Boundary Value Problem for the Langevin Equation and Inclusion with the Hilfer Fractional Derivative

In this work, we discuss the existence and uniqueness of solution for a boundary value problem for the Langevin equation and inclusion with the Hilfer fractional derivative. First of all, we give some definitions, theorems, and lemmas that are necessary for the understanding of the manuscript. Second of all, we give our first existence result, based on Krasnoselskii’s fixed point, and to deal with the uniqueness result, we use Banach’s contraction principle. +ird of all, in the inclusion case, to obtain the existence result, we use the Leray–Schauder alternative. Last but not least, we give an illustrative example.


Introduction
Fractional derivatives give an excellent description of memory and hereditary properties of different processes. Properties of the fractional derivatives make the fractionalorder models more useful and practical than the classical integral-order models. Several researchers in the recent years have employed the fractional calculus as a way of describing natural phenomena in different fields such as physics, biology, finance, economics, and bioengineering (for more details see [1][2][3][4][5][6][7][8][9][10] and many other references).
With the recent outstanding development in fractional differential equations, the Langevin equation has been considered a part of fractional calculus, and thus, important results have been elaborated [11][12][13][14][15].
An equation of the form md 2 x/dt 2 � λdx/dt + η(t) is called Langevin equation, introduced by Paul Langevin in 1908. e Langevin equation is found to be an effective tool to describe the evolution of physical phenomena in fluctuating environments [8]. For some new developments on the fractional Langevin equation, see, for example, [16][17][18].
In this study, we investigate the existence and uniqueness criteria for the solutions of the following nonlocal boundary value problem: where H D α i ,β i , i � 1, 2 is the Hilfer fractional derivative of order α i , 0 < α i < 1, and parameter β i , 0 ≤ β i ≤ 1, i � 1, 2, 1 < α 1 + α 2 ≤ 2, λ ∈ R, a ≥ 0, I ] i is the Riemann-Liouville fractional integral of order ] i > 0, μ i ∈ R, i � 1, 2, and f: In order to study problem (1), we transform it into a fixed-point problem and then use Krasnoselskii's fixed-point theorem to prove the existence results.
As a second problem, we study the multivalued case of (1) by considering the inclusion problem: is the family of all nonempty subjects of R ). We prove the existence of solution for problem (2) by applying the nonlinear alternative of the Leray-Schauder [19]. Finally, in the last part, we give an example to support our study.

Preliminaries
Let us recall some basic definitions and notations of fractional calculus and multivalued analysis which are needed throughout this study.
e Riemann-Liouville fractional integral of order α > 0 for a continuous function f: [a, ∞) ⟶ R can be defined as provided that the right-hand side exists on (a, ∞).
e Caputo fractional derivative of order α > 0 of a continuous function f is defined by where n − 1 < α < n, n � [α] + 1, and [α] denotes the integer part of the real number α.

Remark 1.
When β � 0, the Hilfer fractional derivative corresponds to the Riemann-Liouville fractional derivative: When β � 1, the Hilfer fractional derivative corresponds to the Caputo fractional derivative: e following lemma plays a fundamental role in establishing the existence results for the given problem.
. en, the function x is a solution of the boundary value problem: International Journal of Differential Equations if and only if where Proof. Applying the Riemann-Liouville fractional integral of order α 1 to both sides of (10), we obtain by using Lemma 1 where c 0 is the constant and c 1 � α 1 + β 1 − α 1 β 1 . Applying the Riemann-Liouville fractional integral of order α 2 to both sides of (13), we obtain by using Lemma 1 From using the boundary condition x(a) � 0 in (14), we obtain that c 1 � 0. en, we get From using the boundary condition (15), we find Substituting the value of (c 0 ) in (13), we obtain the solution (11).
Conversely, suppose that x is the solution of the fractional integral (11). By applying fractional derivetive H D α 2 ,β 2 on both sides of (11) and then applying fractional derivative, we obtain that It follows that Now, we will prove that x satisfies the boundary conditions; for that, we have x(a) � 0, and from (11), we have By (12), we get International Journal of Differential Equations 3 is completes the proof.
For the basic concepts of multivalued analysis, we refer to ( [2,3]).
for all x ∈ R with ‖x‖ ≤ ρ and for a.e. t ∈ [a, b]. Fixed-point theorems play a major role in establishing the existence theory for problem (1) and problem (2). We collect here some well-known fixed-point theorems used in this study. [19]. Let M be a closed, bounded, convex, and nonempty subset of a Banach space. Let A, B be the operators such that (i) Ax + By ∈ M whenever x, y ∈ M (ii) A is compact and continuous (iii) B is contraction mapping en, there exists z ∈ M, such that z � Az + Bz.

