Double resonance in Sturm-Liouville planar boundary value problems

We provide some existence results for Sturm-Liouville boundary value problems associated with the planar differential system $J{z}^{\mathrm{\prime }}=g(t,z)+r(t,z)$ where g is suitably controlled by the gradient of two positively homogeneous functions of degree 2 and r is bounded. We study the existence of solutions when a double resonance phenomenon occurs by the introduction of Landesman-Lazer type of conditions. Applications to scalar second order differential equations are given.