Existence and Ulam–Hyers Stability of a Fractional-Order Coupled System in the Frame of Generalized Hilfer Derivatives

Abdulkafi M Saeed,Mdi Begum Jeelani,Mohammed S Abdo

doi:10.3390/math9202543

Abstract

In this research paper, we consider a class of a coupled system of fractional integrodifferential equations in the frame of Hilfer fractional derivatives with respect to another function. The existence and uniqueness results are obtained in weighted spaces by applying Schauder’s and Banach’s fixed point theorems. The results reported here are more general than those found in the literature, and some special cases are presented. Furthermore, we discuss the Ulam–Hyers stability of the solution to the proposed system. Some examples are also constructed to illustrate and validate the main results.

Highlights

The theory of fractional differential equations (FDEs) has become an active space of exploration
Fractional calculus has been improving the mathematical modeling of sundry phenomena in science and engineering, for more details, refer to the monographs [1,2,3,4,5]
The fundamental benefit of using fractional-order derivatives (FODs) rather than integer-order derivatives (IODs) is that IODs are local in nature, whereas FODs are global in nature

Summary

Introduction

The theory of fractional differential equations (FDEs) has become an active space of exploration. By considering physical phenomena which are modeled by utilizing classical FDs, the importance of ψ-Hilfer FD can be discussed by redesigning and remodeling such models under ψ-Hilfer FD In this regard, the most relaxing technique for stability for functional equations was presented by Ulam [25] and Hyers [26] which is famous for Hyers–Ulam (in short H-U). The existence and stability of solutions of the following θ-Hilfer type FDE: Dρa+1,,ρθ2(κ)υ(κ) =. Abdo and Panchal in [36] proved the existence, uniqueness and Ulam–Hyers stability of the following θ-Hilfer type fractional integrodifferential equation:. Motivated by the above discussion, we investigate the existence, uniqueness, and H-U stability of the solutions of a coupled system involving aθ-Hilfer FD of the type: DDρaρa+1+1,,,ρ,ρθθ22((κκ))ωυ((κκ)). We epitomize our study in the Conclusion section

Preliminaries

Existence Result

Uniqueness Result

Special Cases

Examples Consider the θ-Hilfer type system

Conclusions