A variational approach to resonance for asymmetric oscillators

We consider in this note the equation x" + ax(+) - beta x(-) + g(x) - p(t), where x(+) = max{x, 0} is the positive part of x, x(-) = max{-x, 0} its negative part and alpha, beta are positive parameters. We assume that g : R --> R is continuous and bounded on R, p : R --> R is continuous and 2 pi-periodic. We provide some sufficient conditions of Ahmad, Lazer and Paul type for the existence of 2 pi-periodic solutions when (alpha, beta) belongs to one of the curves of the Fucik spectrum corresponding to 2 pi-periodic boundary conditions.