Interval oscillation theorems for asecond-order linear differential equation

Abstract

Interval oscillation criteria are given for the forced second-order linear differential equation Ly(t) = (p(t)y′)′ + q(t)y = ƒ(t), tε (0, ∞), where p, q, ƒ are locally integrable functions and p(t) > 0, for t > 0. No restriction is imposed on ƒ( t) to be the second derivative of an oscillatory function as assumed by Kartsatos [1). Our results also allow both q and f to change sign in the neighborhood at infinity. In particular, we show that all solutions of y″ + c(sin t)y = t β cos t with β ≥ 0 are oscillatory, for c ≥ 1.3448. This improves an estimate given by Nasr [2] for the linear equation.