Multi-Point Boundary Value Problems for ( k , φ ) -Hilfer Fractional Differential Equations and Inclusions

: In this paper we initiate the study of boundary value problems for fractional differential equations and inclusions involving ( k , φ ) -Hilfer fractional derivative of order in ( 1,2 ] . In the single-valued case the existence and uniqueness results are established by using classical fixed-point theorems, such as Banach, Krasnoselski˘i and Leray-Schauder. In the multivalued case we consider both cases, when the right-hand side has convex or non-convex values. In the first case, we apply the Leray– Schauder nonlinear alternative for multivalued maps, and in the second, the Covit–Nadler fixed-point theorem for multivalued contractions. All results are well illustrated by numerical examples.


Introduction and Preliminaries
Fractional calculus and fractional differential equations have cashed substantial consideration owing to the broad applications of fractional derivative operators in the mathematical modelling, describing many real world processes more accurately than the classicalorder differential equations. For a systematic development of the topic, see the monographs [1][2][3][4][5][6][7][8][9]. Fractional derivative operators are usually defined via fractional integral operators. In the literature, many fractional derivative operators have been proposed, such as Riemann-Liouville, Caputo, Hadamard, Erdélyi-Kober and Hilfer fractional operators, to name a few. The Riemann-Liouville fractional integral operator of order α > 0 is one of the most used and studied operators, defined by The Riemann-Liouvile and Caputo fractional derivative operators of order α > 0 are defined in light of the above definition by and C D α a+ f(w) = I n−α a+ D n f(w) = 1 Γ(n − α) w a (w − u) n−α−1 f (n) (u)du, w > a, respectively, where n − 1 < α ≤ n and n ∈ N. In [10], the Riemann-Liouville fractional integral operator was extended to k-Riemann-Liouville fractional integral of order α > 0 (α ∈ R) as k I α a+ h(w) = where h ∈ L 1 ([a, b], R), k > 0 and Γ k is the k-Gamma function for w ∈ C with (w) > 0 and k ∈ R, k > 0 which is defined in [11] by Γ k (w) = ∞ 0 s w−1 e − s k k ds.
The following relations are well known.
In [16] the authors proved several properties of (k, φ)-Hilfer fractional derivative operator. Moreover they studied the following nonlinear initial value problem involving (k, φ)-Hilfer fractional derivative of the form where k,H D α,β;φ denotes the (k, φ)-Hilfer fractional derivative operator of order α, 0 < α ≤ 1 and parameter β, 0 ≤ β ≤ 1, and f : [a, b] × R → R is a continuous function. By applying Banach's fixed point theorem they proved the existence of a unique solution for the problem (15). In the present work, motivated by the paper [16], we study boundary value problems involving (k, φ)-Hilfer fractional derivative operator of order α and parameter β, where 1 < α ≤ 2 and 0 ≤ β ≤ 1. To be more precisely, we consider in this paper the following (k, φ)-Hilfer fractional boundary value problem with nonlocal multipoint boundary conditions of the form where k,H D α,β;φ denotes the (k, φ)-Hilfer fractional derivative operator of order α, 1 < α < 2 and parameter β, . . , m. Our aim in this paper is to establish results concerning existence and uniqueness, by using Banach's and Krasnoselskiȋ's fixed point theorems, as well as a Leray-Schauder nonlinear alternative. Next, we also study the multivalued problem in which F : [a, b] × R → P (R) is a multivalued map and the other parameters are as in problem (16). Here, P (R) denotes the family of all nonempty subsets of R. We will study both cases, when the right-hand side is convex or nonconvex valued, and we will establish existence results by using Leray-Schauder nonlinear alternative for multivalued maps and the Covitz-Nadler fixed-point theorem for multivalued contractions, respectively. Numerical examples are constructed illustrating the applicability of our obtained theoretical results.
The rest of our paper is organized as follows. In Section 2, we prove an ancillary result toward a linear variant of the (k, φ)-Hilfer fractional nonlocal boundary value problem (16). This lemma is important to transform the nonlinear boundary value problem (16) into an equivalent fixed-point problem. The main results for the single valued (k, φ)-Hilfer fractional nonlocal boundary value problem (16) are included in Section 3, while the results for the multivalued (k, φ)-Hilfer fractional nonlocal boundary value problem (17) are presented in Section 4. Finally, Section 5 is dedicated to illustrative examples.

An Auxiliary Result
In this section an auxiliary result is proved, which is the basic tool in transforming the nonlinear problem (16) into a fixed-point problem, and dealing with a linear variant of the problem (16). First we recall two useful lemmas.
Then the function ϑ ∈ C([a, b], R) is a solution of the boundary value problem if and only if Proof. Assume that ϑ is a solution of the boundary value problem (19). Operating fractional integral k I α;φ on both sides of equation in (19) and using Lemmas 1 and 2, we obtain Consequently where From the boundary condition ϑ(a) = 0 we get c 2 = 0, since θ k k − 2 < 0 by Remark 1. From the second boundary condition Replacing the values of c 0 and c 1 in (21), we get the solution (20). We can prove easily the converse by direct computation. The proof is finished.

