Oscillation And Stability Of Fractional Impulsive Differential Equation

Fractional differential equations are the generalizations of integral differential equations.Besides mathematics,viscoelasticity,electrochemistry,physics,control system,porous media,electromagnetics and so on all involve fractional differential equations.Many researches are devoted to the qualitative properties of this kind of equations,especially for its oscillation and stability.Meanwhile,impulsive differential equations are recognized as adequate mathematical models for studying evolution processes that are subject to abrupt changes in their states at certain moments.Many applications in theoretical physics,biotechnology,economy,pharmacokinetics,population ecology and other application fields exhibit impulsive effects.Due to the intensive development about the theory of impulsive differential equations and its widely applications in diverse fields,impulsive fractional differential equations have become a new hot topic.In this paper,the oscillation and stability of several kinds of fractional order impulsive equations are studied by means of inequality technique,Riccati transformation,analysis of real roots of characteristic equation,etc.This paper is structured as follows:In Chapter 1,the significance,application and research background of fractional order impulsive differential equations are introduced.In Chapter 2,we study the generalized zero distribution of the solutions of second order neutral difference equations.By using inequality technique,special function sequence and non–increasing solutions of the corresponding first order difference inequalities,we give some new estimates of the generalized zero distribution of the oscillation solutions,and generalize and improve some known results.In Chapter 3,the oscillation of neutral differential equations are considered.Firstly,we consider the oscillation of third order neutral differential equations with non–canonical operators.The method used is to establish a sufficient condition for the nonexistence of Kneser solution.Combined with the result of almost oscillation of the equations,we establish a sufficient condition for the oscillation of the equations.Secondly,the oscillation of secondorder mixed Emden–Fowler type differential equations is studied by using inequality principle,comparison principle and Riccati transformation,and the sufficient conditions for the oscillation of the equation are obtained.In Chapter 4,we study the oscillation of the Conformable fractional differential equations by establishing the properties of Conformable fractional calculus.We study the oscillation of three kinds of fractional differential equations,which are the fractional order differential equations with finite delay arguments,the fractional order neutral differential equations and the fractional order differential equations with damping term by Gronwall inequality,Riccati transformation and Comparison principle.And establish sufficient conditions for the oscillation of three kind of equations.In Chapter 5,the oscillation of impulsive differential equations is considered.Firstly,Caputo fractional impulsive differential equations are considered.By using inequality principle and Bihari lemma,the sufficient conditions for the oscillation of the equation are obtained.Then,by using fractional Ricatti transform,the oscillation of Riemann–Liouville fractional impulsive differential equation is studied,and the sufficient conditions for the oscillation of the equation are given,as well as find out the impulse condition that makes the oscillation of the system change.Finally,the interval oscillation of impulsive differential equations is studied.By estimating the ratio of unknown function y(t)and y(t-?(t)),the sufficient conditions for the oscillation of the equation are givenIn Chapter 6,the stability and oscillation of Caputo fractional differential equations with distributed delay are studied.By using the constant variation formula of Caputo fractional order differential equations and the semigroup property of Mittag–Leffler function,the study of fractional order differential equations is transformed into the study of higher order difference equations,and the sufficient and necessary conditions for the stability and oscillation of the formula are obtained.In Chapter 7,we summarize the main results of this paper and clarify the future research objectives.