A Poincaré-Birkhoff theorem for multivalued successor maps with applications to periodic superlinear Hamiltonian systems

We provide a new version of the Poincaré-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.