Ulam-Hyers stabilities of mild solutions of the fractional nonlinear abstract Cauchy problem

TL;DR

This paper investigates the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of mild solutions to fractional nonlinear abstract Cauchy problems using Banach fixed point theorem, extending stability analysis in fractional differential equations.

Contribution

It provides new results on the stability of mild solutions for fractional nonlinear Cauchy problems, a less explored area in fractional differential equations.

Findings

Established Ulam-Hyers stability conditions for fractional Cauchy problems.

Extended stability analysis to both finite and infinite intervals.

Applied Banach fixed point theorem to prove stability results.

Abstract

Since the main work on Ulam-Hyers dependable stabilities of differential equations to date, numerous significant and applicable papers have been published, both in the sense of integer order and fractional order differential equations. However, when we enter the field of fractional differential equations, the path that is still long to be traveled, although there is a range of published works. In this sense, in this paper, we will investigate the Ulam--Hyers and Ulam--Hyers--Rassias stabilities of mild solutions of the fractional nonlinear abstract Cauchy problem on the intervals $[0,T]$ and $[0,\infty)$, by means of Banach fixed point theorem.