Study On The Oscilation And Existence Of Positive Solutions Of Several Kinds Of Neutral Difference Equations

In recent years, with the development of science and technology, the difference equation theory not only has important applications in such fields as physics, space satellite, etc., but also has become an indispensable mathematics tool in such natural science and domain of the social sciences as economics, biology, control theory, etc.. People have put forward a lot of mathematics modle that described by neutral difference equations which need us to study. For oscillation thory, existence of positive solution theory and stability theory being the key content of qualitative study of the neutral difference equations, it is of great theoretical significance and practical value to research the oscillatory properties, stability and existence of positive solution.This thesis firstly discusses the problem of qualitative behavior of a delay difference equations with maxima, a second order and a high-order neutral difference equations with continuous argument and a high-order nonlinear neutral difference equation. The relative topics extend the problems studied in the literatures, and gives separately some sufficient conditions for oscillation, asymptotic behavior of solutions and existence of positive solutions.As to the difference equations with maxima, firstly two sufficient conditions of oscillation to first order equations are given. Then we study the asymptotic and oscillation behavior of second order equations, and several sufficient conditions are given.For the difference equations with continuous argument, we study the oscillation behavior of second order equations with continuous argument, and several sufficient conditions are given. Meanwhile, we discuss the oscillation and asymptotic behavior to high-order equations, using reduction to absurdity and mathematics induction, these results about oscillation in the prevenient literatures were expanded to the high order case. In the last paper of this thesis, by using the fixed point theorem, we study the existence of final positive solution of a high-order nonlinear neutral delay difference equation. The range of parameter is extended to the whole real number field, and two sufficient conditions is obtained.