Oscillation And Non-oscillation For Delay Difference And Partial Difference Equations

The delay difference and partial difference equations arise in numbers of important applications including problems in population dynamics, chemical reactions, electrical network, mathematical physics problems, as well as finite difference schemes. In the past decade, oscillation theory of delay difference and partial difference equations has developed very rapidly. With the further study of this direction, the studing contents and methods continuously get abundant. A lots of results have been obtained whether for linear equations or for nonlinear equations.This dissertation focuses on the research of the oscillation, existence, asymptotication and classification of positive solutions of linear or nonlinear delay difference equations, partial difference equations and functional equations ect. The paper consists of six chapters. Main contents are as follows:In Chapter 1, we give a survey to the development and situation of the oscillation of delay difference and partial difference equations and introduces main results in this paper. We also introduces some preliminary material, including some basic concepts from the oscillation theory of delay difference equations, some results from calculus of finite differences and some importment fixed point theorems.In Chapter 2, We establish some oscillation and nonoscillation criteria for first order difference equations with variable delays. Our results improve and extend most results in the literature. We also discuss the semicycles of the oscillatory solutions of delay difference equations for the first time.In Chapter 3, we obtain the oscillation and nonoscillation results for second order quasilinear difference equations. We also give a classification of nonoscillatory solutions of second nonlinear neutral difference equations. Some existence results for each kind of nonoscillatory solutions are established.In Chapter 4, we begin with the study of higher order linear delay difference equations and give a series of sufficient conditions for the oscillation. We then discuss higher order nonlinear neutral difference equations and present three comparison theorems for oscillation which are important in studying oscillation of neutral difference equations. We also obtain several global results which are some sufficient condition for the existence of nonoscillatory solution of nonlinear neutral difference equations with forcing terms. Our results improve and extend a number of results in the literature. In last section, we investigates the classification of nonoscillatory solutions of superlinear and sublinear neutral difference equations according to its asymptotic behaviour and establish necessary and sufficient conditions for the existence of each kind of nonoscillatory solutions.In Chapter 5, we establish new sufficient conditions for the oscillation of partial difference equations in critical states and nonexistence of positive solutions of neutral partial difference equations. These results are illustrated by some examples. In the oscillation theory of partial difference equations, it is very difficult to study the existence of partial difference equations. This chapter has studied this field. By using Krasnoselskii's, Knaster's and Shauder's fixed point theorems, we obtain sufficient conditions for existence of positive solutions of linear or nonlinear higher order neutral partial difference equations.The last chapter, Chapter 6 concerns oscillation and nonoscillation of one-variables and two-variables functional equations. A serial of "sharp" conditions for oscillation are obtained.