Existence and Uniqueness Results of Coupled Fractional-Order Differential Systems Involving Riemann–Liouville Derivative in the Space Wa+γ1,1(a,b)×Wa+γ2,1(a,b) with Perov’s Fixed Point Theorem

Noura Laksaci,Taqi A M Shatnawi,Kamaleldin Abodayeh,Ahmed Boudaoui,Wasfi Shatanawi

doi:10.3390/fractalfract5040217

Abstract

This paper is devoted to studying the existence and uniqueness of a system of coupled fractional differential equations involving a Riemann–Liouville derivative in the Cartesian product of fractional Sobolev spaces E=Wa+γ1,1(a,b)×Wa+γ2,1(a,b). Our strategy is to endow the space E with a vector-valued norm and apply the Perov fixed point theorem. An example is given to show the usefulness of our main results.

Highlights

Academic Editor: Enzo OrsingherReceived: 25 October 2021Accepted: 11 November 2021Published: 13 November 2021The beauty of fractional calculus lies in finding the derivative and integration of an operator for any order
The subject of fractional calculus has become the center of attractive research
Let (E, dG ) be a complete generalized metric space and let N be an operator from E

Summary

Introduction

The beauty of fractional calculus lies in finding the derivative and integration of an operator for any order. To ensure a solution of some nonlinear problems, researchers utilize some suitable fixed point theorems. One of these theorems is the Banach contraction principle. Perov’s fixed point theorem is one of the crucial methods to prove an existence solution of systems of differential equations, fractional differential equations, and integral equations in N variables; see [7,8,9,10], and the references cited therein. A number of interesting papers [11,12,13] on the solvability of mathematical problems in Sobolev spaces W n,p (R+ ) [14] with the help of fixed point theory have been iations. We give an example to show the applicability of our main result

Preliminaries

Main Results

Methods