Frequent Oscillatory Solutions For Several Kinds Of Difference Equations And Dynamic Equations On Time Scales

With the increasing development of science and technology, in the field of naturalscience and edging field including physics, population dynamics, theory of control,biology, medicine, many mathematical models wich are described by difference equationsand partial difference equations are proposed. It is starved for us to solve them with relatedmathematical theories.Partial difference equations are difference equations that involve functions with twoor more independent integer variables. The partial difference equations occur frequently inthe approximation of solutions of partial differential equations by finite differencemethods, random walk problems, the study of molecular orbits, mathematical physicsproblems and etc. But the oscillatory theories get rapid development only in recent years.So it is a new investigative realm with blooming life-force. Because of the progress inmodern science and technology, the studies about this new academic branch are not only ademand of the development of mathematics theories themselves, but also a demand ofactual applications.The theory of time scales can unify continuous and discrete cases, which pioneers anew mathematical area. This theory not only unifies differential and difference equations,reveals the essence of continuous and discrete cases, avoids repeat study, but also containsmany other cases. The prominent peculiarity of time scales is unification andgeneralization, so the study of time scales has important significance in theory andapplication.This dissertation focuses on three sides: frequent oscillation of difference equations,frequent oscillation of partial difference equations and frequent oscillation for dynamicequations on time scales. This paper is made up of four chapters. Main contents are asfollows:Firstly, we give a survey to the development and current state of the differenceequations, partial difference equations and dynamic equations on time scales.Secondly, we investigate frequently oscillatory solutions of second-order nonlinear neutral difference equations based on the theory of ferquency measures. Some sufficientconditions to satisfy frequent oscillations of these equations are gained. At the same time,an example is given.Thirdly, we study frequent oscillatory solutions for neutral partial differenceequations with variable delay and nonlinear partial difference equation with continuousvariables. These results are illustrated by examples.Finaly, based on frequency measures and the theory of time scales, this paperconsiders frequent oscillations of second-order neutral dynamic equations. At the sametime, examples are given.