Compact and Noncompact Solutions to Generalized Sturm–Liouville and Langevin Equation with Caputo–Hadamard Fractional Derivative

Faris Alzahrani,Shahram Rezapour,Moustafa El-Shahed,Noorah Mshary,Ahmed Salem

doi:10.1155/2021/9995969

Abstract

In this work, through using the Caputo–Hadamard fractional derivative operator with three nonlocal Hadamard fractional integral boundary conditions, a new type of the fractional-order Sturm–Liouville and Langevin problem is introduced. The existence of solutions for this nonlinear boundary value problem is theoretically investigated based on the Krasnoselskii in the compact case and Darbo fixed point theorems in the noncompact case with aiding the Kuratowski’s measure of noncompactness. To demonstrate the applicability and validity of the main gained findings, some numerical examples are included.

Highlights

Over the past few years, the fractional calculus (which is basically an expansion of the traditional calculus) has provided its remarkable contribution in addressing lots of physical and biological phenomena as well as describing many engineering dilemmas [1]
Over the past few years, the fractional calculus has provided its remarkable contribution in addressing lots of physical and biological phenomena as well as describing many engineering dilemmas [1]. It has been confirmed in several literature studies that many real-world problems associated with many applied science fields can be described more convenient using the fractional-order differential equations (FoDEs) rather than that of using the ordinary differential equations (ODEs) [2]
For further facts on the basic principles of fractional calculus and the FoDEs, the reader may return to the references [4,5,6,7]. e fractional calculus, new interesting research field, is attracting the interest of mathematicians and researchers

Summary

Introduction

Over the past few years, the fractional calculus (which is basically an expansion of the traditional calculus) has provided its remarkable contribution in addressing lots of physical and biological phenomena as well as describing many engineering dilemmas [1]. E remaining of this paper is ordered as follows: Section 2 presents briefly some fundamental concepts related to the fractional calculus and fixed point theorems. E Caputo-Hadamard derivative operator of fractional-order μ for at least n-times differentiable function f: [a, ∞) ⟶ R is outlined as

Results

Conclusion