Studies On Oscillation For High Order Differential Equations

The oscillation theory of differential equation is one of important branch of differential equations. In recent years,many scholars take on the research of this field,they have achieved many good results. (see [1]-[42]).The present paper employs a generalized Riccati transformation,Integral average technique and the integral inequality to investigate the oscillation cri-teria for some class of high order nutral differential equations, the results of which generalized and improved some known oscillation criteria.The thesis is divided into four sections according to contents.In Chapter 1, Preface, we introduce the main contents of this paper.In Chapter 2, we are interested in oscillation of solutions of even order neutral differential equations of the form: where t≥t0≥0, n≥2 is even integer. we mainly employed a generalized Riccati transformation and developing ideas exploited by Yuri V. Rogovchenko and Tuncay [1], we establish some new oscillation criteria shall further the investigation and improve the main results of Meng and Xu [2], we obtained several new oscillation criteria at the end of this section.In Chapter 3, we study the oscillation criteria for even order neutral type nonlinear differential equations, We mianly used the generalized Riccati transformation and the integral in-equality to establish the oscillation criterias of the euqtion(3.1.1)by two differ-ent approaches.In Chapter 4, we study the oscillation of certain even order delay differ-ential equations, where n is even integer, is constant, and In this chapter, we introduced a new auxiliary functionΦ(t, s, l), improved the proof range of the oscillation of the eqution,obtained several new oscillation criteria at the end of this section.