A note on a nonresonance condition at zero for first-order planar systems

Abstract

We introduce a Landesman-Lazer type nonresonance condition at zero for planar systems and discuss its rotational interpretation. We then show an application concerning multiplicity of T-periodic solutions to unforced Hamiltonian systems like $$ Ju'=\nabla H(t, u), \quad \nabla H(t, 0) \equiv 0, $$ for which the nonlinearity is resonant both at zero and at infinity, refining and complementing some recent results.

Keywords

• Landesman-Lazer condition
• rotation number
• multiplicity