Asymptotically periodic behavior of solutions to fractional non-instantaneous impulsive semilinear differential inclusions with sectorial operators

In this paper, we prove two results concerning the existence of S-asymptotically ω-periodic solutions for non-instantaneous impulsive semilinear differential inclusions of order 1<α<2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1<\alpha <2$\end{document} and generated by sectorial operators. In the first result, we apply a fixed point theorem for contraction multivalued functions. In the second result, we use a compactness criterion in the space of bounded piecewise continuous functions defined on the unbounded interval J=[0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$J=[0,\infty )$\end{document}. We adopt the fractional derivative in the sense of the Caputo derivative. We provide three examples illustrating how the results can be applied.

uniqueness of solutions for a nonlocal problem with integral transmitting condition for mixed parabolic-hyperbolic type equations with Caputo fractional derivative. Agarwal et al. [5] provided a detailed description of impulsive fractional differential equations using Lyapunov functions and overviewed results for the stability in Caputo's sense. Khan et al. [6] focused on the existence and uniqueness of solutions and Hyers-Ulam stability for ABC-fractional DEs with p-Laplacian operator. Khan et al. [7] studied the stability and numerical simulation of a fractional order plant-nectar-pollinator model. Khan et al. [8] proved the existence and Hyers-Ulam stability of solutions to a class of hybrid fractional differential equations with p-Laplacian operator.
The problem of existence of non-constant periodic solutions for fractional order models has became one of the most interesting topics to conduct research on. This is particularly due to the differences between systems of integers order and systems of fractional orders in terms of the existence of non-constant periodic solutions. In much work, such as [9][10][11][12][13][14], the authors have shown that non-constant periodic solutions of fractional order systems do not exist contrary to the case where the order of the system is an integer. Therefore, the concept of an asymptotically periodic solution for fractional differential equations or inclusions is introduced and discussed in much work. For example, in [12,[15][16][17], the authors considered semilinear differential equations of order α ∈ (0, 1) generated by a C 0semigroup, while the papers [18][19][20] addressed semilinear differential equations of order α ∈ (0, 1) generated by sectorial operators. Moreover, the asymptotically periodic solutions for delayed fractional differential equations with almost sectorial operator of order α ∈ (0, 1) are examined in [21]. Rogovchenko et al. [22] studied the asymptotic properties of solutions for a certain classes of second order nonlinear differential equations. Very recently, Wang et al. [23] discussed the asymptotic behavior of solutions to time-fractional neutral functional differential equations of order α ∈ (0, 1).
For more information regarding this subject, we refer the reader to [24,25]. It is worth noting that the problems discussed in all cited work above do not contain impulse effects, whether it is instantaneous or non-instantaneous, and the nonlinear term is a single-valued function.
To the best of our knowledge, there is no work on S-asymptotic ω-periodic behavior of solutions to fractional non-instantaneous impulsive differential inclusions with order α ∈ (1, 2) and generated by sectorial operators, and this fact is the main goal in the present paper.
To clarify the advantage of this study, we mention that two methods have been provided to demonstrate the existence of S-asymptotic ω-periodic solutions for semilinear fractional differential inclusions in the presence of non-instantaneous impulse effects, and in which the nonlinear part is a multivalued function, and the linear part is a sectorial operator. Moreover, the technique presented in this paper can be used to generalize the work in [12,[15][16][17][18][19][20][21][23][24][25] to the case where the linear part is a sectorial operator, the nonlinear part is a multivalued function, and there is impulse effects. In addition, Problem (1) can be investigated on time scales using the arguments in [1], and using the arguments in [3,6,8], one can examine the asymptotic periodic solutions for Problem (1) when the Caputo derivative is replaced by the ψ-Caputo derivative, ψ-RL derivative, Atangana-Baleanu derivative or p-Laplacian operator. Also, the technique used in this paper can be applied to study the asymptotic periodic solutions for many fractional differential equations or inclusions generated by sectorial operators or almost sectorial operators.
The paper is organized as follows. Section 2 includes definitions and basic information that we need to prove our results. In Sect. 3, we provide two existence results of Sasymptotic ω-periodic solutions for Problem (1). In Sect. 4, we give three examples to illustrate our theoretical results.

Preliminaries and notations
It is known that the vector spaces and are Banach spaces endowed with the norm where γ is a suitable path and δ α / ∈ S ϕ + μ for δ ∈ γ .
Remark 2 ( [19], Remark 3) In view of (6), we get: (iii) As in (ii), we derive Based on Definition 2, we can give the definition of an S-asymptotically ω-periodic mild solution for Problem (1).

Existence of S-asymptotically ω-periodic mild solutions for Problem (1)
In order to give the first result, we need the following lemma which is due to [46].

