Study On The Oscilation, Asym-Ptotic Behavior And Existence Of Positive Solutions For Difference Equations

In recent years, with the development of science and technology, the difference equations theory is extensively applied in such fields as modern phycics, biology, ecomics, and engineering control and so on. For oscillation theory, asymptotic theory and existence of positive solution theory being the key content of qualitative study of the difference equations, it is of great theoretical and practical value to research the oscillatory properties, asymptotic behavior and existence of positive solutions.This paper is devoted to the problems of qualitative behavior of the nonlinear neutral difference equations with several delays, the nonlinear neutral difference equations with continuous arguments and the high order neutral difference equations.Firstly, this paper focues on the research of the oscillation and asymptotic behavior of solutions of the first order, seconder order nonlinear neutral difference equations with several delays, some criteria for the oscillation of solutions of the equations are established, a sufficient condition for the asymptotic behavior of the non-oscillation solutions of the equations is also given; it discusses the nonlinear neutral delays difference equations with positive and negative coefficients and obtain some sufficient conditions for the oscillation of the equations.Then, this thesis discusses the nonlinear neutral difference equations with continuous arguments. Two sufficient conditions for the oscillation of bounded solutions of the second order equations are obtained; it discusses the oscillatory problem of the other second order equations, three sufficient conditions for the oscillation of solutions and difference operator are given, at the some time, we expand the equtions to even order case and discuss the oscillation of bounded solutions for the high order difference equations with continous arguments. Finally, the high order neutral difference equations are considered. This paper discusses the high order neutral difference equations with variable delays, two sufficient conditions for the oscillation of equations are established; by using the fixed point theorem, it studies the existence of final positive solutions of the high order nonautonomous neutral difference equations, a compacter sufficient conditions than the previous papers is obtained.