Three-Point Boundary Value Problems for the Langevin Equation with the Hilfer Fractional Derivative

Department of Mechanical Engineering Technology, College of Industrial Engineering Technology, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand Nonlinear Analysis and Applied Mathematics (NAAM)–Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand


Introduction
Fractional calculus is an emerging field in applied mathematics that deals with derivatives and integrals of arbitrary orders. For details and applications, we refer the reader to the texts in [1][2][3][4][5][6]. In the literature, there exist several definitions of fractional integrals and derivatives, from the most popular Riemann-Liouville and Caputo-type fractional derivatives to others such as the Hadamard fractional derivative and the Erdeyl-Kober fractional derivative. A generalization of both the Riemann-Liouville and Caputo derivatives was given by Hilfer in [7], which is known as the Hilfer fractional derivative D α,β xðtÞ of order α and type β ∈ ½0, 1. One can observe that the Hilfer fractional derivative interpolates between the Riemann-Liouville and Caputo derivatives as it reduces to the Riemann-Liouville and Caputo fractional derivatives for β = 0 and β = 1, respectively. Some properties and applications of the Hilfer derivative can be found in [8,9] and references cited therein.
One of the important equations governing several phenomena occurring in physical sciences and electrical engineering is the Langevin differential equation, first formulated by Langevin in 1908 [10]. In recent years, several fractional variants of the Langevin equation have been introduced and studied; see, for example, [11][12][13][14][15][16][17][18][19] and the references cited therein.
Initial value problems involving the Hilfer fractional derivatives were studied by several authors; see for example [20][21][22]. Nonlocal boundary value problems for the Hilfer fractional differential equation have been discussed in [23]. In [24], the authors proved some results for initial value problems of the Langevin equation with the Hilfer fractional derivative.
Exploring the literature on fractional order boundary value problems, we find that there does not exist any work on boundary value problems of the Langevin equation with the Hilfer fractional derivative. Motivated by this observation, we fill this gap by introducing a new class of boundary value problems of the Hilfer-type Langevin fractional differential equation with three-point nonlocal boundary conditions. In precise terms, we investigate the existence and uniqueness criteria for the solutions of the following nonlocal boundary value problem: x a ð Þ = 0, where H D α i ,β i , i = 1, 2 is the Hilfer fractional derivative of In order to study problem (1)-(2), we convert it into an equivalent fixed-point problem and then use Banach's fixed-point theorem to prove the uniqueness of its solutions. We also obtain two existence results for problem (1)-(2) by applying the nonlinear alternative of the Leray-Schauder type [25] and Krasnoselskii's fixed-point theorem [26].
As a second problem, we switch onto the multivalued analogue of (1) and (2) by considering the inclusion problem: x a ð Þ = 0, where F : ½a, b × ℝ ⟶ P ðℝÞ is a multivalued map (P ðℝÞ is the family of all nonempty subjects of ℝ).
Existence results for problem (3)-(4) with convex and nonconvex valued maps are respectively derived by applying the nonlinear alternative for Kakutani's maps and Covitz and Nadler's fixed-point theorem for contractive maps.
The rest of the paper is organized as follows: Section 3 contains the main results for problem (1)-(2), while the existence results for problem (3)-(4) are presented in Section 4. We recall the related background material in Section 2.

Preliminaries
In this section, we introduce some notations and definitions of fractional calculus and multivalued analysis and present preliminary results needed in our proofs later [1]. Definition 1. The Riemann-Liouville fractional integral of order α > 0 for a continuous function u : ½a,∞Þ ⟶ ℝ is defined by provided that the right-hand side exists on ða, ∞Þ.
Definition 2. The Riemann-Liouville fractional derivative of order α > 0 of a continuous function u is defined by where n = ½α + 1, ½α denotes the integer part of real number α, provided that the right-hand side is point-wise defined on ða, ∞Þ.
Definition 3. The Caputo fractional derivative of order α > 0 of a continuous function u is defined by provided that the right-hand side is point-wise defined on ða, ∞Þ.
Then, the function x is a solution of the boundary value problem: x a ð Þ = 0, if and only if where it is assumed that Proof. Applying the Riemann-Liouville fractional integral of order α 1 to both sides of (12), we obtain by using Lemma 6 where c 0 is an arbitrary constant and ð2 − α 1 Þð1 − β 1 Þ = 2 − γ 1 . Applying the Riemann-Liouville fractional integral of order α 2 to both sides of (16), we obtain Applying Lemma 6 to (17), we obtain Using xðaÞ = 0 in (18), we obtain c 1 = 0, and hence we get Next, combining the condition xðbÞ = θxðηÞ with (19), we have Substituting the value of c 0 in (19) yields the solution (14). The converse follows by direct computation. This completes the proof.

