Resonance and rotation numbers for planar Hamiltonian systems: Multiplicity results via the Poincaré–Birkhoff theorem

In the general setting of a planar first order system equation(0.1)u′=G(t,u),u∈R2, with G:[0,T]×R2→R2G:[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)≡0G(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 TT-periodic solutions of unforced planar Hamiltonian systems Ju′=∇uH(t,u)Ju′=∇uH(t,u) and unforced undamped scalar second order equations x″+g(t,x)=0x″+g(t,x)=0. In particular, by means of the Landesman–Lazer condition, we obtain sharp conclusions when the system is resonant at infinity.