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

Abstract

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. Third 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.

A New Result for ψ‐Hilfer Fractional Pantograph‐Type Langevin Equation and Inclusions

“…Substituting the value of ðd 0 Þ in (16), we obtain the solution (11). The converse follows by direct computation.…”

“…Over the past few years, fractional differential equations have attracted the interest of many mathematicians due to their ability to describe several complex problems in different scientific and engineering fields such as physics, biology, chemistry, and control theory (for more details, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14]).…”

Topological Degree Method for a Class of Ψ-Caputo Fractional Differential Langevin Equation

“…Faree and Panchal [9], investigated the existence and uniqueness of solutions to boundary value problems involving the Caputo fractional derivative in Banach space by topological structures with some appropriate conditions. Hilal et al [10], discussed the existence and uniqueness of solution for a boundary value problem for the Langevin equation and inclusion, based on Krasnoselskii's fixed point theorem, Banach's contraction principle and Leray-Schauder's alternative. Rizwan [26], considered a non local boundary value problem of nonlinear fractional Langevin equation with non-instantaneous impulses.…”

Analytic Solutions for Hilfer Type Fractional Langevin Equations with Variable Coefficients in a Weighted Space

“…Moreover, we refer to some recent works [17,18] that deal with a qualitative analysis of FLEs with Hilfer fractional derivatives.…”

“…In [18], the authors investigated the existence and uniqueness of solutions to the following system of Hilfer FLEs:…”