A Poincar\'e-Birkhoff theorem for multivalued successor maps with applications to periodic superlinear Hamiltonian systems

We provide a new version of the Poincar\'e-Birkhoff theorem for possibly multivalued successor maps associated with planar non-autonomous Hamiltonian systems. As an application, we prove the existence of periodic and subharmonic solutions of the scalar second order equation $\ddot x + \lambda g(t,x) = 0$, for $\lambda>0$ sufficiently small, with $g(t,x)$ having a superlinear growth at infinity, without requiring the existence of an equilibrium point.