Hilfer fractional differential inclusions with Erdélyi–Kober fractional integral boundary condition

In this article, we debate the existence of solutions for a nonlinear Hilfer fractional differential inclusion with nonlocal Erdélyi–Kober fractional integral boundary conditions (FIBC). Both cases of convex- and nonconvex-valued right-hand side are considered. Our obtained results are new in the framework of Hilfer fractional derivative and Erdélyi–Kober fractional integral with FIBC via the fixed point theorems (FPTs) for a set-valued analysis. Some pertinent examples demonstrating the effectiveness of the theoretical results are presented.

In the last few years, many researchers have started to discuss the qualitative properties of solutions of fractional FDEs and FDIs, such as existence, uniqueness, stability, controllability, and optimization, etc., see [1-3, 7-10, 20, 43, 44, 49]. Some other researchers have devoted their works to discussing further analytical properties of solutions of such equations and inclusions, while others already have oriented their investigations towards numerical applications and solutions. For further specialized articles on the existence, uniqueness, and stability of FDEs involving various types of FDs, we refer to [4,6,11,16,33,37,38,46].
Recently, Asawasamrit et al. [17] have initiated the study of Hilfer FDEs with nonlocal IBCs of the type where 1 < r 1 < 2, 0 ≤ r 2 ≤ 1, η i > 0, θ i ∈ R, H D r 1 ,r 2 a + is the Hilfer FD of order r 1 and type r 2 , I η i a+ is the Riemann-Liouville FI of order η i . Later, the authors of [42] have investigated the existence and stability results of an implicit problem for FDE (1.1) involving ψ-Hilfer FD.
On the other hand, Abbas [5] investigated the existence and Ulam-Hyers stability results for the FDE of type (1.1) with the consideration of Erdélyi-Kober FI instead of Riemann-Liouville FI. The set-valued case of problem (1.1) has been studied by Wongcharoen et al. [48].
In order to enhance the work and fill the gap on BVPs of fractional order involving more IBCs, we consider a nonlinear Hilfer-type FDI with Erdélyi-Kober fractional IBC, that is, where H D r 1 ,r 2 is the Hilfer FD of order r 1 ∈ (1, 2) and type r 2 ∈ [0, 1], Our main concern in this manuscript is to obtain the existence results for the Hilfer inclusion problem (1.2) involving convex, nonconvex set-valued maps via some FPTs of Leray-Schauder type, as well as those of Covitz and Nadler, where some pertinent examples are built for the demonstration of our findings.
, then the inclusion problem for it has been studied by Wongcharoen et al. [48]. ii) If r 2 = 0 in (1.2), then our problem reduces to Riemann-Liouville inclusion problem considered by Ahmad and Ntouyas in [12]. iii) If r 2 = 1 in (1.2), then our problem also covers other problems, including those of Caputo type.
This paper is structured as follows. In Sect. 2, we give some fundamental concepts of fractional calculus, set-valued analysis, and FP techniques. In Sect. 3, we study some existence results for the Hilfer inclusion problem (1.2) relying on some FPTs of Leray-Schauder, Covitz, and Nadler. At the end, some examples are given in Sect. 4.

Fractional Calculus (FC)
In this part, we give some essential ideas of FC and axiom outcomes that are prerequisites in our analysis.

Definition 1 ([35])
The Riemann-Liouville FI of a function κ of order r 1 is described by provided the integral exists.

Definition 2 ([35])
The Riemann-Liouville FD of a function h of order r 1 is described by where n = [r 1 ] + 1, n ∈ N.

Definition 4 ([35])
The Erdélyi-Kober FI of a function κ of order ξ > 0 with γ > 0 and η ∈ R is formulated by provided the integral exists.
where B(·, ·) is the beta function determined by

Lemma 4 ([5]) Let
and consider any h ∈ C. Then the solution of the nonlocal BVP is obtained as

Set-valued analysis
We recall some concepts concerning the theory of set-valued maps. For this, let (υ, · ) be a Banach space and N : is measurable.
For the definitions of completely continuous and u.s.c., we refer to [13]. Moreover, a collection of selections of Q at a point ρ ∈ C is defined by Next, we denote the categories of all compact, bounded, closed, and convex subsets of υ, respectively. Also, O cp,c denotes the category of all convex and compact subsets of υ.
for all υ ≤ w and for a.e. t ∈ P.
We will use the following lemmas that will play an important role in the achievement of the desired results in this research.
If N is completely continuous and has a closed graph, then it is u.s.c.

