Periodic solutions of Hamiltonian systems coupling twist with generalized lower/upper solutions

The Hamiltonian systems considered in this paper are obtained by weakly coupling two systems having completely different behaviors. The first one satisfies the twist assumptions usually considered for the application of the Poincaré–Birkhoff Theorem, while the second one presents the existence of some well-ordered lower and upper solutions. In the higher dimensional case, we also treat a coupling situation where the classical Hartman condition is assumed.