Existence and Uniqueness Results for Single-Valued Problem (1)-(2)
In view of Lemma 7, we define an operator A : C ⟶ C associated with problem (1)-(2) by where C = Cð½a, b, ℝÞ denotes the Banach space of all continuous functions from ½a, b into ℝ with the norm kxk ≔ sup fjxðtÞj: t ∈ ½a, bg. One can observe that the existence of a fixed point of operator A implies the existence of a solution for problem (1)- (2). For computational convenience, we introduce the following notations:

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Now, we present our main results for boundary value problem (1)- (2). Our first existence result is based on the well-known Krasnoselskii's fixed-point theorem [26].
Theorem 8. Assume that the following conditions hold: where Ω 2 is given by (23) Then, there exists at least one solution for problems (1) and (2) on ½a, b: Proof. In order to verify the hypothesis of Krasnoselskii's fixed-point theorem [26], we split operator A defined by (21) into the sum of two operators A 1 and A 2 on the closed and For any x, y ∈ B ρ , we have This shows that A 1 x + A 2 y ∈ B ρ . By using (H 2 ), it is easy to establish that A 2 is a contraction mapping.
The Leray-Schauder Nonlinear Alternative [25] is used for our next existence result.
Proof. Let us verify that operator A defined by (21) satisfies the hypothesis of the Leray-Schauder Nonlinear Alternative [25]. In our first step, we establish that operator A maps bounded sets (balls) into a bounded set in C. For a number r > 0, let B r = fx ∈ C : kxk ≤ rg be a bounded ball in C. Then, for t ∈ ½a, b, we have and consequently, Next, we will show that A maps bounded sets into equicontinuous sets of C. Let τ 1 , τ 2 ∈ ½a, b with τ 1 < τ 2 and x ∈ B r . Then we have Observe that the right-hand side of the above inequality tends to zero independently of x ∈ B r as τ 2 − τ 1 ⟶ 0. Thus, the set AB r is equicontinuous. Therefore, the Arzelá-Ascoli theorem applies and hence operator A is completely continuous.
Finally, we show that the set of all solutions to equations x = λAx is bounded for λ ∈ ð0, 1Þ.
Following the computation in the first step, we obtain which yields According to (H 4 ), there exists M > 0 satisfying kxk ≠ M. Introduce a set and notice that A : U ⟶ C is continuous and completely continuous. Then, the choice of U implies that there is no x ∈ ∂U, such that x = λAx for some λ ∈ ð0, 1Þ. In consequence, we deduce by the nonlinear alternative of the Leray-Schauder type [25] that A has a fixed-point x ∈ U, which corresponds to a solution of problem (1)-(2). This completes the proof.

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In the following result, we apply Banach's fixed-point theorem to prove the existence of a unique solution of the problem at hand.
Proof. Let us first show that A defined by (21) satisfies which implies that AB r ⊂ B r . Next, we let x, y ∈ C. Then for t ∈ ½a, b, we have which implies that kAx − Ayk ≤ ðLΩ 1 + Ω 2 Þkx − yk. As L Ω 1 + Ω 2 < 1, A is a contraction. Therefore, by Banach's fixed-point theorem, operator A has a fixed point which is indeed a unique solution of problem (1)- (2). The proof is finished.

Existence Results for Multivalued Problems
(3) and (4) Definition 14. A continuous function x is said to be a solution of problem (3)-(4) if xðaÞ = 0, xðbÞ = θxðηÞ and there exists a function v ∈ L 1 ð½a, b, ℝÞ with v ∈ Fðt, xÞ, a.e., on ½a, b such that For each y ∈ Cð½a, b, ℝÞ, define the set of selections of F by Lemma 15 (see [27]). Let X be a separable Banach space. Let F : ½a, b × ℝ ⟶ P cp,c ðXÞ be an L 1 -multivalued map and let Θ be a linear continuous mapping from L 1 ð½a, b, XÞ to Cð½a, b, XÞ. Then the operator 6 Advances in Mathematical Physics is a closed graph operator in Cð½a, b, XÞ × Cð½a, b, XÞ.
Our first existence result, dealing with the convex valued F, is based on the nonlinear alternative of the Leray-Schauder type for (Kakutani) multivalued maps [25] with the assumption that F is Carathéodory. Proof. Let us transform problem (3)-(4) into a fixed-point problem by introducing an operator ℱ : Cð½a, b, ℝÞ ⟶ P ðCð½a, b, ℝÞÞ as for t ∈ ½a, b and v ∈ S F,x . Notice that the existence of a fixed point of ℱ ensures the existence of a solution of problems (1) and (2). This will be achieved by establishing that operator ℱ satisfies the hypothesis of the Leray-Schauder nonlinear alternative for the Kakutani maps [25]. We do it in several steps.
Step 1. Since S F,x is convex (F has convex values), therefore ℱ ðxÞ is convex for each x ∈ Cð½a, b, ℝÞ.
Let B r = fx ∈ Cð½a, b, ℝÞ: kxk ≤ rg be a bounded set in Cð½a, b, ℝÞ. Then, for each h ∈ ℬðxÞ, x ∈ B r , there exists v ∈ S F,x such that