S-asymptotically ω-periodic solutions in the p-th mean for a Stochastic Evolution Equation driven by Q-Brownian motion

In this paper, we study the existence (uniqueness) and asymptotic stability of the p-th mean S-asymptotically ω-periodic solutions for some non-autonomous Stochastic Evolution Equations driven by a Q-Brownian motion. This is done using the Banach fixed point Theorem and a Gron-wall inequality.


Introduction
Let (Ω, F , P) be a complete probability space and (H, ||.||) a real separable Hilbert space. We are concerned in this paper with the existence and asymptotic stability of p-th mean S-asymptotically ωperiodic solution of the following stochastic evolution equation where (A(t)) t≥0 is a family of densely defined closed linears operators which generates an exponentially stable ω-periodic two-parameter evolutionary family. The functions f : R + × L p (Ω, H) → L p (Ω, H), g : R + × L p (Ω, H) → L p (Ω, L 0 2 ) are continuous satisfying some additional conditions and (W (t)) t≥0 is a Q-Brownian motion. The spaces L p (Ω, H), L 0 2 and the Q-Brownian motion are defined in the next section.
The concept of periodicity is important in probability especially for investigations on stochastic processes. The interest in such a notion lies in its significance and applications arising in engineering, statistics, etc. In recent years, there has been an increasing interest in periodic solutions (pseudo-almost periodic, almost periodic, almost automorphic, asymptotically almost periodic, etc) for stochastic evolution equations. For instance among others, let us mentioned the existence, uniqueness and asymptotic stability results of almost periodic solutions, almost automorphic solutions, pseudo almost periodic solutions studied by many authors, see, e.g. ([1]- [11]). The concept of Sasymptotically ω-periodic stochastic processes, which is the central question to be treated in this paper, was first introduced in the literature by Henriquez, Pierri et al in ( [12,13]). This notion has been developed by many authors.
In the literature, there has been a significant attention devoted this concept in the deterministic case; we refer the reader to ([14]- [20]) and the references therein. However, in the random case, there are few works related to the notion of S-asymptotically ω-periodicity with regard to the existence, uniqueness and asymptotic stability for stochastic processes. To our knowledge, the first work dedicated to S-asymptotically ω-periodicity for stochastic processes is due to S. Zhao and M. Song ( [21,22]) where they show existence of square-mean S-asymptotically ω-periodic solutions for a class of stochastic fractional functional differential equations and for a certain class of stochastic fractional evolution equation driven by Levy noise. But until now and to the best our knowledge,there is no investigations for the exis-S. M. Manou-Abi et al. / Advances in Science, Technology and Engineering Systems Journal Vol. 2, No. 5, 124-133 (2017) tence (uniqueness), asymptotic stability of p-th mean S-asymptotically ω-periodic solutions when p > 2.
This paper is organized as follows. Section 2 deals with some preliminaries intended to clarify the presentation of concepts and norms used latter. We also give a composition result, see Theorem 1. In section 3 we present theoretical results on the existence and uniqueness of S-asymptotically ω-periodic solution of equation (1), see Theorem 2. We also present results on asymptotic stability of the unique S-asymptotically ω-periodic solution of equation (1), see Theorem 3.

Preliminaries
This section is concerned with some notations, definitions, lemmas and preliminary facts which are used in what follows.

