| With the development of modern society, the research of the qualitative properties of ordinary differential equations has been prosperous in the field of mathematics. Neutral differential equations usually arise in the natural science and engineering areas. Because it can well explain the various natural phenomenons, many researchers have paid attention to the problems for a long time. In recent years, a large number of mathematicians worked on the analysis of differential equations with delay and nonlinear differential equations. At present, there are few research results about neutral differential equations with multiple delays. Based on the above situation, oscillation of several categories neutral differential equations with multiple delays will be further discussed in this article.The thesis consists of four sections.Chapter 1 is the preface, which contains some results of other people and the main content of this paper.In Chapter 2, we consider the oscillation of the second order neutral differential equation with multiple delays where r(t)> 0 and r(t) € C1([t0,∞)).The obtained results are based on the new comparison theorems, that enable us to reduce the problem of the oscillation of the second order equation to the oscillation of the first order equation. The obtained comparison principles essentially simplify the examination of the studied equation.In Chapter 3, we consider the oscillation of the second order nonlinear neutral differen-tial equation with multiple delays where γ and β are the ratios of two positive odd integers,α(t)>0 and α(t)∈C([t0,∞)). The oscillation of second order nonlinear equation is judged by solution of the first order neutral differential inequality.In Chapter 4, we consider the oscillation of the third order neutral differential equation with multiple delays where γ is the ratio of two positive odd integers, a(t)> 0 and a(t) ∈ C([to,∞)). The oscillation of the third order neutral differential equation with multiple delays is judged by an integral inequality.The two equations study in the second chapter, the third chapter satisfy 0≤pi(t)≤ p0<∞, qj(t)>0, τi(t) E C1([t0,∞)),σj(t) ∈C1([t0,∞)), and τi(t)>0,σj(t)>0, and satisfy limt-∞τi(t)=∞,limt→∞ σj(t)=∞, τo’(t)≥τ0>0, τi○σj=σj○τi. The equation study in the four chapter, pi(t) satisfy 0< pi(t)≤p<1, and τi(t)≤t,σj(t)≤ t,limt→∞ τi(t)=00, limt→∞ σj(t)=∞. Where i∈{1,2,…,m},j∈{1,2,…,n}. |
