Coupled System of Nonlinear Fractional Langevin Equations with Multipoint and Nonlocal Integral Boundary Conditions

Faris Alzahrani,Ahmed Salem,Mohammad Alnegga

doi:10.1155/2020/7345658

Faris Alzahrani, Ahmed Salem + Show 1 more

Open Access

https://doi.org/10.1155/2020/7345658

Copy DOI

Abstract

This research paper is about the existence and uniqueness of the coupled system of nonlinear fractional Langevin equations with multipoint and nonlocal integral boundary conditions. The Caputo fractional derivative is used to formulate the fractional differential equations, and the fractional integrals mentioned in the boundary conditions are due to Atangana–Baleanu and Katugampola. The existence of solution has been proven by two main fixed-point theorems: O’Regan’s fixed-point theorem and Krasnoselskii’s fixed-point theorem. By applying Banach’s fixed-point theorem, we proved the uniqueness result for the concerned problem. This research paper highlights the examples related with theorems that have already been proven.

Highlights

Many mathematical fields have been developed rapidly via fractional calculus
Fractional differential equations have obtained a remarkable reputation among the mathematicians due to rapid development which is applicable in many fields such as mathematics, chemistry, and electronics
Recent paper [31] has discussed the existence and uniqueness of solutions obtained from boundary value conditions for nonlinear fractional differential equations for Riemann–Liouville type under the generalized nonlocal integral boundary condition

Summary

Introduction

Many mathematical fields have been developed rapidly via fractional calculus. Recent paper [31] has discussed the existence and uniqueness of solutions obtained from boundary value conditions for nonlinear fractional differential equations for Riemann–Liouville type under the generalized nonlocal integral boundary condition. We modify the boundary value conditions of coupled systems of Langevin fractional differential equations of Caputo type into new boundary value conditions. We deal with the following coupled systems of nonlinear fractional Langevin equations of α and β fractional orders: Mathematical Problems in Engineering. To the extent of our knowledge, this is the first paper that discusses the existence and uniqueness of the solutions to coupled systems of fractional Langevin equations involving the nonlocal integro-multipoint of Atangana–Baleanu type and the nonlocal integral of Katugampola type as boundary value conditions. Examples have been considered in order to cover all theorems clearly

Basic Concepts and Relevant Lemmas

Main Results

Conclusion