The class Omega subset of L(loc)(1) (R) of Besicovitch almost-periodic functions is the closure of the set of all finite trigonometric polynomials with the Besicovitch seminorm. Consider the half-linear second order differential equation (E) d/dt phi(u'(t)) - lambda c(t)phi(u(t)) = 0, where phi(s) = \s\(p-2) s with p > 1 a fixed number and c(t) epsilon Omega. We show that if M{c} := lim(t-->infinity)(1/t) integral(0)(t) c(s + alpha) ds = 0 and M{\C\} > 0, then (E) is oscillatory at +infinity and -infinity for every lambda epsilon R - {0}.
