An extension of the Poincaré–Birkhoff Theorem coupling twist with lower and upper solutions

In 1983, Conley and Zehnder proved a remarkable theorem on the periodic problem associated with a general Hamiltonian system, giving a partial answer to a conjecture by Arnold. Their pioneering paper has been extended in different directions by several authors. In 2017, Fonda and Ureña established a deeper relation between the results by Conley and Zehnder and the Poincaré–Birkhoff Theorem. In 2020, Fonda and Gidoni pursued along this path in order to treat systems whose Hamiltonian function includes a nonresonant quadratic term. It is the aim of this paper to further extend this fertile theory to Hamiltonian systems which, besides the periodicity-twist conditions always required in the Poincaré–Birkhoff Theorem, also include a term involving a pair of well-ordered lower and upper solutions. Phase-plane analysis techniques are used in order to recover a saddle-type dynamics permitting us to apply the above mentioned results.