Oscillation And Forced Oscillation Of Impulsive Differential Equations

With the development of modern science and technology, people have proposed a great many new problems about functional differential equations and functional difference equations in natural science and social science, which starve for us to solve them with related mathematical theories. The oscillation theory of impulsive differential equations is the important constituent parts of the oscillation theory of functional differential equations. It is valuable both in theory and practice to investigate such theories widely. The problem of oscillation of impulsive differential equations can open out the things'essence more accurately and make the character of the oscillatory solutions more quantitative. It enriches the oscillation theory of differential equations greatly. This theory got development only in the 90's of the twentieth century and it is a new investigative realm with blooming life-force. So it is not only a demand of the development of mathematics theories themselves, but also a demand of practical applications to investigate such a theory thoroughly, systemically and widely.This paper focuses on the research of the oscillation of first order impulsive differential equations, second order nonlinear neutral impulsive differential equations and higher-order impulsive difference equations. Using impulsive differential inequality, Green's formula and Jensen inequality, new oscillation criteria are obtained.Firstly, the problems of the oscillations of first order nonlinear impulsive differential equations with delay and first order nonlinear neutral impulsive differential equations with delay are studied respectively in this paper. The oscillations of two classes of the equations are proved. And some sufficient conditions to satisfy oscillations of the equations are gained.Secondly, this paper discusses the oscillation of second nonlinear impulsive differential equations with delay; second nonlinear delay impulsive differential equations with force and second nonlinear neutral delay impulsive differential equations with force through establishing the relationship between the second order neutral differential equation and the corresponding first order differential inequality. And some new oscillation criteria of the equations are obtained.Finally, consider the oscillation of higher-order linear impulsive difference equations and nonlinear impulsive difference equations. By making use of the known results, some new oscillatory criteria are obtained.