Resonant Sturm–Liouville Boundary Value Problems for Differential Systems in the Plane

Abstract

We study the Sturm-Liouville boundary value problem associated with the planar differential system $J{z}^{\prime }=\nabla V(z)+R(t,z)$, 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.