S-asymptotically $\omega$-periodic solutions in distribution for a class of stochastic fractional functional differential equations

In this paper, we introduce the concepts of S-asymptotically $\omega$-periodic solutions in distribution for a class of stochastic fractional functional differential equations. The existence and uniqueness results for the S-asymptotically $\omega$-periodic solutions in distribution are obtained by means of the successive approximation and the Banach contraction mapping principle, respectively.


Introduction
For the potential applications in theory and applications, the properties about almost automorphic, asymptotically almost automorphic, almost periodic, asymptotically almost periodic and S-asymptotically ω-periodic solutions of various determinate differential systems have been investigated by many researchers (see e.g.[1][2][3][4][5][6][7][8][9] and references therein).In the mean while, the corresponding concepts of stochastic differential equations are also interesting topics in mathematical analysis, for example, Fu and Liu [10] introduced the concept of square-mean almost automorphy for stochastic processes and they studied square mean almost automorphic solutions to some linear and nonlinear stochastic differential equations.Cao et al. [11] introduced the concept of p-mean almost automorphy for stochastic processes.Moreover, Fu in [12] introduced the concept of distributional almost automorphy for stochastic processes and obtained the existence and uniqueness of distributionally almost automorphic solutions to nonautonomous stochastic equations on any real separable Hilbert space.Liu and Sun [13] introduced the concepts of Poisson square-mean almost automorphy and almost automorphy in distribution and established the existence results of solutions that are almost automorphic in distribution for some semilinear stochastic differential equations with infinite dimensional Lévy noise and Li [14] considered the weighted pseudo almost automorphic solutions for nonautonomous SPDEs driven by Lévy noise.
Henríquez et al. [15] made an initial contribution to develop work in [16][17][18] and references therein to the theory of S-asymptotically ω-periodic functions with values in a Banach space.
Cuevas et al. [19] considered the S-asymptotically ω-periodic solution of the semilinear integrodifferential equation of fractional order Moreover, in [20], Cuevas et al. considered the S-asymptotically ω-periodic solution of the following form, where B is some abstract phase space.Dimbour et al. [21] considered the S-asymptotically ωperiodic solutions of the differential equations with piecewise constant argument of the form Inspired by the work mentioned above, in this paper, we investigate the existence of the Sasymptotically ω-periodic solutions in distribution in an abstract space for a class of stochastic fractional functional differential equations driven by Lévy noise of the form where of sectorial type on a Banach space X, x t be defined by x t (θ) = x(t + θ) for each θ ∈ [−τ, 0], and f, g, F, G are functions subject to some additional conditions.The convolution integral in (1.1) is known as the Riemann-Liouville fractional integral [22,23].We introduce the concept of Poisson square-mean S-asymptotically ω-periodic solution for (1.1) in order to correspond to the effect of the Lévy noise.Furthermore, we make an initial consideration of the S-asymptotically ω-periodic solution in distribution in an abstract space C for (1.1).
The paper is organized as follows.In Section 2, we review and introduce some concepts about square mean S-asymptotically ω-periodic solutions in distribution for (1.1) and some of their basic properties.We show the existence and uniqueness of the mild solution and the S-asymptotically ω-periodic solution in distribution to (1.1) in Section 3 and Section 4, respectively.