Lemma 2 Let (X, d) be a metric space and G be a contraction multivalued function from X to the family of non-empty closed subsets of X. Then G has a fixed point.
For notations about multivalued functions we refer the reader to [47].
Theorem 1 Suppose the following assumptions are satisfied.
(ii) For any x ∈ PC(J, E), the set where h is the Hausdorff distance. (iv) There is a continuous function L 2 : J → (0, ∞) such that (v) The function σ (τ ) := F(τ , 0) = sup z∈F(τ ,0) z is continuous, bounded on J and satisfies the relation (ii) There is N > 0 such that for any i ∈ N (iii) There is N > 0 such that for any i ∈ N (iv) There is κ 1 > 0 such that Then Problem (1) has an S-asymptotically ω-periodic mild solution provided that the following conditions are verified: and Proof Due to (HF)(ii), one can define a multivalued function on SAP ω PC(J, E) in the following manner: an element y ∈ (x) if and only if . We do this in the following steps.
Step 1. We demonstrate that, if y ∈ (x), we have Since x ∈ SAP ω PC(J, E), Now, we consider two cases.
Step 2. In this step, we show that, if x ∈ SAP ω PC(J, E) and y ∈ (x), then y is bounded.
Step 3. The values of are closed.
To show this, let x ∈ SAP ω PC(J, E) and y n ∈ (x), ∀n ≥ 1, with y n → y in SAP ω PC(J, E).
Step 4. is a contraction.
Remark 3 If there is no impulse effect, then N = N = 0, and hence inequality (17) becomes 2Lξ < 1. Now, we present another result concerning the existence of S-asymptotically ω-periodic solutions for Problem (1).
We need the following fixed point theorem for multivalued functions and a compactness criterion in PC(J, E).  N) is uniformly continuous on bounded sets and for any z ∈ E, the function θ → g i (θ , z) is continuously differentiable at s i such that (10), (11), (14) and the following conditions are satisfied: (i) There is a bounded continuous function h * : J → J with lim θ→∞ h * (θ ) = 0 and
Proof Due to (HF) * (iii), we can consider a multioperator on SAP ω PC(J, E) defined as in (18).
In the following steps we show that satisfies the assumptions of Lemma 3.
Due to Eq. (59), λ is well defined. In this step, we show that, if x ∈ D λ and y ∈ (x), then y ≤ λ.

Now, as a result of Steps 1 and 2, is a multivalued function from D λ ⊆ SAP ω PC(J, E)
to the non-empty subsets of D λ .
Step 3. is closed (its graph is closed) on D λ .
Let (x n ) n≥1 , (y n ) n≥1 be two sequences in D λ with x n → x, y n → y and y n ∈ (x n ), ∀n ≥ 1. Then we have f n ∈ S 1 F(·,x n (·)) such that Let θ be a fixed point in [0, θ 1 ] and J θ = [0, θ ]. In view of (HF) * (iv), we have Using similar arguments to Step 3 in the proof of Theorem 1, one can show, by (73), that f n f weakly in L 2 (J θ , E) and there is a sequence of convex combinations (z n ) of (f n ) such that z n → f , a.e. t ∈ J θ , and Now, due to the continuity of g i (θ , ·), g i (s i , ·), S α (θs i ) and K α (θs i ), we arrive at and Noting that (z n ) is a subsequence of (f n ), and hence by (74)-(76), there is a subsequence of (y n ) that converge to Because y n → y, we arrive at y = y * . Moreover, (HF) * (ii) ensures that f (τ ) ∈ F(τ , x(τ )), a.e. τ ∈ J. So, y ∈ (x).
Step 6. Our goal in this step is showing that, for any i ∈ N and any θ ∈ J i , the set Z i θ := {y(θ ) : y ∈ D = (D λ )} is relatively compact in E.
Conclusion Two existence results of S-asymptotically ω-periodic of mild solutions to non-instantaneous impulsive semilinear differential inclusions of order 1 < α < 2 and generated by sectorial operators are given This work generalizes much recent work such as [18][19][20] to the case when there are impulse effects and the right-hand side is a multivalued function. Moreover, our technique can be used to develop the work in [12, 15-17, 21, 23-25] to the case when the linear part is a sectorial operator, the nonlinear part is a multivalued function and we have impulse effects. There are many directions for future work, for example: 1-With the help of technique in [1], we study the existence of solutions for Problem (1) on a time scales. 2-Investigation an existence theorem for a nonlinear singular-delay-fractional differential equation considered in [42,43], when it contains a sectorial operator as a linear term and the nonlinear term becomes a multivalued function instead of single-valued function. 3-With the help of technique in [3], discuss the numerical solutions for Problem (1) on a closed bounded interval. 4-Study the S-asymptotically periodic solutions to Problem (1) when the sectorial operator is replaced by almost sectorial. 5-Study the S-asymptotically periodic solutions to Problem (1) when it involves p-Laplacian operator ϕ p as well as when the Caputo derivative is replaced by the ψ-Caputo or ψ-Riemann-Liouville derivative. For contributions on BVP involving the ψ-Riemann-Liouville derivative, see [3] and for references on BVP containing the p-Laplacian operator ϕ p , see [6,8].