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 \begin{equation}\label{sistemaabstract} u'=G(t, u), \quad u \in \mathbb{R}^2, \end{equation} with $G: [0, T] \times \mathbb{R}^2 \to \mathbb{R}^2$, we study the relationships between some classical nonresonance conditions (including the Landesman-Lazer one) - at infinity and, in the unforced case, i.e. $G(t, 0) \equiv 0$, at zero - and the rotation numbers of ``large'' and ``small'' solutions of (\ref{sistemaabstract}), respectively. Such estimates are then used to establish, via the Poincar\'e-Birkhoff fixed point theorem, new multiplicity results for $T$-periodic solutions of unforced planar Hamiltonian systems $Ju'=\nabla_u H(t, u)$ 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.