Analysis of Caputo Impulsive Fractional Order Differential Equations with Applications

Abstract

We use Sadavoskii's fixed point method to investigate the existence and uniqueness of solutions of Caputo impulsive fractional differential equations of order α ∈ (0, 1) with one example of impulsive logistic model and few other examples as well. We also discuss Caputo impulsive fractional differential equations with finite delay. The results proven are new and compliment the existing one.

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

“…Since is a solution to (51), then it satisfies the integral equation (19). Let V be a solution to problem (51) with initial value V( ) = V 0 − (V).…”

“…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.…”

Stepanov-like weighted pseudo almost automorphic solutions to fractional order abstract integro-differential equations

“…A wonderful book on fractional differential equation is written by Podlubny [36]. The existence and uniqueness of solutions of such kind of differential equations have been shown by many authors, we refer to [1,2,4,10,13,14,24,25,30] and references therein. A very natural question in the field of differential equations is to see whether the solution follows the same pattern of forcing term or not.…”

New Integral Inequalities with Weakly Singular Kernel for Discontinuous Functions and Their Applications to Impulsive Fractional Differential Systems

“…The theory of impulsive fractional differential equations is a new topic of research which involve both the fractional order integral (or differentiation) and the impulsive effect; most of the results related to this topic are the existence of solutions (see [16][17][18][19] and the references therein). To our best knowledge, there is no result on other qualitative properties (such as boundedness and stability), and impulsive fractional differential equations involving the Caputo fractional derivative have not been studied very perfectly, so we set up a new kind of integral inequalities with weakly singular kernel for discontinuous functions and use the new inequalities to study the qualitative properties of the solutions to certain impulsive fractional differential systems.…”