p-th mean S asymptotically omega periodic process
Assume that the probability space (Ω, F , P) is equipped with some filtration (F t ) t≥0 satisfying the usual conditions. Let p ≥ 2. Denote by L p (Ω, H) the collection of all strongly measurable p-th integrable H-valued random variables such that The collection of p-mean S-asymptotically ωperiodic stochastic process with values in H is then denoted by SAP ω L p (Ω, H) .
A continuous bounded stochastic process X, which is 2-mean S-asymptotically ω-periodic is also called square-mean S-asymptotically ω-periodic.
Remark 1 Since any p-mean S-asymptotically ωperiodic process X is L p (Ω, H) bounded and continuous, the space SAP ω L p (Ω, H) is a Banach space equipped with the sup norm : which is jointly continuous, is said to be p-mean Sasymptotically ω periodic in t ∈ R + uniformly in X ∈ K where K ⊆ L p (Ω, K) is bounded if for any > 0 there exists L > 0 such that for all t ≥ L and all process X : R + → K Definition 5 A function F : R + ×L p (Ω, H) → L p (Ω, H) which is jointly continuous, is said to be p-mean asymptotically uniformly continuous on bounded sets K ⊆ L p (Ω, H), if for all > 0 there exists δ > 0 such that H) is bounded and p-mean asymptotically uniformly continuous on bounded sets. Assume that X : R + → L p (Ω, H) is a p-mean S asymptotically ω-periodic process. Then the stochastic process (F(t, X(t))) t≥0 is p-mean S-asymptotically ω periodic.
Proof 1 Since X : R + → L p (Ω, H) is a p-mean Sasymptotically ω-periodic process, for all > 0, there exists T > 0 such that for all t ≥ T : In addition X is bounded that is We have : Taking into account (2) and using the fact that F is pmean asymptotically uniformly continuous on bounded sets, there exists δ = and L = T such that for all t ≥ T : Similarly, using the p-mean S-asymptotically ω periodicity in t ≥ 0 uniformly on bounded sets of F it follows that for all t ≥ T : Bringing together the inequalities (3) and (4), we thus obtain that for all t ≥ T > 0 E ||F(t + ω, X(t + ω)) − F(t, X(t))|| p ≤ so that the stochastic process t → F(t, X(t)) is p-mean Sasymptotically ω-periodic.
is p-mean uniformly S-asymptotically ω-periodic in t ∈ R + uniformly on bounded sets and satisfies the Lipschitz condition, that is, there exists constant L(F) > 0 such that . Let X be an p-mean S asymptotically ω-periodic proces, then the process (F(t, X(t))) t≥0 is pmean S-asymptotically ω-periodic.
For the proof, the reader can refer to [22] whenever p = 2. The case p > 2 is similar. Now let us recall the notion of evolutionary family of operators.
is called an evolutionary family of operators whenever the following conditions hold: For additional details on evolution families, we refer the reader to the book by Lunardi [23].

Q-Brownian motion and Stochastic integrals
Let (B n (t)) n≥1 , t ≥ 0 be a sequence of real valued standard Brownian motion mutulally independent on the filtered space (Ω, F , P, F ). Set where λ n ≥ 0, n ≥ 1, are non negative real numbers and (e n ) n≥1 the complete orthonormal basis in the Hilbert space (H, ||.||). Let Q be a symmetric nonnegative operator with finite trace defined by Let (K, ||.|| K ) be a real separable Hilbert space. Let also L(K, H) be the space of all bounded linear operators from K into H. If K = H, we denote it by L(H).
the space of all Hilbert-Schmidt operators from H 0 to H equipped with the norm In the sequel, to prove Lemma 4 and Theorem 2 we need the following Lemma that is a particular case of Lemma 2.2 in [24] (see also [25]).
(ii) There exists some constant C p > 0 such that the following particular case of Burkholder-Davis-Gundy inequality holds : In the sequel, we'll frequently make use of the following inequalities :

Main results
In this section, we investigate the existence and the asymptotically stability of the p-th mean Sasymptotically ω-periodic solution to the already defined stochastic differential equation : S. M. Manou-Abi et al. / Advances in Science, Technology and Engineering Systems Journal Vol. 2, No. 5, 124-133 (2017) where A(t), t ≥ 0 is a family of densely defined closed linear operators and 2 ) are jointly continuous satisfying some additional conditions and (W (t)) t≥0 is a Q-Brownian motion with values in H and F t -adapted.
Throughout the rest of this section, we require the following assumption on U (t, s) : (H1): A(t) generates an exponentially ω-periodic stable evolutionnary process (U (t, s)) t≥s in L p (Ω, H), that is, a two-parameter family of bounded linear operators with the following additional conditions : (1), satisfies the following relation Thus dg(s) = U (t, s)f (s, X(s))ds + U (t, s)g(s, X(s))dW (s). (6) Integrating (6) on [0, t] we obtain that Therefore, we define Definition 7 An (F t )-adapted stochastic process (X(t)) t≥0 is called a mild solution of (1) if it satisfies the following stochastic integral equation : U (t, s)g(s, X(s))dW (s).

