We provide a geometric assumption which unifies and generalizes the conditions proposed in [11], [12], so to obtain a higher dimensional version of the Poincaré–Birkhoff fixed point Theorem for Poincaré maps of Hamiltonian systems.

VL - 262 UR - http://www.sciencedirect.com/science/article/pii/S0022039616303278 ER - TY - JOUR T1 - Resonance and rotation numbers for planar Hamiltonian systems: Multiplicity results via the Poincaré–Birkhoff theorem JF - Nonlinear Analysis: Theory, Methods & Applications Y1 - 2011 A1 - Alberto Boscaggin A1 - Maurizio Garrione KW - Multiple periodic solutions KW - Poincaré–Birkhoff theorem KW - Resonance KW - Rotation number AB -In the general setting of a planar first order system (0.1)u′=G(t,u),u∈R2, with G:[0,T]×R2→R2, we study the relationships between some classical nonresonance conditions (including the Landesman–Lazer one) — at infinity and, in the unforced case, i.e. G(t,0)≡0, at zero — and the rotation numbers of “large” and “small” solutions of (0.1), respectively. Such estimates are then used to establish, via the Poincaré–Birkhoff fixed point theorem, new multiplicity results for T-periodic solutions of unforced planar Hamiltonian systems Ju′=∇uH(t,u) and unforced undamped scalar second order equations x″+g(t,x)=0. In particular, by means of the Landesman–Lazer condition, we obtain sharp conclusions when the system is resonant at infinity.

VL - 74 UR - http://www.sciencedirect.com/science/article/pii/S0362546X11001817 ER -