Oscillatory And Nonoscillatory Behavior Of Several Classes Of Functional Differential Equations

The content of this paper is composed of four chapers. In the first part of this paper, we mainly introduce the background, the status of recent researches and the tendency of development of oscillation theory for functional differetial equations. We also introduce some basic notions on oscillation. In addition, we briefly introduce researches and innovations of this paper. In the second part of this paper, we discuss the first order linear delay differential equations, x′(t)+p(t)x(t-τ(t))=0, 　　　　　　　(0.1) and advanced differential equations, x′(t)-p(t)x(t+τ(t))=0,　　　　　　　(0.2) where p (t),τ(t)∈C(R~+,R~+) and for Eq. (0.1). Comparison theorems for oscillation of Eq.(0.1)and Eq.(0.2) are respectively establishe-d. By applying these comparison theorems new sufficient conditions for os-cillation of all solutions of equations are obtained . In the third part of this paper, we consider the existence of nonoscilla-tory solutions of a class of n-th order neutral functional d ifferential equati-ons, are the same sign with y and f j (t ,y1 )? fj(t,y2)≤Ly1?y2 when ? δ