On the Impulsive Implicit $\Psi$--Hilfer Fractional Differential Equations with Delay

TL;DR

This paper studies the existence, uniqueness, and stability of solutions for impulsive implicit $ ext{ extPsi}$--Hilfer fractional differential equations with delay, extending stability concepts and employing advanced mathematical tools.

Contribution

It introduces new stability results for impulsive implicit $ ext{ extPsi}$--Hilfer fractional equations with delay, using extended Gronwall inequalities and Picard operator theory.

Findings

Established existence and uniqueness of solutions.

Derived Ulam--Hyers--Mittag--Leffler stability results.

Provided an illustrative example.

Abstract

In this paper, we investigate the existence and uniqueness of solutions and derive the Ulam--Hyers--Mittag--Leffler stability results for impulsive implicit $\Psi$--Hilfer fractional differential equations with time delay. It is demonstrated that the Ulam--Hyers and generalized Ulam--Hyers stability are the specific cases of Ulam--Hyers--Mittag--Leffler stability. Extended version of Gronwall inequality, abstract Gronwall lemma and Picard operator theory are the primary devices in our investigation. We give an example to illustrate the obtained results.