Multiplicity of Periodic Solutions for Nearly Resonant Hamiltonian Systems

We prove a multiplicity result for the periodic problem associated with a Hamiltonian system whose Hamiltonian function has a twisting part and a nonresonant part. The possible approach to resonance together with some kind of Landesman–Lazer conditions is also analyzed. We propose a new version of this condition, and we also treat the so-called double resonance situation.