Consider the second order nonlinear neutral differential equation with dela
ys: (E) d(2)/dt(2)[y(t) - py(t - tau)] + q(t)f(y(t - sigma)) = 0, far t is
an element of [0, infinity), where q(t),f(x) are continuous functions, q(t)
greater than or equal to 0, yf(y) > 0 if y not equal 0, and 0 < p < 1, pi
> 0, sigma > 0. When f(y) satisfies either the superlinear or sublinear con
ditions which include the special case f(y) = y/y/(y-1) of gamma > 1 and 0
< <gamma> < 1, respectively, we give necessary and sufficient conditions fo
r the oscillation of all continuable solutions of (E). When p = <tau> = sig
ma = 0 in (E), these results reduce to the well known theorems of Atkinson
and Belohorec in the special case when f(y)= y/y/(y-1), gamma not equal 1.
(C) 2000 Academic Press.