New Existence Results for Nonlinear Fractional Integrodifferential Equations

Lahcen Ibnelazyz,Khalid Hilal,Karim Guida,Said Melliani,Soheil Salahshour

doi:10.1155/2021/5525591

Abstract

This paper discusses a boundary value problem of nonlinear fractional integrodifferential equations of order 1 < α ≤ 2 and 1 < β ≤ 2 and boundary conditions of the form x 0 = x 1 = D c β x 1 = D c β x 0 = 0 . Some new existence and uniqueness results are proposed by using the fixed point theory. In particular, we make use of the Banach contraction mapping principle and Krasnoselskii’s fixed point theorem under some weak conditions. Moreover, two illustrative examples are studied to support the results.

Highlights

Fractional differential equations are relevant in many fields of science, such as chemistry, fluid systems, and electromagnetic; for more details about the theory of fractional differential equations and their applications, we invite the readers to see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] and the references therein
Fractional integrodifferential equations are used as an important tool to gain insight into some emerging problems from several science areas, for more details, we give the following references [18,19,20,21,22,23]
The existence of solutions for nonlinear fractional integrodifferential equations has been studied in [26]. Motivated by all these works and by the fact that there are no papers dealing with the new existence results for nonlinear fractional integrodifferential equations, in this work, we consider the existence and uniqueness of solutions for the following problem: 8 < cDα cDβ xðtÞ = f ðt, xðtÞ, ΦxðtÞ, ψxðtÞÞ, t ∈ 1⁄20, 1

Summary

Introduction

Fractional differential equations are relevant in many fields of science, such as chemistry, fluid systems, and electromagnetic; for more details about the theory of fractional differential equations and their applications, we invite the readers to see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] and the references therein. Fractional integrodifferential equations are used as an important tool to gain insight into some emerging problems from several science areas, for more details, we give the following references [18,19,20,21,22,23]. In [25], the authors discussed the existence and uniqueness of solutions for nonlinear integrodifferential equations of fractional order with three-point nonlocal fractional boundary conditions. The existence of solutions for nonlinear fractional integrodifferential equations has been studied in [26] Motivated by all these works and by the fact that there are no papers dealing with the new existence results for nonlinear fractional integrodifferential equations, in this work, we consider the existence and uniqueness of solutions for the following problem:. In the last section, we give two examples to illustrate the results

Preliminaries and Notations

ΓðαÞ ðt ðt

Main Results

ΓðαÞΓðβ ð1 eυsds kθkυ ð1 !

Examples

Conclusion