Existence of Solutions and Hyers–Ulam Stability for a Coupled System of Nonlinear Fractional Differential Equations with p-Laplacian Operator

Jing Shao,Boling Guo

doi:10.3390/sym13071160

Abstract

In this paper, the existence and uniqueness of solutions to a coupled formally symmetric system of fractional differential equations with nonlinear p-Laplacian operator and nonlinear fractional differential-integral boundary conditions are obtained by using the matrix eigenvalue method. The Hyers–Ulam stability of the coupled formally symmetric system is also presented with certain growth conditions. By using the topological degree theory and nonlinear functional analysis methods, some sufficient conditions for the existence and uniqueness of solutions to this coupled formally symmetric system are established. Examples are provided to verify our results.

Highlights

Symmetry is an important form of many things in nature and society; many of the differential equations we studied are symmetric
Mathematical models of fractional differential equations are at the heart of quantitative descriptions of a large number of physical systems, including engineering, plasma physics, aerodynamics, electrical circuits and many other fields
We use the coincidence degree method and nonlinear functional analysis theory to deal with the existence and uniqueness of solutions and the matrix eigenvalue method in order to investigate Hyers–Ulam stability

Summary

Introduction

Symmetry is an important form of many things in nature and society; many of the differential equations we studied are symmetric. A. Khan et al [6] discuss the existence, uniqueness and Hyers–Ulam stability of solutions to a coupled system of fractional differential equations with nonlinear p-Laplacian operator. Khan [14,15], this paper is devoted to study the existence, uniqueness and Hyers–Ulam stability of solutions to nonlinear coupled fractional differential equations with p-Laplacian operator of the form. Fi : T → R are closed bounded and linear operators for any t ∈ T = [0, 1], and Φ, Ψ : T × R × R → R are continuous functions, i = 1, 2 For this purpose, we use the coincidence degree method and nonlinear functional analysis theory to deal with the existence and uniqueness of solutions and the matrix eigenvalue method in order to investigate Hyers–Ulam stability.

Auxiliary Results

Existence Results

Hyers–Ulam Stability

Examples

Conclusions