Theorem 1 Krasnoselskii's fixed-point theorem
Theorem 2 (Leray-Schauder nonlinear alternative [19]). Let X be a Banach space, C a closed, convex subset of X, U an open subset of C, and 0 ∈ U. Suppose that F: U ⟶ C is a continuous, compact (F(U) is a relatively compact subset of C ) map. en, either (i) F has a fixed point in U or (ii) ere exists x ∈zU (the boundary of U in C) and θ ∈ (0, 1) with x � θF(x).

Existence and Uniqueness Results for Problem (1)
In this section, we deal with the existence and uniqueness of solution for the boundary value problem (1). By Lemma 2, we define an operator A: C ⟶ C by where To simplify the computations, we use the following notations: Our first result is an existence result, based on wellknown Krasnoselskii's fixed-point theorem.

Theorem 3 Assume that
where Ω 2 is given by (19) en, there exists at least one solution for the boundary value problem (1) on [a, b].
Proof. We will show that the operator A defined by (17) satisfies the assumptions of Krasnoselskii's fixed-point theorem. We split the operator A into the sum of two operators A 1 and A 2 on the closed ball For any x, y ∈ B ρ , we have and hence, ‖A 1 x + A 2 y‖ ≤ ρ, which implies that Next, by using (H2), we show that A 2 is a contraction mapping. Let x, y ∈ C, for t ∈ [a, b], is shows that ‖A 2 x + A 2 y‖ ≤ Ω 2 ‖x − y‖; then, by using (H2), A 2 is a contraction mapping. e operator A 1 is continuous, since f is continuous. It is uniformly bounded on B ρ as Now, we prove that the operator A 1 is compact. Setting sup (t,x)∈[a,b]×B ρ |f(t, x)| � f < ∞, and let t 1 , t 2 ∈ [a, b], t 1 < t 2 , we obtain International Journal of Differential Equations where the right-hand side tends to zero as t 2 − t 1 ⟶ 0, independently of x ∈ B ρ . en, A 1 is equicontinuous, and hence, A 1 is relatively compact on B ρ . By the Arzelá-Ascoli theorem, A 1 is compact on B ρ . It follows by Krasnoselskii's fixed-point theorem that problem (1) has at least one solution on [a, b].
To deal with the uniqueness of solution for our problem (1), we use Banach's contraction principle.
] and x, y ∈ R.
If LΩ 1 + Ω 2 < 1, where Ω 1 , Ω 2 are, respectively, given by Proof. Consider the operator A defined in (17). e problem (1) is then transformed into a fixed-point problem x � Ax. By using Banach contraction principle, we will show that A has a unique fixed point.

Existence Results for Problem (2)
Definition 6. A continuous function x is said to be a solution of problem (2) For each x ∈ C[a, b], R, define the set of selections of F by Lemma 3 (see [5] Our second existence result is based on the nonlinear alternative of the Leray-Schauder for multivalued maps [19]. where Ω 1 , Ω 2 are, respectively, given by (24)

Conclusion
In the current article, we study and investigate the existence and uniqueness of solution for a boundary value problem for the Langevin equation and inclusion with the Hilfer fractional derivative. e novelty of this work is that it is more general than the works based on the derivative of Caputo and Riemann-Liouville because when β � 0, we find the Riemann-Liouville fractional derivative, and when β � 1, we find the Caputo fractional derivative. In this study, we established the existence and uniqueness results for the first problem, by using the fixed-point theorems (Banach' fixedpoint theorem and Krasnoselskii's fixed-point theorem), and for the inclusion case, we use Leray-Schauder alternative to prove the existence of solution. In the end, we give an example to illustrate our results.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.