Oscillation and existence of positive solutions in a class of higher order neutral equations

Consider the higher order neutral differential equation [x(t) - x(t - tau)]((n)) + Q(t)x(t - sigma) = 0, t greater than or equal to t(0) with Q(t) continuous, tau > 0, sigma greater than or equal to 0, and n odd. We establish several new sufficient conditions for the oscillation of all solutions and the existence of a positive solution by an associated ordinary differential equation. Some of these conditions are sharp and improve the known results in the literature which include the newest one by J. S. Yu (Chinese Ann. Math. Ser: A 16, 1995, 33-42). (C) 1997 Academic Press.