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 \(\ddot{x} + \lambda g(t,x) = 0\) $\ddot{x}+\lambda g(t,x)=0$ , for \(\lambda >0\) $\lambda >0$ sufficiently small, with g(t, x) having a superlinear growth at infinity, without requiring the existence of an equilibrium point.