A new study on the mild solution for impulsive fractional evolution equations

TL;DR

This paper introduces a new definition of mild solutions for impulsive fractional evolution equations, improving upon existing methods by modifying the impulse term operator using fractional resolvent operators.

Contribution

It proposes a novel definition of mild solutions for impulsive fractional evolution equations, replacing the impulse operator with a product involving the inverse of the fractional solution operator.

Findings

New definition enhances the analytical framework for impulsive fractional equations

Replaces traditional impulse operator with a more appropriate fractional resolvent-based operator

Improves the mathematical understanding of mild solutions in fractional evolution equations

Abstract

In this article, we consider mild solutions to a class of impulsive fractional evolution equations of order $0<\alpha<1$. After analyzing analytic results reported in the literature using Mittag-Leffer function, $\alpha$-resolvent operator theory, we propose a more appropriate new definition of mild solutions for impulsive fractional evolution equations by replacing the impulse term operator $S_\alpha(t-t_i)$ with $S_\alpha(t)S_\alpha^{-1}(t_i)$, where $S_\alpha^{-1}(t_i)$ denotes the inverse of the fractional solution operator $S_\alpha(t)$ at $t=t_i, (i=1,2,\cdots m)$.