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.

Approximate controllability of impulsive fractional evolution equations of order 1<α<2$$ 1&amp;amp;amp;lt;\alpha &amp;amp;amp;lt;2 $$ with state‐dependent delay in Banach spaces

“…Later, Wang et al 11,12 extended this model to two general classes of impulsive differential equations, which are very useful to characterize certain dynamics of evolutionary processes in pharmacotherapy. For the study of noninstantaneous impulsive systems, one may refer to previous works [13][14][15][16][17][18] and the references therein.…”

“…for more details). In the past two decades, a good number of results came out on the existence of solutions and approximate controllability for fractional systems of order $$$ \alpha \in \left(0,1\right) $$$ in Hilbert and Banach spaces; see, for example, previous works 15,16,33,36–40 and references therein. Recently, many authors established the existence of mild solutions and approximate controllability for fractional evolution equations of order $$$ \alpha \in \left(1,2\right) $$$ by invoking the theory of sectorial operators via suitable fixed point theorems; see, for instance, previous studies 41–43 …”

Analysis of Existence and Stability Results for Impulsive Fractional Integro-Differential Equations Involving the Atangana–Baleanu–Caputo Derivative under Integral Boundary Conditions

“…In addition, BVPs with integral boundary conditions have numerous contributions of mathematical modeling to the heat conduction process, hemic conduction process, and hydrodynamics issues. Many authors have investigated FDEs with boundary conditions [14][15][16][17][18][19][20][21][22][23].…”