n ∈ N   So, the operator Ξ is a contraction for all n ∈ N. From the choice of X there is no x ∈ ∂X n such that x = λΞ(x) for some λ ∈ (0, 1).Then the statement (C2) in Theorem 2.1 does not hold.A consequence of the nonlinear alternative of Frigon and Granas shows that (C1) holds.We deduce that the operator Ξ has a unique fixed point x, which is the unique mild solution of the problem (4.1)-(4.2).The proof is completed.
Qe i = λ i e i , i = 1, 2, • • • .So, actually, w(t) is defined by Then the statement (C2) in EJQTDE, 2010 No. 47, p. 7 theorem 2.1 does not hold.A consequence of the nonlinear alternative of Frigon and Granas shows that (C1) holds.We deduce that the operator Γ has a unique fixed point x, which is the unique mild solution of the problem (1.1)-(1.2).The proof is completed.Example 3.1 Consider the following nonlinear stochastic functional differential equations Definition 4.1 An F t -adapted stochastic process x : [0, +∞) → H is called a mild solution of the problem (4.1)-(4.2) if x(0) = x 0 ∈ L 0 2 (Ω, H), x(t) is continuous and satisfies the following integral equation EJQTDE, 2010 No. 47, p. 8 Ml n (t)[MM 1 M 2 n 2 + 1] and l n is the function from (H2).Then B +∞ is a Fréchet space with the family of semi-norms • n∈N .We transform (4.1)-(4.2) into a fixed point problem.Consider the operator Ξ : B +∞ → B +∞ defined by Ξ