On the concept of general solution for impulsive differential equations of fractional order q ∈(0, 1)

Keywords: Fractional differential equations Impulsive fractional differential equations Impulse General solution Existence a b s t r a c t In this paper, for impulsive differential equations with fractional-order q 2 ð0; 1Þ, we show that the formula of solutions in cited papers are incorrect. Secondly, we find out a formula of the general solution for impulsive Cauchy problem with Caputo fractional derivative q 2 ð0; 1Þ. Further, for a kind of impulsive fractional differential equations system with special initial value, we come to an existence result for it by applying fixed point methods. Fractional differential equations have proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. And the subject of fractional differential equations is gaining much attention. For detail, see [1–13] and the references therein. Fractional differential equation was extended to impulsive fractional differential equations, since Agarwal and Benchohra published the first paper on the topic [14] in 2008. Recently, in [15], Fečkan and Zhou pointed out that the formula of solutions for impulsive fractional differential equations in [16–19] is incorrect and gave their correct formula. In [20,21], the authors established a general framework to find the solutions for impulsive fractional boundary value problems and obtained some sufficient conditions for the existence of the solutions to a kind of impulsive fractional differential equations respectively. In [22], the authors illustrated their comprehensions for the counterexample in [15] and criticized the viewpoint in [15,20,21]. Next, in [23], Fečkan et al. expounded for the counterexample in [15] and provided further five explanations in the paper. This paper is motivated from some recent papers which treated the problem of the existence of the solutions for impulsive differential equations with fractional derivate q 2 ð0; 1Þ and directly or indirectly used an unfit integral equation. To prove our claim, we consider a general impulsive fractional system 0 D