Necessary and sufficient conditions for oscillation of second order neutral differential equations

Consider the second order nonlinear neutral differential equation with delays: (E) d²/dt²[y(t)-py(t-τ)]+q(t)f(y(t-σ))=0, for t∈[0,∞), where q(t),f(x) are continuous functions, q(t)≥0, yf(y)0 if y≠0, and 0<p<1, τ0, σ0. When f(y) satisfies either the superlinear or sublinear conditions which include the special case f(y)=y|y|^γ-1 of γ1 and 0<γ<1, respectively, we give necessary and sufficient conditions for the oscillation of all continuable solutions of (E). When p=τ=σ=0 in (E), these results reduce to the well known theorems of Atkinson and Belohorec in the special case when f(y)=y|y|^γ-1, γ≠1. © 2000 Academic Press.