On a new class of abstract impulsive functional differential equations of fractional order

Abstract

In this paper, we prove the existence and uniqueness of mild solutions for the impulsive fractional differential equations for which the impulses are not instantaneous in a Banach space H. The results are obtained by using the analytic semigroup theory and the fixed points theorems.

Caputo fractional differential equations with non-instantaneous impulses and strict stability by Lyapunov functions

“…The study of fractional differential equations with non-instantaneous impulses is quite recent and only some qualitative properties are investigated ( [18], [19], [20], [28], [35], [46]). The goal of the paper is to study strict stability of the NIFrDE (7).…”

p-Moment exponential stability of Caputo fractional differential equations with noninstantaneous random impulses

“…and s k = T k + d which are values of the random variables ξ k and ξ k + d, respectively. Then the corresponding function x(t; T 0 , x 0 , {T k }) is a sample path solution of the IVP for RIFrDE (9) and the corresponding function u(t) = u(t; T 0 , V (T 0 , x 0 ), {T k }) is a sample path solution of the IVP for RIFrDE (33), i.e., x(t) = x(t; T 0 , x 0 , {T k }) is a solution of the IVP for the IFrDE with fixed points of impulses (4).…”

“…Since the introduction of the drugs in the bloodstream and the consequent absorption for the body are gradual and continuous processes, we can interpret the situation as an impulsive action which starts abruptly and stays active on a finite time interval. Recently results concerning noninstantaneous impulses are obtained for differential equations [4,14,38], delay integro-differential equations [16], abstract differential equations [24,39], and fractional differential equations, FrDEs [2,33,37].…”

Existence Results for an Impulsive Fractional Integro-Differential Equations with a Non-compact Semigroup

“…A lot of models about fractional impulsive differential equations were studied recently, for more details we give the references [19,12,21] and the references therein.…”