An extension of the Poincare-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 Urena established a deeper relation between the results by Conley and Zehnder and the Poincare-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 Poincare-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. (c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).