ASYMPTOTICS OF NONOSCILLATORY SOLUTIONS OF SOME 2ND-ORDER LINEAR-DIFFERENTIAL EQUATIONS

In this paper we prove a theorem on the existence and asymptotic behav iour of nonoscillatory solutions of the equation x''+p(t)x=0, where p( t)=(-lambda(2)+h(t))t(-2 alpha) with lambda > 0 and 0 < alpha < 1. The coefficient p need not be onesigned. Examples show that the same asym ptotic formula can hold either when p(t) is eventually negative, or wh en it oscillates. Moreover, the result can be applied to cases where p is negative, but is such that the classical Lioville-Green approximat ion formula cannot be used.