Preliminaries
Let (Ω, F, P ) be a complete probability space equipped with some filtration {F t } t≥0 which satisfy the usual conditions, (
Proposition 2.1.(see [24]) (Lévy-Itô decomposition).If L is a U -valued Lévy process, then there exist a ∈ U, a U -valued Wiener process w with covariance operator Q, the so-called Q-wiener process, and an independent Poisson random measure N on R + × (U − {0}) such that, for each t ≥ 0, where the Poisson random measure N has the intensity measure ν which satisfies U (|y| 2 U ∧1)ν(dy) < ∞ and Ñ is the compensated Poisson random measure of N.
The detail properties of Lévy process and Q-Wiener processes, we refer the readers to [26] and [27].Throughout the paper, we assume the covariance operator Q of w is of trace class, i.e.
T rQ < ∞ and the Lévy process L is defined on the filtered probability space (Ω, F, P, (F t ) t∈R + ).
(2) A function g : R + × C → L(U, L 2 (P, H)), (t, ϕ) → g(t, ϕ) is said to be square-mean Sasymptotically ω-periodic in t for each ϕ ∈ C if g is continuous in the following sense and and that Remark 2.2.Any square-mean S-asymptotically ω-periodic process x(t) is L 2 -bounded and, by [15], SAP ω (L 2 (P, H)) is a Banach space when it is equipped with the norm For the sequel, we introduce some definitions about square-mean S-asymptotically ω-periodic functions with parameters.
Definition 2.3.(1) A function f : R + × C → L 2 (P, H) is said to be uniformly square-mean S-asymptotically ω-periodic in t on bounded sets if for every bounded set K of C, we have (2) A function g : R + ×C → L(U, L 2 (P, H)), is said to be uniformly square-mean S-asymptotically ω-periodic on bounded sets if for every bounded set K of C, we have uniformly on φ ∈ K.
(3) A function F : R + ×C ×U → L 2 (P, H) with U F (t, φ, u) 2 ν(du) < ∞ is said to be uniformly Poisson square-mean S-asymptotically ω-periodic in t on bounded sets if for every bounded ) be uniformly square-mean S-asymptotically ω-periodic in t on bounded sets of C and assume that f satisfies the Lipschitz condition in the sense C for all φ, ϕ ∈ C and t ∈ R, where L is independent of t.Then for any square-mean S-asymptotically ω-periodic process Y : R → L 2 (P, H), the stochastic process We get which completes the proof.
Lemma 2.4.Let F : R + × C × U → L 2 (P, H) be uniformly Poisson square-mean S-asymptotically ω-periodic in t on bounded sets of C and F satisfies the Lipschitz condition in the sense for all φ, ϕ ∈ C and t ∈ R, where L is independent of t.Then for any square-mean S-asymptotically ω-periodic process Y (t) : R → L 2 (P, H), the stochastic process F : R × U → L 2 (P, H) given by Proof.Since F is uniformly Poisson square-mean S-asymptotically ω-periodic in t on bounded For any ǫ > 0, we can find T (ǫ) > 0 such that when t ≥ T (ǫ), we have so for the above ǫ, when t ≥ T (ǫ), we have We deduce that lim t→∞ U which means that F (t, u) is Poisson square-mean S-asymptotically ω-periodic.
Let P(C) be the space of Borel probability measures on C, for P 1 , P 2 ∈ P(C), denote metric d L as follows where Definition 2.4.A stochastic process x t : R → C is said to be S-asymptotically ω-periodic in distribution if the law µ(t) of x t is a P(C)-valued S-asymptotically ω-periodic mapping, i.e. there is a positive number ω such that Lemma 2.5.Any square-mean S-asymptotically ω-periodic solution of (1.1) is necessarily Sasymptotically ω-periodic in distribution.
Proof.Let x(t) ∈ SP A ω (L 2 (P, H)) be a solution of (1.1), then there exists ω > 0 such that We need to show that the law µ(t) of x t satisfies lim which is equivalent to show for any ǫ > 0, there is a T > 0 such that sup Since for any f ∈ L, For the arbitrary of f ∈ L, we get sup The proof is completed.

Sectorial operators
We recall some definition about sectorial operators which have been studied well in the past decades, for details, see [28,29].
Definition 2.5.Let X be an Banach space, A : D(A) ⊆ X → X is a close linear operator.A is said to be a sectorial operator of type µ and angle θ if there exist 0 < θ < π/2, M > 0 and µ ∈ R such that the resolvent ρ(A) of A exists outside the sector µ Definition 2.6.(see [23]) Let A be a closed and linear operator with domain D(A) defined on a Banach space X.We call A the generator of a solution operator if there exist µ ∈ R and a strongly continuous function S α : R + → L(X, X) such that {λ α : Re(λ) > µ} ⊂ ρ(A) and In this case, S α (•) is called the solution operator generated by A.
In order to establish our main result, we impose the following conditions.
Set x n 0 = φ, for n = 1, 2, ..., ∀ T ∈ (0, ∞), we define the sequence of successive approximations to (1.1) as follows: Obviously, I 1 ≤ 5c ′ , and It follows from Itô's isometry that By using the properties of integrals for poisson random measures, we get Note that by taking the expectation on both sides of the above inequality, we get Combining the estimations for I 1 − I 5 , we get Then for any arbitrary positive integer k, we have By the Gronwall inequality, we get Due to the arbitrary of k, we have namely, and By the fact and we get Hence < ∞, by using the Borel-Cantelli lemma, we can get a stochastic process x(t) on [0, T ] such that x n (t) uniformly converges to x(t) as n → ∞ almost surely.
It is easy to check that x(t) is a unique mild solution of (1.1).The proof of the theorem is complete.
The proof process is similar to that of Lemma 1 in [32], so we omit it.

4
Existence of almost automorphic mild solutions Lemma 4.1.If x(t) ∈ SAP ω (L 2 (P, H) and T 3, Lemma 2.4 and the similar discussion as that for Lemma 4.2.