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

• Published in: Mathematical methods in the applied sciences Vol. 44; no. 18; pp. 13979 - 13988
• Main Authors: Fernandez, Arran, Ali, Sartaj, Zada, Akbar
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 01.12.2021
• Subjects: Caputo fractional derivative; Differential equations; Equivalence; fractional differential equations; fractional integrals; impulsive differential equations; Integral equations; non‐instantaneous impulses; Operators (mathematics); Volterra integral equations
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.7669

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.