The existence of p-th mean Sasymptotically ω-periodic solution
We require the following additional assumptions: (H. 2) The function f : R + × L p (Ω, H) → L p (Ω, H) is p-mean S-asymptotically ω periodic in t ∈ R + uniformly in X ∈ K where K ⊆ L p (Ω, H) is a bounded set.
Moreover the function f satisfies the Lipschitz condition, that is, there exists constant L(f ) > 0 such that 2 ) is a bounded set. Moreover the function g satisfies the Lipschitz condition, that is, there exists constant L(g) > 0 such that
It is easy to check that F is bounded and continuous. Now we have : Let p and q be conjugate exponents. Using Hölder inequality, we obtain that Using Hölder inequality, we obtain that J(t) where

Proof 3
We define h(s) = g(s, φ(s)). Since the hypothesis (H.3) is satisfied, using Lemma 1, we deduce that the function h is p-mean S asymptotically ω periodic.
It is easy to check that F is bounded and continuous. We have : where www.astesj.com S. M. Manou-Abi et al. / Advances in Science, Technology and Engineering Systems Journal Vol. 2, No. 5, 124-133 (2017) where the constant C p will be precised in the next lines. We have EJ(t) where Note that for all t ≥ 0,

Estimation of EJ 1 (t).
Assume that p > 2. Using Hölder inequality between conjugate exponents p p−2 and p 2 together with Lemma 2, part (ii), there exists constant C p such that : Assume that p = 2. By Lemma 2, part (i) we get :

Assume that p = 2. By Lemma 2, part (i) and Cauchy-Schwarz inequality we have
Note also that for t ≥ T : www.astesj.com S. M. Manou-Abi et al. / Advances in Science, Technology and Engineering Systems Journal Vol. 2, No. 5, 124-133 (2017) so that This implies that

Then the stochastic evolution equation (1) has a unique p-mean S-asymptoticaly ω-periodic solution.
Proof 4 We define the nonlinear operator Γ by the expression According to the hypothesis (H1) we have : According to Lemma 3 and Lemma 4, the operators ∧ 1 and ∧ 2 maps the space of p-mean S-asymptotically ωperiodic solutions into itself. Thus Γ maps the space of p-mean S-asymptotically ω periodic solutions into itself.
Case p > 2 : By Lemma 2,part (ii) and Hölder in- Consequently, if Θ < 1, then Γ is a contraction mapping. One completes the proof by the Banach fixed-point principle.

Stability of p-mean S asymptotically ω periodic solution
In the previous section, for the non linear SDE, we obtain that it has a unique p-mean S-asymptotically ωperiodic solution under some conditions. In this section, we will show that the unique p mean S asymptotically ω periodic solution is asymptotically stable in the p mean sense.

Recall that
Definition 8 The unique p-mean S asymptotically ω periodic solution X * (t) of (1) is said to be stable in p-mean sense if for any > 0, there exists δ > 0 such that stands for a solution of (1) with initial value X(0).

Definition 9
The unique p-mean S asymptotically ω periodic solution X * (t) is said to be asymptotically stable in p-mean sense if it is stable in p-mean sense and lim t→∞ E||X(t) − X * (t)|| p = 0 The following Gronwall inequality is proved to be useful in our asymptotical stability analysis.
Lemma 5 Let u(t) be a non negative continuous functions for t ≥ 0, and α, γ be some positive constants. If whenever p > 2. (ii) whenever p = 2.
Then the p-mean S-asymptotically solution X * t of (1) is asymptotically stable in the p-mean sense.
Assume that p > 2. Using Hölder inequality we have which is equivalent to our condition (14). Therefore X * is asymptotically stable in the p-mean sense.