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

Abstract

In this paper the statement of initial value problems for fractional differential equations with noninstantaneous impulses is given. These equations are adequate models for phenomena that are characterized by impulsive actions starting at arbitrary fixed points and remaining active on finite time intervals. Strict stability properties of fractional differential equations with non-instantaneous impulses by the Lyapunov approach is studied. An appropriate definition (based on the Caputo fractional Dini derivative of a function) for the derivative of Lyapunov functions among the Caputo fractional differential equations with non-instantaneous impulses is presented. Comparison results using this definition and scalar fractional differential equations with non-instantaneous impulses are presented and sufficient conditions for strict stability and uniform strict stability are given. Examples are given to illustrate the theory.

Existence and finite-time stability of solutions for a class of nonlinear fractional differential equations with time-varying delays and non-instantaneous impulses

“…Compared with some recent results in the literature, such as [6,[8][9][10][11][12][13] and some others, the chief contributions of our study contain at least the following four issues:…”

“…(1) In [6,[8][9][10], authors discussed several types of stability except the finite-time stability, and we first introduce the definition of finite-time stability into the ψ-Hilfer fractional differential equation with non-instantaneous impulses. (2) Compare with [6,[8][9][10][11][12][13], in system (1.4), we study the equation with time-varying delays, which is a significant breakthrough in dealing with a non-instantaneous impulsive ψ-Hilfer fractional differential system. 3The model we are concerned with is more generalized, and some ones in the references are the special cases of it.…”

“…Recently, we found that many researchers have set out to study a new type of impulsive (called non-instantaneous impulsive) fractional differential equations, where the impulsive action starts at an arbitrary fixed point and remains active on a finite time interval, which is very different from the classical instantaneous impulsive case that changes are relatively short compared to the overall duration of the process. For an extensive collection of non-instantaneous impulsive results, we refer the reader to the related literature, such as the monograph [5] and the papers [6][7][8][9][10][11][12][13][14][15][16][17].…”

Application of two different algorithms to the approximate long water wave equation with conformable fractional derivative

“…There are two major approaches in the theoretical formulation of initial value problems for fractional differential equations (Dai, Wang, & Liu, 2016;Imran, Khan, Ahmad, Shah, & Nazar, 2017;Saqib, Ali, Khan, Sheikh, & Jan, in press;Shah & Khan, 2016). One of them is based on the interpretation of the initial condition of fractional systems as a distributed initial condition (Agarwal, O'Regan, Hristova, & Cicek, 2017).…”

On non‐instantaneous impulsive fractional differential equations and their equivalent integral equations

“…As far as we can ascertain, the first studies of these appear to have been in 2014 23,24 and 2015. 25 Research into non-instantaneous impulsive fractional differential equations (NIFDEs) has continued in the last few years, as can be seen from the papers of Agarwal, Hristova, O'Regan, et al [26][27][28][29][30] and the papers of Zada, Ali, et al [31][32][33] The purpose of this work is to investigate the different approaches that have been used to study NIFDEs in the works cited above. We discover that the ways of rewriting the differential equation as an equivalent integral equation fall into two general approaches, according to the lower bound used in the integral equation, and that these approaches have been applied to two different general types of NIFDEs, according to the lower limit of the fractional derivative.…”