OSCILLATION AND NONOSCILLATION CRITERIA FOR DIFFERENTIAL EQUATIONS OF SECOND ORDER

Abstract

We give necessary and sufficient conditions such that

the homogeneous differential equations of the type: (r(t)x′(t))′ + q(t)x′(t) + p(t)x(t) = 0

are nonoscillatory where r(t) > 0 for t ∈ I = [α, ∞), α > 0. Under the suitable conditions we show that the above equation is nonoscillatory if and only if for γ > 0,

(r(t)x′(t))′ + q(t)x′(t) + p(t)x(t − γ) = 0
is nonoscillatory. We obtain several comparison theorems.