Abstract
Fractional differential equations with impulses have been studied over the past decades. A methodology often used for the Caputo fractional derivative case is to change the problem into an equivalent integral equation in a space of piecewise continuous functions. One approach is to change the fractional derivative at each impulse point, thereby studying a sequence of problems, one on each subinterval. A second approach is to have one fixed fractional derivative on the whole interval. These approaches have often been confused. In the second case an integral equation was proposed by Fečkan, Zhou and Wang (Commun. Nonlinear Sci. Numer. Simul., 2012). This paper and some successors have been cited a large number of times. However, it is shown here, with simple counter-examples, that the proposed piecewise continuous solutions of the integral equation do not give solutions of the fractional differential equation with an impulse. Moreover, it is proved that no solution of this type is actually possible. A similar approach for the non-instantaneous impulse case is also shown to be not correct. An impulse problem involving the Riemann–Liouville fractional derivative is also discussed.