Oscillation For Two Class Of Second Order Neutral Differential Equations

For the widely applicative background and rich theory foundation, this paper, consisting three chapters, studies oscillation for two class of second order neutral dif-ferential equations by the method of existent literatures. A series of new results are obtained, and some of them improve or extend the related results in literatures. The main contents of this paper as following:Chapter1introduces firstly the development process and the research significance of functional differential equation and differential equation with impulses. Then,we summarize the main research results of the two classes of differential equations. At last, contents and methods of this paper are presented.Chapter2introduces firstly the main results of oscillation for the neutral func-tional differential equations,Second, by the methods in existent papers, this chapter consider oscillation for the second order nonlinear neutral differential equation (γ(t)[x(t)+p(t)x(τ(t))]’)’+q(t)f(x(σ(t)))=0, t≥0. wherer γ∈C1([t0,∞)),p,σ,τ∈([t0,∞),R+),f∈C(R,R),q∈C([T0,∞),R+) and q(t) is not equal to zero finally. The chapter extends the linear neutral differen-tial equation to nonlinear differential equation. Under the different conditions, several comparison theorems are established for oscillation of the equation and first order differential inequalities, which extend and improve the corresponding results on the literature.Chapter3introduces some research conclusions of second order impulsive differ-ential equations. Then, this chapter proves oscillation of the second order impulsive neutral functional differential equation is equivalent to the second order non-impulsive functional differential equation (r{t)[y(t)+P(t)y(τ(t))]’)’+Q(t)y(σ(t))=0,t≥γ,a.e. where The chapter extends the non-impulsive neutral differential equation to the impulsive differential equation. Several comparison theorems are established for oscillation of the equation and first order differential inequalities. And the results in this chapter extends the results in the existent paper.