An Alternative Method for the Study of Impulsive Differential Equations of Fractional Orders in a Banach Space

This paper is concerned with the existence, uniqueness, and stability of the solution of some impulsive fractional problem in a Banach space subjected to a nonlocal condition. Meanwhile, we give a new concept of a solution to impulsive fractional equations of multiorders. The derived results are based on Banach's contraction theorem as well as Schaefer's fixed point theorem. 1. Introduction It is well known that the theory of fractional calculus deals with the concepts of differentiation and integration of arbitrary orders, real and complex. Actually, the real importance of fractional derivatives lies in their nonlocal character which gives rise to a long memory effect and thus to a better insight into the modelled processes. On the other hand, since models using classical derivatives are just a special case of those using fractional derivatives, then most of the investigators in different areas such as electronics, viscoelasticity, satellite guidance, medicine, anomalous diffusion, signal processing, and many other branches of science and technology have revisited some classical dynamic systems in the framework of fractional derivatives to get better results; see the references [1–8]. We point out that most of dynamic systems are naturally governed by fractional differential equations; for further applications of fractional derivatives in other areas and useful backgrounds we refer the reader to the works [1–5, 7–12]. As far as we are concerned with impulsive fractional differential equations, we intend to improve and correct in this paper some existence results established earlier in [4, 13–18] for impulsive fractional differential equations. There have been in the last couple of years several concepts of solutions satisfying some fractional equations subjected to impulsive conditions, see [13, 14, 18, 19], while the authors of [18] claimed that their new concept is the more realistic than the existing ones. Actually, we believe that nobody holds all the truth about this subject and a lot of dark sides of these approaches are not yet well elucidated. Regarding the concept of a solution for impulsive fractional equations introduced by [18] we point out that Lemma 2.6 which has been used by the authors to obtain the equivalence between an impulsive fractional problem and an integral equation is false as we see in the following counterexample. In the famous book of Nagy and Riesz [20, page 48], there is an example of monotonic continuous function which is not constant in any subinterval of and satisfies , almost everywhere in . So, in terms of Caputo’s