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 x ̈+λg(t,x)=0, for λ>0 sufficiently small, with g(t, x) having a superlinear growth at infinity, without requiring the existence of an equilibrium point.