Behavior For Solutions Of Differential And Difference Equations

With the increasing development of natural science, the differential and difference equations arise in numbers of important applications, including physics, population dynamics, theory of controls, biologies, medicines and economies. The differential and difference equations are a kind of powerful means to investigate the rule of natural phenomena. Since it is difficult to find their general solutions, therefore, there has been an increasing interest in the study of the nature of solutions of differential equations and difference equations in theory.This dissertation focuses on three sides: The first one is the oscillation and existence of nonoscillatory solutions of differential equations; The second one is the oscillatory theory of differential dynamic systems on time scales; The last one is the periodicity and asymptotic behavior of the solutions of the difference equation. The paper is made up of five chapters. Main contents are as follows:In chapter one, we give a survey to the development and current state of the oscillation of differential equations and difference equations. We also introduce some preliminary material, including some basic concepts of the oscillation theory and some important fixed point theorems.In chapter two, we consider the higher order nonlinear neutral differential equation, by establishing some important differential inequalities, we obtain sufficient conditions of oscillation of the solutions of the equation.In chapter three, through establishing mappings and using fixed point theorems, we get the sufficient conditions for the existence of the bounded nonoscillatory solutions of the higher order neutral differential equations.In chapter four, we investigate the oscillation of the two order equations on time scales. We obtain the sufficient and necessary conditions of oscillation of the bounded solution, our results improve and extend a number of results in the literature.In chapter five, we consider a kind of rational difference equation, by establishing a function which satisfies some conditions, we obtain the results for periodicity and asymptotic behavior of the solutions.