On the asymptotic behaviour of solutions of x″ + a(t)f(x) = 0

Abstract

We consider the equation:where a: [0, ∞) → R1, a(t) > 0, a'(t) is continuous,f:( −∞, +∞) → R1, f is continuous, and xf(x) > 0 for x ≠ 0. The problem is to give conditions on a(t) and f(x) to ensure that all solutions of (1) tend to zero as t → ∞. First, however, we give some sufficient conditions and some necessary and sufficient conditions to ensure that all solutions of (1) are oscillatory or bounded.