Subharmonic solutions of the forced pendulum equation: a symplectic approach

Using the Poincaré-Birkhoff fixed point theorem, we prove that for every β > 0 and for a large (both in the sense of prevalence and of category) set of continuous and T-periodic functions f: ℝ → ℝ with ∫0 T f(t)dt = 0, the forced pendulum equation x″ + β sin x = f(t) has a subharmonic solution of order k for every large integer number k. This improves the well known result obtained with variational methods, where the existence when k is a (large) prime number is ensured