Existence results for (1.2)
Definition 6 A function υ ∈ C is considered a solution of (1.2), if there is an integrable functionṽ ∈ L 1 (P, R) withṽ(t) ∈ Q(t, υ) for all t ∈ P satisfying the nonlocal fractional IBC and

The U.S.C. case
The first result deals with a convex-valued Q relying on Leray-Schauder principle for setvalued maps [29].
and assume that: There is a nondecreasing function ϑ ∈ C(R + , R + ) and a continuous function P : P → R + such that (As3) There is a constant L > 0 such that Then problem (1.2) has at least one solution on P.
Proof Initially, to write problem (1.2) as an FP problem, we consider the operatorS : forṽ ∈ R Q,υ . Obviously, the solution of (1.2) is an FP of the operatorS. The proof steps will be presented as follows: Step 1. The set-valued mapS(υ) is convex for any υ ∈ C.
Step 2.S is bounded on bounded sets of C.
For a constant r > 0, let B r = {υ ∈ C : υ ≤ r} be a bounded set in C. Then for each φ ∈S(υ) and υ ∈ B r , there existsṽ ∈ R Q,υ such that Under the hypothesis (As2) and for any t ∈ P, we obtain .
Step 3.S sends bounded sets of C into equicontinuous sets. Let υ ∈ B r and φ ∈S(υ). Then there is a functionṽ ∈ R Q,υ such that Let t 1 , t 2 ∈ P, t 1 < t 2 . Then As t 1 → t 2 , we obtain HenceS(B r ) is equicontinuous. From the above-mentioned steps 2-3, along with Arzela-Ascoli theorem, we infer thatS is completely continuous.
Step 4. We prove that the graph ofS is closed.
Let υ n → υ * , φ n ∈S(υ n ) and φ n tends to φ * . We show that φ * ∈S(υ * ). Since φ n ∈S(υ n ), there existsṽ n ∈ R Q,υ n such that Therefore, we have to prove that there existsṽ * ∈ R Q,υ * such that, for each t ∈ P, Define the continuous linear operator Z : L 1 (P, υ) → C(P, υ) as follows: Notice that when n → ∞. So in view of Lemma 6, the operator Z • R Q,υ has a closed graph. Moreover, we have φ n ∈ Z(R Q,υ n ).
Consequently, we obtain Under the hypothesis (As3), there is an L > 0 such that υ = L. We build the set D as follows: From steps 1-4, the operatorS : D → O(C) is u.s.c. and completely continuous. From the choice of D, there is no υ ∈ ∂D such that υ ∈ μS(υ) for some μ ∈ (0, 1). So, by Leray-Schauder theorem for set-valued maps, we infer that problem (1.2) has at least one solution υ ∈ D.

The Lipschitz case
For further existence investigation of problem (1.2) in this subsection, we deal with another existence criterion under new hypotheses. In what follows, we will demonstrate that our desired existence of solutions in the case of nonconvex-valued right-hand side follows by Covitz and Nadler theorem [24].
In particular, if κ < 1, the set valued operatorS is a contraction.
Then, (1.2) has at least one solution on P if where is defined in (3.1).
Proof By using the hypothesis (As4) and Theorem III.6 in [23], Q has a measurable selectionṽ : P→ R,ṽ ∈ L 1 (P, R), and so Q is integrably bounded. Thus, R Q,υ = ∅. Now, we show thatS : C → O(C) defined in (3.3) satisfies the hypotheses of FPT of Nadler and Covitz. To prove thatS(υ) is closed for any υ ∈ C, let {u n } ∞ n=0 ∈S(υ) be such that u n → u (n → ∞) in C. Then u ∈ C and there isṽ n ∈ R Q,υ n such that As Q has compact values, so there exists a subsequenceṽ n converging toṽ in L 1 (P, R).
Next, we prove that there is a ϑ ∈ (0, 1) (ϑ = ) such that Let υ, υ ∈ C(P, R) and φ 1 ∈S(υ). Then there existsṽ 1 (t) ∈ Q(t, υ(t)) such that By (As5), we have So, there existsw(t) ∈ Q(t, υ) such that We construct a set-valued map E : P → O(R) as follows: We see thatṽ 1 and σ = |υ -υ| are measurable, therefore we can conclude that the setvalued map E(t) ∩ Q(t, υ) is measurable. Now, we choose the functionṽ 2 (t) ∈ Q(t, υ) such that We define As a result, we obtain . Therefore Similarly, interchanging the roles of υ and υ, we get SinceS is a contraction, in the light of Covitz and Nadler theorem, we infer thatS has an FP υ which is a solution of (1.2).

Conclusions
We have studied a class of BVPs for Hilfer FDIs with nonlocal fractional IBC. Indeed, we acquired the existence of solutions by taking into account the cases when the set-valued map has convex or nonconvex values. The Leray-Schauder FPT was applied in the case of a convex set-valued map, whereas the FPT due to Nadler and Covitz concerning setvalued contractions was used in the case of a nonconvex set-valued map. The acquired results have been well demonstrated by numerous pertinent examples. We assert that our obtained findings are novel in the framework of Hilfer FDIs with Erdélyi-Kober fractional nonlocal IBC and they greatly contribute to the existing literature on this topic.