On non‐instantaneous impulsive fractional differential equations and their equivalent integral equations

Abstract

Real‐world processes that display non‐local behaviours or interactions, and that are subject to external impulses over non‐zero periods, can potentially be modelled using non‐instantaneous impulsive fractional differential equations or systems. These have been the subject of many recent papers, which rely on re‐formulating fractional differential equations in terms of integral equations, in order to prove results such as existence, uniqueness, and stability. However, specifically in the non‐instantaneous impulsive case, some of the existing papers contain invalid re‐formulations of the problem, based on a misunderstanding of how fractional operators behave. In this work, we highlight the correct ways of writing non‐instantaneous impulsive fractional differential equations as equivalent integral equations, considering several different cases according to the lower limits of the integro‐differential operators involved.