Oscillation And Nonoscillation Of Delay Dynamic Equations

The research of oscillation theory began in the Newton era in 18 th century.Since the 1980 s,contents and methods of the oscillation theory have been enriched and developed greatly in both linear differential equations and nonlinear differential equations.Numerous famous papers and monographs have been published.Oscillation theory which is regarded as one of the three basic qualitative theories is widely used in areas of control,economics,ecology and life sciences.Therefore,it is very significant and meaningful to study oscillation and control of differential equations.Since delay dynamic equations can take full account of the influences of history and current status on the change of future status,they are more profound and accurate to reflect the changing law and reveal essential features compared with traditional differential equations.Delay dynamic equations arise in areas of natural science and engineering technology,such as dynamic behavior of delay network system,population dynamics and stability theory and so on.Because of their great influences on practical problems and mathematical theory itself,as a challenging research topic,dynamic problems of delay dynamic equations have been paid more and more attentions.Oscillation theory is one of the central contents in the theory of delay dynamic equations and it is also an important part in qualitative theory.Due to the influence of time delay,the oscillation theory of delay dynamic equations will be more complex and significant in both theory and practice.In this paper,we mainly study the qualitative properties of oscillatory and nonoscillatory solutions for several classes of delay dynamic equations by means of fixed point theorems,inequality technique,comparison theorem,Riccati transformation and the method of eigenvalue and eigenfunction.We present some results about the existence and uniqueness of oscillatory and nonoscillatory solutions,oscillation criteria and the distribution of zeros of oscillatory solutions,which generalize and improve some known results.The main contents of this paper are as follows:In the first chapter,we make a brief overview for the background and status of the research and introduce our main work.In chapter 2,we present several new sufficient conditions for the existence of oscillatory solutions for second order neutral delay differential equations by Krasnoselskii's fixed point theorem and inequality technique.In chapter 3,we consider the existence and classification of nonoscillatory solutions for several different kinds of delay dynamic equations.On the one hand,we establish some criteria for the existence and classification of nonoscillatory solutions and some sufficient and necessary conditions for the existence of oscillatory and nonoscillatory solutions for a class of second order superlinear Emden-Fowler type dynamic equations by means of SchauderTychonoff fixed point theorem and H?lder inequality.On the other hand,we present respectively several sufficient conditions for the existence of nonoscillatory solutions for second order mixed neutral delay differential equations with positive and negative terms,higher order nonlinear mixed neutral delay differential equations and higher order mixed differential equations with distributed delay by Banach contraction mapping principle.In chapter 4,we study the second order nonlinear neutral delay differential equations and nonlinear neutral fractional partial system with forcing term.By using comparison theorem,Riccati transform,the corresponding properties of related first order differential inequality,inequality technique and the method of eigenvalue and eigenfunction,some oscillatory criteria are established,which improve and generalize certain known results.In chapter 5,we discuss the distribution of zeros for a class of second order nonlinear neutral delay differential equations.Some upper bounds of distances between adjacent zeros are obtained by inequality technique,structuring certain function iteration sequences and some properties of related differential inequalities.Results in this chapter are more exact than previous papers since we not only discuss the oscillation of equations but also consider the upper bound between adjacent zeros of oscillatory solutions.The last chapter is mainly dedicated in induction and summary for our main research contents and results.Meanwhile,it makes a prospect for further research work.