Investigation of Ulam Stability Results of a Coupled System of Nonlinear Implicit Fractional Differential Equations

Zeeshan Ali,Kamal Shah,Poom Kumam,Akbar Zada

doi:10.3390/math7040341

Abstract

This manuscript deals with the existence theory, uniqueness, and various kinds of Ulam–Hyers stability of solutions for a class and coupled system of fractional order differential equations involving Caputo derivatives. Applying Schaefer and Banach’s fixed point approaches, existence and uniqueness results are obtained for the proposed problems. Stability results are investigated by using the classical technique of nonlinear functional analysis. Examples are given with each problem to illustrate the main results.

Highlights

Fractional order differential equations (FODE) are generalizations of ordinary differential equations to an arbitrary order
The aforesaid equations arise in many disciplines of science and engineering as the mathematical modeling of systems and processes in the fields of biophysics, blood flow phenomena, signal and image processing, polymer rheology, control theory, the electrodynamics of a complex medium, physics, aerodynamics, economics, chemistry, etc
The theory of boundary value problems for nonlinear FODE remains within the initial stages, and plenty of aspects of this theory have to need to be explored

Summary

Introduction

Fractional order differential equations (FODE) are generalizations of ordinary differential equations to an arbitrary order. Benchohra and Lazreg [42] investigated the existence and different kinds of Ulam–Hyers stability for the following nonlinear implicit FODE involving the Caputo derivative of fractional order: u(t) t=0 = u0 , where 1 < p ≤ 2, J = [0, T] with T > 0, u0 ∈ R, and α : J × R × R → R is continuous. Motivated by the aforesaid discussion, in this manuscript, our target is to study the existence, uniqueness, and different kinds of Ulam–Hyers stability for the following nonlinear implicit FODE involving the Caputo derivative of fractional order: u(t) t=0 = −u(t) t=T , u0 (t) t=0 = −u0 (t) t=T , u00 (t) t=0 = u000 (t) t=0 = 0,.

Preliminaries

Existence and Uniqueness Results

Ulam Stability Results

Tp K α α

Example

Conclusions