Study On The Oscillation Of Several Classes Of Neutral Delay Differential Equations

In real life,there are oscillation phenomena everywhere,such as sound transmission oscillation,bridge oscillation,mechanical oscillation,pendulum swing,stock vibration,oscillation behavior of power systems,etc.The oscillation phenomenon of functional differential equations with time delays is also ubiquitous,such as the following logistic equation of a single population with time delaysx^′（t）=ax（t）{1-_x（t-τ）/K},where a>0,K>0,τ>0.When the time delayτ（?）1,the solution given the ini-tial value oscillates steadily above and below K.With the development of the subject field,the oscillation theory of delay differential equations has developed into an impor-tant branch of the qualitative theory of functional differential equations.The direction usually involved in the oscillation theory of delay differential equations is to explore the sufficient conditions for the oscillation of all solutions and the existence conditions for non-oscillation solutions,etc.Since the 1980s,the oscillation theory of delay differen-tial equations has developed rapidly,and a lot of research work has emerged.Among these researches,there is a very important class of equations is neutral delay differen-tial equations.The so-called neutral delay differential equations is a class of equations with delays in the highest derivatives.Neutral equations have many applications in the natural sciences and technology,for example,in the study of communication networks,the mathematical model of the propagation route of lossless circuits is a neutral delay differential equation by some transformation.Due to the special property of time delay in the highest derivative of neutral delay differential equations,it becomes much more difficult to deal with in terms of ideas and techniques.In 1977,Norkin published an article on the oscillatory of neutral functional differential equations,but the strict conditions it imposed virtually eliminated the effects of neutral terms.In 1980,Zahariev and Baǐnov proved that the oscillatory properties of neutral equations and their standard equations are fundamentally different.In 1980s,Baǐnov,Myshikis,Gy?ri,etc.gave a general method to study the oscillatory and asymp-totic behavior of neutral differential equations.In 1985,Grammatikopoulos et al.gave some sufficient conditions for oscillatory and asymptotic behavior of neutral differential equations with variable coefficients.These works laid the foundation for the oscillation theory of neutral delay differential equations,and created new research ideas and meth-ods.Therefore,more and more scholars have studied the oscillation and asymptotic behavior of neutral delay differential equations.Inspired and motivated by these works,in this paper,we mainly study the oscillatory and non-oscillatory properties of several classes of neutral delay differential equations.We obtain some oscillation criteria of the equations and the existence conditions of the non-oscillatory solutions of the equations,and some examples are used to verify these results.Our conclusions generalize and im-prove the research on the oscillation theory of some neutral delay differential equations.In Chapter 1,we firstly introduce the development history of the oscillation theory,explain the influence of the term with time delay on the condition of the differential equations,and briefly explain the research status and some main results of the oscillation theory of the delay differential equations.Secondly,we briefly introduce the research results of this paper.In Chapter 2,we study the oscillatory and non-oscillatory properties of a class of second-order nonlinear neutral delay differential equations with multiple positive and negative coefficients.On the one hand,when the neutral term coefficients satisfy different ranges,we give the existence conditions of the non-oscillatory solutions of the equations.On the other hand,in the case where the neutral term coefficients are bounded,we give the oscillation criterion for the bounded solution of the equations.In Chapter 3,we explore the oscillatory properties of a class of even-order quasi-linear neutral delay differential equations.When the coefficients of the neutral terms satisfy different ranges,we give oscillation criteria of the equations in the canonical and non-canonical cases,respectively.In Chapter 4,we consider the oscillatory properties of a class of even-order Emden-Fowler type neutral delay differential equations.On the one hand,when the neutral term coefficient is greater than 1,we give the oscillation criteria of the second-order Emden-Fowler type neutral delay differential equations in the canonical and canonical cases,respectively.On the other hand,when the neutral term coefficients satisfy different ranges,we give the oscillation criteria of the even-order Emden-Fowler type neutral delay differential equations whose order is greater than two in the normal case.In Chapter 5,we study the oscillatory properties of a class of third-order neutral delay differential equations.When the coefficient of the neutral term is greater than 1,we give the oscillation criteria of third-order neutral delay differential equations in the canonical and non-canonical cases,respectively.