Resonant Sturm-Liouville boundary value problems for differential systems in the plane

We study the Sturm-Liouville boundary value problem associated with the planar differential system Jz'= ∇V (z) + R(t, z), where V (z) is positive and positively 2-homogeneous and R(t, z) is bounded. Assuming Landesman-Lazer type conditions, we obtain the existence of a solution in the resonant case. The proofs are performed via a shooting argument. Some applications to boundary value problems associated with scalar second order asymmetric equations are discussed.