A study on the mild solution of impulsive fractional evolution equations

Abstract

This paper is concerned with the formula of mild solutions to impulsive fractional evolution equation. For linear fractional impulsive evolution equations [8–25,27,30,31], described mild solution as integrals over (tk,tk+1](k=1,2,…,m) and [0, t1]. On the other hand, in [26,28,29], their solutions were expressed as integrals over [0, t]. However, it is still not clear what are the correct expressions of solutions to the fractional order impulsive evolution equations. In this paper, firstly, we prove that the solutions obtained in [8–25,27,30,31] are not correct; secondly, we present the right form of the solutions to linear fractional impulsive evolution equations with order 0 < α < 1 and 1 < α < 2, respectively; finally, we show that the reason that the solutions to an impulsive ordinary evolution equation are not distinct.