One-Signed Harmonic Solutions and Sign-Changing Subharmonic Solutions to Scalar Second Order Differential Equations

Using a recent modified version of the Poincare-Birkhoff fixed point theorem [19], we study the existence of one-signed T-periodic solutions and sign-changing subharmonic solutions to the second order scalar ODE u" + f(t, u) = 0, being f : R x R -> R a continuous function T-periodic in the first variable and such that f(t, 0) equivalent to 0. Partial extensions of the results to a general planar Hamiltonian systems are given, as well.