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

Athasit Wongcharoen,Jessada Tariboon,Bashir Ahmad,Sotiris K Ntouyas

doi:10.1155/2020/9606428

Abstract

We discuss the existence and uniqueness of solutions for the Langevin fractional differential equation and its inclusion counterpart involving the Hilfer fractional derivatives, supplemented with three-point boundary conditions by means of standard tools of the fixed-point theorems for single and multivalued functions. We make use of Banach’s fixed-point theorem to obtain the uniqueness result, while the nonlinear alternative of the Leray-Schauder type and Krasnoselskii’s fixed-point theorem are applied to obtain the existence results for the single-valued problem. Existence results for the convex and nonconvex valued cases of the inclusion problem are derived via the nonlinear alternative for Kakutani’s maps and Covitz and Nadler’s fixed-point theorem respectively. Examples illustrating the obtained results are also constructed. (2010) Mathematics Subject Classifications. This study is classified under the following classification codes: 26A33; 34A08; 34A60; and 34B15.

Highlights

Fractional calculus is an emerging field in applied mathematics that deals with derivatives and integrals of arbitrary orders
A generalization of both the Riemann-Liouville and Caputo derivatives was given by Hilfer in [7], which is known as the Hilfer fractional derivative Dα,βxðtÞ of order α and type β ∈ 1⁄20, 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

Summary

Introduction

Fractional calculus is an emerging field in applied mathematics that deals with derivatives and integrals of arbitrary orders. 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. 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.

Preliminaries

ΓðαÞ ð t a ðt

C Dα uðt Þ