Existence Of Nonoscillatory Solutions And Oscillation For The Dynamic Equations On Time Scales

Differential equations and difference equations are useful approaches to describe the laws of change about natural phenomena.With the deVelopment of science and technology, differential equations and difference equations have been applied in a number of important fields,including physics,population dynamics,automatic control,biology,medicine and economics. Since it is difficult,even impossible sometimes,to find their general solutions,there has been an increasing interest in studying theoretically the nature of solutions for differential equations and difference equations in recent years.Differential equations lead to difference equations after discretization.Experience has shown that lots of nature of differential equations has yet been retained after discretization. However,there are also specific equations showing that differential equations have a completely different nature with the corresponding difference equations.As for the nonlinear equations,sometimes the two have a greater difference.Such possible differences may result in people's repetition of the study in differential equations and their corresponding difference equations.Sometimes both the continuous and the discrete components will be held in a single problem.It is even not clear for us to decide whether the problem is the continuous or the discrete one.This brings us great inconvenience for our research.On the other hand,the structure of differential operator and that of difference operator are very similar,therefore,it is a urgent task to find in the current scientific research a new theoretical framework,which can unify the continuous systems and the discrete ones for the sake of study in order to not only avoid unnecessary repetition in work but also be able to make better insight into the nature of the differences among different systems.The theory of time scales is a theory unifying the continuous systems and the discrete ones,which pioneers a new area for mathematical study. Since the distinguishing feature of the theory of time scales is to unify and to popularize,the study of this theory has important theoretical significance and practical value.This dissertation focuses on five issues:The first part is the introduction.The second one is the existence of positive solutions for higher order nonlinear neutral dynamic equations on time scales;The third one is the existence of positive solutions for higher order nonlinear neutral partial differential equations on time scales;The fourth one is the oscillation criteria for the second order nonlinear neutral dynamic equations on time scales;The last one is the existence of bounded nonoscillatory solutions for higher order difference equations.Main contents are as follows:In chapter one,we give a survey to the background in which dynamic equations on time scales is applied and the situation of its study both at home and abroad.This part also includes some preparatory knowledge,such as the basic concepts of time scales,lemmas and important fixed-point theorems.In chapter two,we study the existence of positive solution for a kind of higher order nonlinear neutral dynamic equations on time scales.Since the function either is monotonous or meets the Lipschitz condition,starting separately from the two conditions,we establish the criteria for the existence of positive solutions by constructing appropriate bounded closed convex subsets in Banach space and theirs completely continuous mappings,using Krasnoselskii's fixed point theorem and the Banach contraction mapping principle,what's more,we obtain the corresponding examples according to the different conditions the function satisfies.In chapter three,we discuss the existence of positive solution for a kind of higher order nonlinear neutral partial differential equations on time scales.According to the monotonicity of the function,we establish the criteria for the existence of positive solutions by constructing appropriate bounded closed convex subsets in Banach space and theirs completely continuous mappings,using the Krasnoselskii's fixed point theorem.At the same time,examples are given.In chapter four,we investigate the oscillation for a kind of second order nonlinear neutral dynamic equations on time scales.Through the comparison of equations,we obtain the sufficient conditions of oscillation for the equations.In chapter five,we study the existence of bounded nonoscillatory solutions for three kinds of higher order nonlinear neutral difference equations with positive and negative coefficients. By structuring bounded dosed convex subsets in Banach space which is composed of bounded sequence of real numbers and the corresponding continuous mappings,using the Schauder's fixed point theorem and the Krasnoselskii's fixed point theorem,we establish the criteria for the existence of bounded nonoscillatory solutions.