Attractivity for Hilfer fractional stochastic evolution equations

This article is devoted to the study of the attractivity of solutions to a class of stochastic evolution equations involving Hilfer fractional derivative. By employing the semigroup theory, fractional calculus and the fixed point technique, we establish new alternative criteria to ensure the existence of globally attractive solutions for the Cauchy problem when the associated semigroup is compact.


Introduction
In the past three decades, fractional differential equations received much attention. The growing interest in the subject is due to its extensive applications in diverse fields such as physics, fluid mechanics, viscoelasticity, heat conduction in materials with memory, chemistry and engineering. Much of the work is devoted to the existence and uniqueness of solutions for fractional differential equations; see, for example, Kilbas et al. [10], Miller and Ross [13], Podlubny [14], Zhou [22] and [1,5,19,21,23,24] and the references cited therein. Since Hilfer [9] proposed the generalized Riemann-Liouville fractional derivative (Hilfer fractional derivative), there has been shown some interest in studying evolution equations involving Hilfer fractional derivatives (see [2,4,7,8,18,20]).
In this work, we study the attractivity of solutions for the following Hilfer fractional stochastic evolution equations: where D μ,β 0 + denotes the Hilfer fractional derivative of order μ and type β which will be given in next section, 1 2 < μ < 1, 0 ≤ β < 1, A is the infinitesimal generator of a strongly continuous semigroup S(t), (t ≥ 0) on a separable Hilbert space X with inner product · , and norm · . Let {ω(t)} t≥0 denote a K -valued Wiener process with a finite trace nuclear covariance operator Q ≥ 0 defined on the filtered complete probability space (Ω, F, P). The functions f , σ are given functions satisfying some appropriate assumptions. x 0 is an element of the Hilbert space L 0 2 (Ω, X) which will be specified later. The objective of this paper is to discuss the attractivity of solutions for Cauchy problem (1.1). In fact, we establish sufficient conditions for the global attractivity of mild solutions for system (1.1) in cases that semigroup associated with A is compact. The obtained results essentially reveal certain characteristics of solutions for Hilfer fractional evolution equations, in contrast to integer-order evolution equations.
The rest of this paper is organized as follows. Section 2 contains some basic notations and essential preliminary results. In Sect. 3, we obtain alternative sufficient conditions for the attractivity of fractional stochastic evolution equations. Some conclusions are presented in Sect. 4.

Preliminaries
In this section, we provide some basic definitions, notations, lemmas, properties of semigroup theory and fractional calculus which are used throughout this paper.

Definition 2.1 ([10])
The fractional integral of order α with the lower limit 0 for a function f is defined as provided the right side is pointwise defined on [0, ∞), where Γ (·) is the gamma function.

Definition 2.2 ([10])
The left-sided Riemann-Liouville fractional-order derivative of order α with the low limit 0 for a function f : [0, +∞) → R is defined as

Lemma 2.4
By a mild solution of system (1.1) we mean the F t -adapted stochastic progress x : (0, +∞) → L 2 (Ω, H) that satisfies To formulate some essential propositions, we introduce the assumption: ) Under the assumption (H 0 ), {T μ (t)} t>0 and {S μ,β (t)} t>0 are strongly continuous, that is, for any x ∈ X, and 0 < t < t ≤ b, we have We also need the following generalization of the Ascoli-Arzela theorem.

Lemma 2.6
The set H ⊂ C 0 ((0, ∞), X) is relatively compact if and only if the following conditions hold:

