Study On The Oscilation And Nonoscillation For Some Kinds Of Delay Difference Equations

The theory of qualitative behavior of difference equations,which is the important constitute part of the theory of difference equations. Nowadays, with the development of science and technology, a lot of mathematical models, described by delay difference equations, have been put forward in natural science and social science as control theory, modern physics, medical science, economics, bio-science, and so on. Because of applied universality and a mass of mathematics problems relate to itself, which starve for us to solve them with related mathematical theories. For oscillation theory, asymptotic theory and existence of positive solution theory being the key content of qualitative study of the delay 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 first order neutral delay difference equations, the second order nonlinear difference equations with damping term and higher order neutral difference equations with maxima. The relative topics extend the problems studied in the literatures, and give separately some sufficient conditions for oscillation, asymptotic behavior of solutions and existence of positive solutions. Some results which expand and improved the related conclusions in previous literatures are obtained.Firstly, by using formation of functions and reduction to absurdity, the oscillation for a class of first order neutral delay difference equations is studied. Under the different conditions, two sufficient conditions of this equation to be oscillatory are obtained and a sufficient condition for the existence of positive solution of this equation is established.Secondly, for the second order nonlinear difference equations with dampi- ng term, this thesis first studies the oscillation for a class of second order difference equations with nonlinear damping term by using the methods of Riccati, segment difference and difference inequality. Two sufficient conditio- ns for the oscillation of the equation are obtained. At the same time, we expand this conclusion to the cases with maxima and damping term. The oscillation for a second order difference equation with maxima and damping term is studied and two sufficient conditions of this equation to be oscillatory are obtained.Finally, as to the difference equations with maxima, by using formation of functions and reduction to absurdity, the oscillation and asymptotic behavior of solutions of the odd order neutral difference equations are studied and several sufficient conditions are given.