Boundary Value Problems for ψ -Hilfer Type Sequential Fractional Differential Equations and Inclusions with Integral Multi-Point Boundary Conditions

: In the present article, we study a new class of sequential boundary value problems of fractional order differential equations and inclusions involving ψ -Hilfer fractional derivatives, supplemented with integral multi-point boundary conditions. The main results are obtained by employing tools from ﬁxed point theory. Thus, in the single-valued case, the existence of a unique solution is proved by using the classical Banach ﬁxed point theorem while an existence result is established via Krasnosel’ski˘i’s ﬁxed point theorem. The Leray–Schauder nonlinear alternative for multi-valued maps is the basic tool to prove an existence result in the multi-valued case. Finally, our results are well illustrated by numerical examples. conditions. Fractional differential equations and inclusions are considered. Existence and uniqueness results are established in the single-valued case, by using the classical Banach and Krasnosel’ski˘i ﬁxed point theorems. In the multi-valued case, an existence result is proved by using Leray–Schauder nonlinear alternative for multi-valued maps. Illustrative examples are presented to show the validity of our main results. The present work is innovative and interesting, and signiﬁcantly contributes to the available material on ψ -Hilfer fractional differential equations and inclusions.


Introduction
Fractional-order differentiations and integrations are more accurate tools in expressing real-world problems as compared to integer-order differentiations and integrations. Thus, the theory of fractional differential equations has attracted a lot of attention from many researchers for their wide applications to various fields, such as in physics, bioengineering, electrochemistry, and so on; see [1][2][3] and related references therein. The interested reader is referred to the monographs [4][5][6][7][8][9][10][11] for the basic theory of fractional calculus and fractional differential equations.
In the literature, there are several definitions of derivatives and integrals of arbitrary orders. For instance, Kilbas et al. in [5] introduced fractional integrals and fractional derivatives concerning another function. In a recent paper, Almeida [12] introduced the socalled ψ-Caputo fractional operator. Numerous interesting results concerning the existence, uniqueness, and stability of initial value problems and boundary value problems for fractional differential equations with ψ-Caputo fractional derivatives by applying different types of fixed-point techniques were obtained in [13][14][15].
Hilfer in [16] generalized both Riemann-Liouville and Caputo fractional derivatives, known as the Hilfer fractional derivative. We refer to [17,18], and references cited therein, for some properties and applications of the Hilfer fractional derivative and to [19][20][21] for initial value problems involving Hilfer fractional derivatives.
Fractional differential equations involving Hilfer derivative have many applications, and we refer to [22] and the references cited therein. There are actual world occurrences with uncharacteristic dynamics such as atmospheric diffusion of pollution, signal transmissions through strong magnetic fields, the effect of the theory of the profitability of stocks in economic markets, the theoretical simulation of dielectric relaxation in glass forming materials, network traffic, and so on. See [23,24] and references cited therein.
In [25], the authors initiated the study of nonlocal boundary value problems for the Hilfer fractional derivative, by studying the boundary value problem of Hilfer-type fractional differential equations with nonlocal integral boundary conditions x(a) = 0, where H D α,β is the Hilfer fractional derivative of order α, 1 < α < 2 and parameter β, 0 ≤ β ≤ 1, I ϕ i is the Riemann-Liouville fractional integral of order ϕ i > 0, ξ i ∈ [a, b], a ≥ 0 and δ i ∈ R. Several existence and uniqueness results were proved by using a variety of fixed point theorems.
In a series of papers [26][27][28][29][30], nonlocal boundary value problems involving Hilfer fractional derivatives were studied, with a variety of boundary conditions. Thus, the authors in [26] studied Hilfer Langevin three-point fractional boundary value problems, the authors in [27] studied pantograph Hilfer fractional boundary value problems with nonlocal integral boundary conditions, the authors in [28] studied Hilfer fractional boundary value problems with nonlocal integral integro-multipoint boundary conditions, the authors in [29] studied Hilfer fractional boundary value problems with nonlocal multipoint, fractional derivative multi-order, and fractional integral multi-order boundary conditions, and the authors in [30] studied sequential Hilfer fractional boundary value problems with nonlocal integro-multipoint boundary conditions. Systems of Hilfer-Hadamard sequential fractional differential equations were studied in [31].
In the present paper, motivated by the research going on in this direction, we study a new class of boundary value problems of sequential Hilfer-type fractional differential equations involving integral multi-point boundary conditions of the form Here, H D α,β;ψ is the ψ-Hilfer fractional derivative operator of order α, 1 < α < 2 and parameter β, . . , n, j = 1, 2, . . . m and ψ is a positive increasing function on (a, b], which has a continuous derivative ψ (t) on (a, b).
We also cover the multi-valued case of the problem (3) by considering the following inclusion problem: where F : [a, b] × R → P (R) is a multi valued function, and (P (R) is the family of all nonempty subjects of R).
At the end of this section, we mention that the remaining part of the paper will be organized as follows. In Section 2, we recall some basic concepts of fractional calculus. In Section 3, we prove first a lemma relating a linear variant of the problem in (3) with an integral equation. Moreover, the existence and uniqueness results are established, in the single valued case, by using fixed point theorems. We obtain the existence of a unique solution via Banach's contraction mapping principle, while Krasnosel'skiȋ's fixed point theorem is applied to obtain the existence result for the sequential Hilfer fractional boundary value problem (3). In Section 4, an existence result is proved for the sequential Hilfer inclusion boundary value problem (4), via Leray-Schauder nonlinear alternative for multi-valued maps. Illustrative examples for the main results are provided.

