Boundary Value Problem for ψ-Caputo Fractional Differential Equations in Banach Spaces via Densifiability Techniques

Choukri Derbazi,Yong Zhou,Mouffak Benchohra,Zidane Baitiche

doi:10.3390/math10010153

Abstract

A novel fixed-point theorem based on the degree of nondensifiability (DND) is used in this article to examine the existence of solutions to a boundary value problem containing the ψ-Caputo fractional derivative in Banach spaces. Besides that, an example is included to verify our main results. Moreover, the outcomes obtained in this research paper ameliorate and expand some previous findings in this area.

Highlights

The study of fractional differential equations (FDEs) has become a hot topic because they can suitably explain the behavior of a wide range of real-world problems more accurately than integer-order derivatives
Our proposed method is essentially based on the excellent results given by García [28,29] to study the existence of solutions to a boundary value problem (BVP) containing the ψ-Caputo fractional derivative in Banach spaces via densifiability techniques
Under some suitable assumptions on the nonlinear part F and by the application of the new fixed-point theorem combined with densifiability techniques, we obtained the existence of solutions to the considered fractional BVP (1);

Summary

Introduction

The study of fractional differential equations (FDEs) has become a hot topic because they can suitably explain the behavior of a wide range of real-world problems more accurately than integer-order derivatives. Many expressions of fractional calculus have been published, but the most prevalent definitions are Riemann–Liouville and Caputo fractional derivatives. The former has an abstraction mathematically, but the latter is mostly used by engineers. Our proposed method is essentially based on the excellent results given by García [28,29] to study the existence of solutions to a boundary value problem (BVP) containing the ψ-Caputo fractional derivative in Banach spaces via densifiability techniques. Under some suitable assumptions (weak conditions) on the nonlinear part F and by the application of the new fixed-point theorem combined with densifiability techniques, we obtained the existence of solutions to the considered fractional BVP (1);. The paper closes with a brief conclusion and points out some possible future directions of research

Basic Definitions

Main Results

Application

Conclusions