A note on the mild solutions of Hilfer impulsive fractional differential equations

TL;DR

This paper introduces a new Gronwall-type inequality and studies the uniqueness and stability of mild solutions for Hilfer impulsive fractional differential equations with non-instantaneous impulses.

Contribution

It presents a novel inequality and analyzes the stability and uniqueness of solutions in a fractional differential equations context.

Findings

Established a new Gronwall-type inequality.

Proved uniqueness of mild solutions.

Analyzed $ ext{Ulam-Hyers-Rassias}$ stability.

Abstract

In this paper, we present a new class of inequality of the Gronwall type and discuss some particular cases. In this sense, we investigate the uniqueness and $\delta$-Ulam-Hyers-Rassias stability of the mild solution for the fractional differential equation with non-instantaneous impulses in a P$\delta$-normed Banach space.