On a class of differential inclusions in the frame of generalized Hilfer fractional derivative

1 Applied Mathematics Lab, Department of Mathematics, Annaba University, P.O. Box 12, Annaba 23000, Algeria 2 Department of Mathematics, Hodeidah University, P.O. Box 3114, Al-Hudaydah, Yemen 3 Department of Mathematics and Informatics, University of Souk Ahras, P.O. Box 1553, Souk Ahras 41000, Algeria 4 Department of Mathematics and Sciences, Prince Sultan University, Riyadh, Saudi Arabia 5 Department of Medical Research, China Medical University, Taichung 40402, Taiwan

The authors in [46] have started the investigation of the following Hilfer-type FDEs where λ i > 0, δ i ∈ R, H D ϱ 1 ,ϱ 2 and I λ i a+ are the Hilfer FD of order (ϱ 1 , ϱ 2 ) and the Riemann-Liouville FI of order λ i ,respectively. The existence and stability of solutions for implicit-type FDEs (1.1) in the ψ-Hilfer FD sense have been investigated by [47]. In this regard, Wongcharoen et al.,in [48] studied the problem (1.1) with set-valued case, that is where F : [a, b]×R → P (R) is a set-valued map. Motivated by aforesaid works, we prove the existence of solutions for the following nonlinear FDI in the frame of φ-Hilfer FD with nonlocal IBCs where H D ϱ 1 ,ϱ 2 ;φ a+ is the φ-Hilfer FD of order ϱ 1 ∈ (1, 2) and type ϱ 2 ∈ [0, 1], This article is framed as follows. In Section 2, we provide some essentials concepts of advanced fractional calculus, set-valued analysis, and FP methods. The existence results for a φ-Hilfer type inclusion problem (1.3) are obtained in Section 3. The results obtained will be illustrated by examples in the Section 4.

Fractional calculus (FC)
In this portion, we introduce some notations and definitions of FC.
Clearly, C is a Banach space with norm Denote L 1 (℧, R) be the Banach space of Lebesgue-integrable functions g : ℧ → R with the norm Let g ∈ L 1 (℧, R) and φ ∈ C n (℧, R) be increasing such that φ ′ (τ) 0 for each τ ∈ ℧.
). The ϱ th 1 -φ-Riemann-Liouville FI of g is given by
then, the solution of nonlocal BVP is obtained as

Set-valued analysis
We requisition some basics related to the theory of set-valued maps. To this purpose, consider the Banach space (E, ∥.∥) and the multi-valued map M : For other definitions such as completely continuous, upper semi-continuity (u.s.c.), we indicate to [49]. Further, the set of selections of F is given by where P b , P cl , P cp , and P c are the categories of all closed, bounded, compact and convex subsets of E, respectively.
for a.e. τ ∈ ℧, and for all ∥ϕ∥ ≤ w. Now, we offer the next essential lemmas: Conversely, if M is completely continuous and has a closed graph, then it is u.s.c.
. Let E be a separable Banach space. F : ℧×R → P cp,c (E) be an L 1 -Carathéodory set-valued map, and T : L 1 (℧, E) → C (℧, E) be a linear continuous mapping. Then the operator

Existence results for set-valued problem
The first consequence transacts with the convex valued F depending on Leray-Schauder-type for set-valued maps [51].
Then the problem (1.3) has at least one solution on ℧.
Proof. At first, to convert (1.3) into a FP problem, we define the operatorB : C → P (C) bỹ for κ ∈ R F,ϕ . Clearly, the solution of (1.3) is the FP of the operatorB. Proof cases will be given in a number of steps as: Case 1.B (ϕ) is convex for any ϕ∈ C.
So,B (D r ) is equicontinuous. Based on Arzela-Ascoli theorem and above cases (2 − 3), we conclude thatB is completely continuous. Case 4. The graph ofB is closed. Let ϕ n →ϕ * , p n ∈B (ϕ n ) and p n converges to p * . We prove that p * ∈B (ϕ * ). Since p n ∈B (ϕ n ), there exists κ n ∈ R F,ϕ n such that Thus, we need to show that there exists κ * ∈ R F,ϕ * such that, for each τ ∈ ℧, Define T : L 1 (℧, R) → C (℧, R) such that be continuous linear operator by Observe that when n → ∞. So in light of Lemma (2.10) that T • R F,ϕ is a closed graph operator. Besides, we have p n ∈ T R F,ϕ n .
Let δ ∈ (0, 1) and ϕ∈ δB (ϕ). Then there exists κ ∈ R F,ϕ such that Hence, we obtain ∥ϕ∥ From (As3), there is a positive constant K such that ∥ϕ∥ K. We define the set N by From previous cases,B : N → P (C) is completely continuous and u.s.c. Depending on the choice of N, there is no ϕ ∈ ∂N such that ϕ∈ δB (ϕ) for some δ ∈ (0, 1). Therefore, We can infer that problem (1.3) possesses at least one solution ϕ∈ N according to Leray-Schauder theorem for multivalued maps. □

The Lipschitz case
In this part, we give another existence criterion for φ-Hilfer FDI (1.3) according to new assumptions. In what follows, we prove the existence result when F has a non convex-valued using Covitz and Nadler theorem [52].
Then the problem (1.3) has at least one solution on ℧ if where η is defined in (3.1).
Proof. In view of Theorem III.6 in [8] and the assumption (As4), F has a measurable selection κ : ℧→ R, κ ∈ L 1 (℧, R), as well as F is integrably bounded. Thus, R F,ϕ ∅. Now, we prove that B : C → P (C) defined in (3.3) satisfies the assumptions of FPT of Nadler and Covitz. To show that B (ϕ) is closed for any ϕ∈ C. Let {u n } n≥0 ∈B (ϕ) be such that u n → u (n → ∞) in C. Then u ∈ C and there is κ n ∈ R F,ϕ n such that As F possesses compact values, so there is a subsequence κ n → κ in L 1 (℧, R). Consequently. κ ∈ R F,ϕ and we get Hence u ∈B (ϕ).

Examples
In this section, we give some special cases of FDIs to illustrate the obtained outcomes. Consider the FDIs of the following type (4.1) The following instances are special cases of FDIs defined by (4.1).

Conclusions
In this article, we have considered a class of BVP's for φ-Hilfer-type FDIs subjected to nonlocal IBC. The existence results have been proved by considering the kinds when the set-valued map has convex or nonconvex values. In the case of a convex set-valued map, we have applied the Leray-Schauder FPT, whereas the Nadler's and Covitz's FPT concern set-valued contractions are used in the case of a nonconvex set-valued map. The obtained outcomes are well explained through many relevant illustrative examples. We have settled that current results are new in the frame of φ-Hilfer FDIs and it covers many findings in the existing literature as a special case as shown in the Remark 1.1.
In future studies. We will try to expand the problem presented in this article to a general structure using the Mittag-Leffler power law [21] and fractal fractional operators [54].