On the Existence and Stability of Variable Order Caputo Type Fractional Differential Equations

Shahzad Sarwar

doi:10.3390/fractalfract6020051

Abstract

In the theory of differential equations, the study of existence and the uniqueness of the solutions are important. In the last few decades, many researchers have had a keen interest in finding the existence–uniqueness solution of constant fractional differential equations, but literature focusing on variable order is limited. In this article, we consider a Caputo type variable order fractional differential equation. First, we present the existence–uniqueness of a solution of the considered problem. Secondly, By borrowing the idea from the theory of ordinary differential equations, we extend the continuation theorem for the variable order fractional differential equation. Further, we prove the global existence results. Finally, we present different types of Ulam–Hyers stability results, which have never been studied before for the Caputo type variable order fractional differential equation.

Highlights

Over the past few years, the study of fractional calculus [1,2,3,4,5] has broadened, because of its applications in most disciplines of science and engineering
In 1993, reference [6] first introduced the concept of variable order fractional calculus
There is some literature on the existence of a solution of variable order fractional differential equations (FDEs), and the results are interesting

Summary

Introduction

Over the past few years, the study of fractional calculus [1,2,3,4,5] has broadened, because of its applications in most disciplines of science and engineering. The existence–uniqueness of the solution of fractional differential equations (FDEs) is a interesting research area [12,13,14,15]. There is some literature on the existence of a solution of variable order FDEs, and the results are interesting. University—“Under what conditions does there exist an additive mapping near an approximately additive mapping?”İn 1941, Hyers [27] obtained an interesting solution to Ulam’s question, by considering the Banach spaces. This type of stability is called the. To the best of our knowledge, the continuation theorem, global existence, and the Ulam–Hyers type stability of (1) have not previously been studied.

Preliminaries

Global Solution

Ulam Stability Results

Conclusions