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

Abstract

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