Extremal mild solutions for Hilfer fractional evolution equation with mixed monotone Impulsive conditions

TL;DR

This paper develops a method using mixed monotone iterative techniques and fixed point theorems to establish the existence and uniqueness of mild solutions for impulsive Hilfer fractional evolution equations, with an illustrative example.

Contribution

It introduces a novel approach combining measure of non-compactness and Sadovskii's fixed point theorem for impulsive Hilfer fractional systems.

Findings

Existence of mild L-quasi solutions is proven.

Method applies to noncompact semigroups.

An example demonstrates the applicability of the results.

Abstract

The well established mixed monotone iterative technique that is used to study the existence and uniqueness of fractional order system is studied explicitly for impulsive system with Hilfer fractional order in this paper. The procedure of finding mild $L$-quasi solution of such impulsive evolution equation with noncomapct semigroups involves measure of non-compactness and Sadovskii's fixed point theorem as well. An example is provided to illustrate the main results.