Main results
In this section, we establish the attractivity of solutions for system (1.1).
Since T μ (t) = t μ-1 P μ (t), (2.2) takes the form In order to establish the attractivity of solutions for system (1.1), we need the following assumptions: and Observe that x is a mild solution of (1.1) if and only there exists a fixed point x * such that the operator equation Proof Obviously, D 1 is a nonempty bounded closed and convex subset of C 0 ((0, ∞), L 2 (Ω, X)). The proof will be completed in three steps.
For any t > 0, from (H 4 ) and Lemma 2.5, we obtain Note that the above inequity is restricted to the integrability of s -η , which is indeed true for η < 1.
For t > T 1 , taking into account the fact that η > 2μ -1 and inequality (3.5), we infer that Thus Step 2. F 2 is continuous.
Applying Hölder's inequality, we have which implies that F 2 is continuous.
On the other hand, if T 1 ≤ t < T < t and tt → 0, then t → T and t → T. Thus one can easily get
Hence, {F 2 x, x ∈ D 1 } is a relatively compact set in L 2 (Ω, X) by the Arzola-Ascoli theorem. As a consequence of Lemmas 3.1-3.3 and Theorem 2.1, there exists a y ∈ D 1 such that y = F 1 y + F 2 y, that is, H has a fixed point in D 1 which is a solution of system (1.1) for t ≥ T 1 . Now, we are well prepared to present our first attractivity result for system (1.1).

Theorem 3.1 Suppose that assumptions (H 1 )-(H 4 ) hold. Then the zero solution of system (1.1) is globally attractive.
Proof From Lemmas 3.1-3.3 and the properties of D 1 , for t ≥ T 1 , we know that the solution of (1.1) does exist which is still in D 1 . Moreover, all functions in D 1 tend to 0 as t → ∞. Therefore the solution of system (1.1) tends to zero as t → ∞. The proof is completed.
To give our second attractivity result, we require the following hypothesis: for ∀t ∈ (0, ∞) and each x ∈ C((0, ∞), L 2 (Ω, X)), where L ≥ 0, 2μ -1 < η 1 < μ. Proof Set D 2 = {y(t) | y(t) ∈ C((0, ∞), L 2 (Ω, X)), E|y(t)| 2 ≤ t -δ 1 for t ≥ T 2 }, where δ 1 = 1μ. We choose constant T 2 large enough such that the following inequality holds: One can easily see that operator F 1 is contraction. In addition, for each fixed y ∈ D 2 and for ∀x ∈ L 2 (Ω, X), Since η 1 < μ and δ 1 = 1μ, we have -η 1δ 1 > -1, therefore s -η 1 -δ 1 in (3.7) is integrable. Furthermore, for t > T 2 , we have From inequality (3.6) and ν -1 < δ 1 , we have which implies that x(t) ∈ D 2 for t ≥ T 2 . Moreover, taking (3.6) and (3.9) into account, we can also get E|F 2 y(t)| 2 ≤ t -δ 1 , which implies that F 2 D 2 ⊂ D 2 for t ≥ T 2 . By applying a similar argument to the one used in Lemma 3.1, we can deduce that the operator F 2 is continuous and F 2 D 2 resides in a compact subset of L 2 (Ω, X) for t ≥ T 2 . Using Theorem 2.1 and Krasnoselskii's theorem, there exists some y ∈ D 2 such that y = F 1 y + F 2 y, that is, F has a fixed point in D 2 which is indeed a solution of (1.1). Moreover, it is obvious that all functions in D 2 tend to 0 as t → ∞, therefore, the solution of (1.1) tends to zero as t → ∞ which implies the zero solution of (1.1) is globally attractive. This completes the proof.

Corollary 3.1 Suppose that assumptions (H 1 )-(H 3 ) hold and that
which, together with Lemma 3.3 implies thatV (t) = {(Fx)(t), x ∈D} is relatively compact for t > 0. In view of the foregoing arguments, it follows by Schauder's fixed point theorem that system (1.1) has a mild solution x ∈D and that x(t) tends to zero as t → ∞. This completes the proof.
To end this section, we now present the last result of our paper.

Conclusion
In this paper, we have revealed that a certain class of Hilfer fractional stochastic evolution equations with fast decaying nonlinear term have global attractive solutions, whereas the integer-order evolution equations do not have such attractivity. Our future work will be focused on addressing the attractivity for Hilfer fractional stochastic evolution equations with fractional Brownian motion and Poisson jumps.