Abstract
This paper deals with the oscillation of the first-order differential equation with several delay arguments x' (t) + Sigma(m)(i=1) p(i) (t) x (tau(i) (t)) = 0, t >= t(0), where the functions p(i), tau(i) is an element of C ([t(0), infinity), R+), for every i = 1, 2, ... , m, tau(i) (t) <= t for t >= t(0) and lim(t ->infinity) tau(i) (t) = infinity. In this paper, the state-of-the-art on the sharp oscillation conditions are presented. In particular, several sufficient oscillation conditions are presented and it is shown that, under additional hypotheses dealing with slowly varying at infinity functions, some of the "liminf" oscillation conditions can be essentially improved replacing "liminf" by "limsup". The importance of the slowly varying hypothesis and the essential improvement of the sufficient oscillation conditions are illustrated by examples.