A oscillation theorem for a class of even order neutral differential equations

The even order neutral differential equation dn dtn x(t) + h(t)x(t − τ) + f t,x g(t) =0 (1) is considered under the following conditions: n 2 is even; τ > 0; h ∈ C(R); g ∈ C[t0,∞), limt→∞g(t)=∞; f ∈ C([t0,∞) × R), uf (t , u) 0 for (t ,u) ∈ [t0,∞) × R, and f (t,u) is nondecreasing in u ∈ R for each fixed t t0. It is shown that (1) is oscillatory if and only if the certain non-neutral differential equation is oscillatory, for the case where 0 μ h(t) λ<1 or 1<λ h(t) μ.  2002 Elsevier Science (USA). All rights reserved.