The Single Valued Problem
It should be noticed that the solutions of the nonlocal (k, φ)-Hilfer fractional boundary value problem (16) will be fixed points of A.
For convenience we put:

Existence of a Unique Solution
In our first result we will prove the existence of a unique solution of the problem (16). The basic tool is the Banach's contraction mapping principle [24]. Theorem 1. Assume that: Then the (k, φ)-Hilfer nonlocal multi-point fractional boundary value problem (16) has a unique solution on [a, b], provided that LG < 1, where G is defined by (23).
Proof. We transform the (k, φ)-Hilfer nonlocal multipoint fractional boundary value problem (16) into a fixed-point problem, with the help of the operator A defined in (22). Then, we shall show that the operator A has a unique fixed point.
In the first step we will show that AB r ⊂ B r . We have, for ϑ ∈ B r , using (H 1 ), that For any ϑ ∈ B r , we have Consequently Aϑ ≤ r and thus AB r ⊂ B r . Now we will show that A is a contraction. For w ∈ [a, b] and ϑ, Hence Ax − Ay ≤ LG x − y which implies that A is a contraction, since LG < 1. By the Banach's contraction-mapping principle, the operator A has a unique fixed point, which is the unique solution of (k, φ)-Hilfer nonlocal multipoint fractional boundary value problem (16). The proof is finished.

Existence Results
In the forthcoming theorems we will prove existence results for the (k, φ)-Hilfer nonlocal multipoint fractional boundary value problem (16), utilizing Krasnoselskiȋ's fixed point theorem [25] and nonlinear alternative of Leray-Schauder type [26].
Theorem 2. Let f : [a, b] × R → R be a continuous function satisfying (H 1 ). In addition we assume that: Then the (k, φ)-Hilfer nonlocal multi-point fractional boundary value problem (16) has at least one solution on [a, b], if G 1 L < 1, where For any ϑ, y ∈ B ρ , we have Therefore (A 1 ϑ) + (A 2 y) ≤ ρ, which shows that A 1 ϑ + A 2 y ∈ B ρ . Next we show that A 2 is a contraction mapping. We omit the details since it is easy by using (26).
The operator A 1 is continuous, since f is continuous. Moreover, A 1 is uniformly bounded on B ρ as To prove the compactness of the operator A 1 , we consider w 1 , which tends to zero as w 2 − w 1 → 0, independently of ϑ. Thus, A 1 is equicontinuous. By the Arzelá-Ascoli theorem, A 1 is completely continuous. By Krasnoselskiȋ's fixed-point theorem the (k, φ)-Hilfer nonlocal multipoint fractional boundary value problem (16) Then the (k, φ)-Hilfer nonlocal multipoint fractional boundary value problem (16) has at least one solution on [a, b].

Proof.
In the first step we will show that the operator A maps bounded sets into bounded set in C([a, b], R), where A is defined by (22). For r > 0, let and consequently, Ax ≤ χ(r) σ G.

Now we will show that A maps bounded sets into equicontinuous sets of
As w 2 − w 1 → 0 the right-hand side of the above inequality tends to zero independently of ϑ ∈ B r . Hence, the operator A : C([a, b], R) → C([a, b], R) is completely continuous, by the Arzelá-Ascoli theorem.
Finally we will show the boundedness of the set of all solutions to equations ϑ = λAϑ for λ ∈ (0, 1).
Let ϑ be a solution. Then, for w ∈ [a, b], and working as in the first step, we have In view of (H 4 ), there exists K such that ϑ = K. Let us set We see that the operator A :Ū → C([a, b], R) is continuous and completely continuous. There is no ϑ ∈ ∂U such that ϑ = λAϑ for some λ ∈ (0, 1), from the choice of U. By the nonlinear alternative of Leray-Schauder type, we deduce that A has a fixed point ϑ ∈Ū, which is a solution of the (k, φ)-Hilfer nonlocal multipoint fractional boundary value problem (16). This completes the proof.
For details of multivalued analysis we refer the reader to [27,28]. See also [7]. The set of selections of F, for each ϑ ∈ C([a, b], R), is defined by

