An Estimate For Distance Between Adjacent Zeroes Of Solutions Of Functional Differential Equations And Difference Equations

In recent years, with the more development of natural science and social science such as medicine, biology, economics, cybernetic and so on, people have proposed many material mathematical models which are described by functional differential equations and are needed to research. The distribution of zeroes theory of the oscillatory solutions of functional differential equations is the important constituent parts of the oscillation theory of functional differential equations. It can open out the things' essence more accurately and enrich 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 distribution of zeros of the oscillatory solutions of first order neutral delay differential equations, differential equations with a variable delay and difference equations with continuous arguments. Meantime, some estimates for the distance between adjacent zeroes of oscillatory solutions of the equations are established and some sufficient conditions for oscillations of the all solutions of the equations are obtained. These results are illustrated by some examples.Firstly, the distribution of zeroes problems of the oscillatory solutions of first order linear and nonlinear neutral delay differential equations are studied respectively in this paper. Some estimates for the distance between adjacent zeroes of the oscillatory solutions of two classes of the equations under different conditions are established. And some sufficient conditions to satisfy oscillations of the all solutions of the equations are gained.Secondly, first order linear and nonlinear differential equations with a variable delay are considered. Some upper boundary estimates for the distance between adjacent zeroes of the oscillatory solutions of the equations are obtained. Some sufficient conditions to satisfy oscillations of the solutions of the equations are derived.Finally, this paper discusses the distribution of zeroes of solutions of linear, nonlinear single delay and linear, nonlinear several delays difference equations with continuous arguments. Some estimates for the distance between adjacent zeroes of solutions of several different kinds of difference equations by establishing the relationship between difference equations and corresponding delay differential equations are attained. And some sufficient conditions for oscillations of the all solutions of the equations are obtained.