Preliminaries
This section is assigned to recall some notation in relation to fractional calculus. Throughout the paper, C([a, b], R) denotes the Banach space of all continuous functions from [a, b] into R with the norm defined by x = sup{|x(t)| : t ∈ [a, b]}. We denote by AC n ([a, b], R) the n-times absolutely continuous functions given by be a finite or infinite interval of the half-axis (0, ∞) and α > 0. In addition, let ψ(t) be a positive increasing function on (a, b], which has a continuous derivative ψ (t) on (a, b). The ψ-Riemann-Liouville fractional integral of a function f with respect to another function ψ on [a, b] is defined by where Γ(·) represents the Gamma function.

Definition 2 ([5]
). Let ψ (t) = 0 and α > 0, n ∈ N. The Riemann-Liouville derivatives of a function f with respect to another function ψ of order α correspondent to the Riemann-Liouville is defined by where n = [α] + 1, [α] represents the integer part of the real number α. This is the greatest integer n such that n ≤ α.

Lemma 1 ([5]
). Let α, ρ > 0. Then, we have the following semigroup property given by Next, we present the ψ-fractional integral and derivatives of a power function.

Existence and Uniqueness Results for Problem (3)
The following auxiliary lemma concerning a linear variant of the sequential Hilfer boundary value problem (3) plays a fundamental role in establishing the existence and uniqueness results for the given nonlinear problem.
Lemma 3. Let a ≥ 0, 1 < α < 2, 0 ≤ β ≤ 1, γ = α + 2β − αβ be given constants and For a given h ∈ C([a, b], R), the unique solution of the sequential Hilfer linear fractional boundary value problem is given by Proof. Applying the operator I α;ψ a+ on both sides of Equation (12) and using Lemma 2, there exist real numbers c 0 and c 1 such that From the boundary condition x(a) = 0, we see c 0 = 0. Then, we get From Substituting the values of c 1 in (15), we obtain the solution (14). That the function x(t), as defined in formula (14), solves the boundary value problem in (12), (13) can be proved by direct computation. This finishes the proof of Lemma 3.

Remark 1.
If ψ(t) = t and β = 0, then (12) reduces to which is the Riemann-Liouville fractional differential equation, where If ψ(t) = log e t and β = 0, then (12) is transformed to the Hadamard fractional differential equation of the form: where Next, in view of Lemma 3, we define an operator The continuity of f shows that A is well defined and fixed points of the operator equation x = Ax are solutions of the integral Equation (14) in Lemma 3. In the sequel, we use the following abbreviations: and By using classical fixed point theorems, we establish in the following subsections existence, as well as existence and uniqueness results, for the sequential ψ-Hilfer fractional boundary value problem (3).
In our first result, we prove the existence of a unique solution of the sequential ψ-Hilfer fractional boundary value problem (3) based on Banach's fixed point theorem [33].

Theorem 1. Assume that:
(H 1 ) There exists a finite number L > 0 such that, for all t ∈ [a, b] and for all x, y ∈ R, the following inequality is valid: Then, the sequential ψ-Hilfer fractional boundary value problem (3) has a unique solution on [a, b] provided that where Ω and Ω 1 are defined by (17) and (18), respectively.
Proof. With the help of the operator A defined in (16), we transform the sequential ψ-Hilfer fractional boundary value problem (3) into a fixed point problem, x = Ax. By applying the Banach contraction mapping principle, we shall show that A has a unique fixed point.
For any x ∈ B r , we have and consequently Ax ≤ r, which implies that AB r ⊂ B r .
which implies that Ax − Ay ≤ (LΩ + Ω 1 ) x − y . As LΩ + Ω 1 < 1, A is a contraction. Therefore, by the Banach's contraction mapping principle, we deduce that A has a fixed point. Obviously, this is the unique solution of the sequential ψ-Hilfer fractional boundary value problem (3). The proof is complete now.
The next existence result is based on the a classical fixed point theorem due to Krasnosel'skiȋ's [34].

