Planar Hamiltonian systems at resonance: The Ahmad-Lazer-Paul condition

We consider the planar Hamiltonian system Ju′ = ∇F(u) + ∇uR(t,u), t ∈ [0,T], u ∈ ℝ2, with F(u) positive and positively 2-homogeneous and ∇uR(t, u) sublinear in u. By means of an Ahmad-Lazer-Paul type condition, we prove the existence of a T-periodic solution when the system is at resonance. The proof exploits a symplectic change of coordinates which transforms the problem into a perturbation of a linear one. The relationship with the Landesman-Lazer condition is analyzed, as well. © 2012 Springer Basel AG