Resonance and rotation numbers for planar Hamiltonian systems: Multiplicity results via the PoincarBirkhoff theorem

In the general setting of a planar first order systemu′= G(t,u),u∈R2, with G:[0,T] ×R2→R2, we study the relationships between some classical nonresonance conditions (including the LandesmanLazer 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 PoincarBirkhoff 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 equationsx″+g(t, x)=0. In particular, by means of the LandesmanLazer condition, we obtain sharp conclusions when the system is resonant at infinity. © 2011 Elsevier Ltd. All rights reserved.