Definition 1. A function ϑ ∈ C([a, b], R) is said to be a solution of the (k, φ)-Hilfer nonlocal multipoint fractional boundary value problem (17) if there exists a function
In the first existence result, which concern the case when F has convex values, we apply nonlinear alternative of Leray-Schauder type [26] with the assumption that F is is upper semicontinuous for almost all w ∈ [a, b] and (iii) for each r > 0, there exists a function m r ∈ L 1 ([a, b], R + ) such that F(w, u) = sup{|v| : v ∈ F(w, u)} < m r (w), for each u ∈ R with |u| ≤ r and for almost every w ∈ [a, b]. Theorem 4. Assume that:

continuous nondecreasing function and a continuous
positive function q such that Then the (k, φ)-Hilfer nonlocal multi-point fractional boundary value problem (17) has at least one solution on [a, b].
Proof. We define an operator F : C([a, b], R) −→ P (C([a, b], R)) by and v ∈ S F,ϑ . It is obvious that the solutions of the (k, φ)-Hilfer nonlocal multipoint fractional boundary value problem (17) are the fixed points of F . We will give the proof in several steps.
We omit the proof, because it is obvious, since F has convex values and thus S F,ϑ is convex.
Step 2. F maps the bounded sets into bounded sets in C([a, b], R).
Then, for w ∈ [a, b], we have and consequently, h ≤ z(r) q G.
Step 3. F maps bounded sets into equicontinuous sets of C([a, b], R).
By virtue of the Proposition 1.2 of [24], it is enough to prove that the F has a closed graph, which will imply that F is upper semicontinuous multivalued mapping.
Step 4. F has a closed graph.
Let ϑ n → ϑ * , h n ∈ F (ϑ n ) and h n → h * . Then we need to show that h * ∈ F (ϑ * ). Associated with h n ∈ F (ϑ n ), there exists v n ∈ S F,ϑ n such that for each w ∈ [a, b], Thus it suffices to show that there exists v * ∈ S F,ϑ * such that for each w ∈ [a, b], Let us consider the linear operator Θ : Observe that h n − h * → 0, as n → ∞. Therefore, it follows by a Lazota-Opial result [29], that Θ • S F is a closed-graph operator. Further, we have h n (w) ∈ Θ(S F,ϑ n ). Since ϑ n → ϑ * , we have for some v * ∈ S F,ϑ * .
Let ν ∈ (0, 1) and ϑ ∈ νF (ϑ). Then there exists v ∈ L 1 ([a, b], R) with v ∈ S F,ϑ such that, for w ∈ [a, b], we have Working as in second step, we have In view of (H 3 ), there exists K such that ϑ = K. Let us set The operator F : U → P (C([a, b], R)) is a compact multivalued map, upper semicontinuous with convex closed values. There is no ϑ ∈ ∂U such that ϑ ∈ νF (ϑ) for some ν ∈ (0, 1), from the choice of U .
By the nonlinear alternative of Leray-Schauder type F has a fixed point ϑ ∈ U which is a solution of the (k, φ)-Hilfer nonlocal multi-point fractional boundary value problem (17). This ends the proof.
In our second result, the existence of solutions for the (k, φ)-Hilfer nonlocal multipoint fractional boundary value problem (17) is showed when F is not necessarily nonconvex valued by using a fixed-point theorem for multivalued contractive maps due to Covitz and Nadler [30].
Theorem 5. Assume that the following conditions hold: Proof. By the assumption (A 1 ), the set S F,ϑ is nonempty for each ϑ ∈ C([a, b], R). Hence F has a measurable selection (see Theorem III.6 [31]). We show that F (ϑ) Then u ∈ C([a, b], R) and there exists v n ∈ S F,ϑ n such that, for each w ∈ [a, b], As F has compact values, we pass onto a subsequence (if necessary) to obtain that v n converges to v in L 1 ([a, b], R). Thus, v ∈ S F,ϑ and for each w ∈ [a, b], we have Hence, u ∈ F (ϑ). Next we show that So, there exists ω ∈ f(w,x(w)) such that Since the multivalued operator U(w) ∩ F(w,θ(w)) is measurable (Proposition III.4 [31]), there exists a function v 2 (w) which is a measurable selection for U. So v 2 (w) ∈ F(w,θ(w)) and for each w ∈ [a, b], we have |v 1 For each w ∈ [a, b], let us define Thus, Hence Analogously, interchanging the roles of x andx, we obtain So F is a contraction and by Covitz and Nadler theorem F has a fixed point ϑ which is a solution of the (k, φ)-Hilfer nonlocal multipoint fractional boundary value problem (17). This completes the proof.

Conclusions
In the present research, we have investigated fractional boundary value problems consisting of (k, φ)-Hilfer fractional differential equations and inclusions, supplemented by nonlocal multipoint boundary conditions. First we considered the single valued case. After transforming the given problem into a fixed-point problem, we applied the Banach contraction-mapping principle, the Krasnoselskiȋ fixed-point theorem and the Leray-Schauder nonlinear alternative and established existence and uniqueness results. After that, we studied the multivalued case. We considered both cases, convex-valued and nonconvex-valued multivalued maps. In the first case, we established an existence result via a Leray-Schauder nonlinear alternative for multivalued maps, while in the second case the Covitz-Nadler fixed-point theorem for contractive multivalued maps was applied. Numerical examples illustrating the theoretical results are also presented. The used methods are standard, but their configuration in (k, φ)-Hilfer nonlocal multipoint fractional boundary value problems is new. To the best of our knowledge, our results in this paper are the only concerning boundary value problems involving (k, φ)-Hilfer fractional differential equations and inclusions of order in (1,2]. Hence our results will enrich the literature on this new research area.