Proof. We consider
For any x, y ∈ B ρ , we have Therefore, A 1 x + A 2 y ≤ ρ, which shows that A 1 x + A 2 y ∈ B ρ . It is easy to see, using the condition Ω 1 < 1, that A 2 is a contraction mapping.
The operator A 1 is continuous because f is continuous. In addition, A 1 is uniformly bounded on B ρ because we have The compactness of the operator A 1 is proved now. Let t 1 , as t 2 − t 1 → 0, and is independent of x.
(i) Let the function f (t, x) be given by Then, we can check the Lipchitz condition of f (t, x) as for all x, y ∈ R, t ∈ [1/4, 11/4]. By setting a constant L = 1/2000, we obtain LΩ + Ω 1 ≈ 0.9750871662 < 1, which claims that inequality (19) is fulfilled. Therefore, by an application of Theorem 1, the boundary value problem for ψ-Hilfer type sequential fractional differential equation with integral multi-point boundary conditions (21) with (22) has a unique solution on [1/4, 11/4]. Note that the Theorem 2 can not be applied to this problem because the given function in (22) It is obvious that the function f (t, x) is bounded by which satisfy condition (H 2 ) in Theorem 2.
, and there exists a function v ∈ L 1 ([a, b], R) such that v(t) ∈ F(t, x(t)) a.e. on [a, b] and In the next theorem, we prove the existence of solutions of the sequential Hilfer inclusion fractional boundary value problem (4) when the multi-valued map F has convex values assuming that it is L 1 -Carathéodory, that is, for all x ∈ R with x ≤ α and for a.e. t ∈ [a, b].
For each x ∈ C([a, b], R), denote the set of selections of F by For a normed space (X, · ), let P cp,c (X) = {Y ∈ P (X) : Y is compact and convex}. The following lemma is used in the sequel.
Then, the sequential Hilfer inclusion fractional boundary value problem (4) has at least one solution on [a, b].
Proof. We define an operator N : C([a, b], R) −→ P (C([a, b], R)) by for v ∈ S F,x , in order to transform the problem (4) into a fixed point problem. Clearly, the solutions of the boundary value problem (4) are fixed points of N . Our proof strategy is to show that all conditions of Leray-Schauder nonlinear alternative for multi-valued maps [39] are satisfied and, consequently, we conclude that sequential Hilfer inclusion fractional boundary value problem (4) has at least one solution on [a, b]. We will give the proof in several steps.
Step 1: N (x) is convex for all x ∈ C([a, b], R).
Step 2: Bounded sets are mapped by N into bounded sets in C ([a, b], R).
Then, for t ∈ [a, b], we have Thus, Step 3: Bounded sets are mapped by N into equicontinuous sets.
Let t 1 , t 2 ∈ [a, b] with t 1 < t 2 and x ∈ B r . For each h ∈ N (x), we obtain as t 2 − t 1 → 0, and is independent of x ∈ B r . By the Arzelá-Ascoli theorem, it follows that In the next step, we will prove that N is upper semicontinuous. In order to reach the desired conclusion, we have to recall from [35], Proposition 1.2 that a completely continuous operator is upper semicontinuous if it has a closed graph. Therefore, we will show the following result.
Step 4: N has a closed graph.
Consider x n → x * , h n ∈ N (x n ) and h n → h * . Then, we will show that h * ∈ N (x * ). From h n ∈ N (x n ), there exists v n ∈ S F,x n such that, for each t ∈ [a, b], h n (t) = I α;ψ We must show that there exists v * ∈ S F,x * such that, for each t ∈ [a, b], Consider the linear operator Θ : Observe that h n (t) − h * (t) → 0 as n → ∞. By Lemma 4 that Θ • S F is a closed graph operator. Moreover, we have h n (t) ∈ Θ(S F,x n ). Since x n → x * , we have that for some v * ∈ S F,x * .
Let x ∈ θN (x) for some θ ∈ (0, 1). Then, there exists v ∈ L 1 ([a, b], R) with v ∈ S F,x such that, for t ∈ [a, b], we have Following the computation as in Step 2, we have for each t ∈ [a, b], In view of (A 3 ), there exists M such that x = M. Consider Note that N : U → P (C([a, b], R)) is a compact, upper semicontinuous multi-valued map with convex closed values, and there is no x ∈ ∂U such that x ∈ θN (x) for some θ ∈ (0, 1), from the choice of U. By the Leray-Schauder nonlinear alternative for multivalued maps [39], we deduce that N has a fixed point x ∈ U, which is a solution of the sequential Hilfer inclusion fractional boundary value problem (4). This completes the proof.

Conclusions
In this work, we studied a new class of ψ-Hilfer sequential boundary value problems of fractional order, supplemented with integral multi-point boundary conditions. Fractional differential equations and inclusions are considered. Existence and uniqueness results are established in the single-valued case, by using the classical Banach and Krasnosel'skiȋ fixed point theorems. In the multi-valued case, an existence result is proved by using Leray-Schauder nonlinear alternative for multi-valued maps. Illustrative examples are presented to show the validity of our main results. The present work is innovative and interesting, and significantly contributes to the available material on ψ-Hilfer fractional differential equations and inclusions.