Double resonance in Sturm-Liouville planar boundary value problems

TL;DR

This paper investigates the existence of solutions for planar Sturm-Liouville boundary value problems exhibiting double resonance, using Landesman-Lazer conditions, with applications to scalar second order differential equations.

Contribution

It introduces new existence results for double resonance problems in planar Sturm-Liouville systems under Landesman-Lazer conditions, extending previous work.

Findings

Existence of solutions under double resonance conditions.

Application of Landesman-Lazer conditions to planar systems.

Results applicable to scalar second order differential equations.

Abstract

We provide some existence results for Sturm-Liouville boundary value problems associated with the planar differential system $Jz' = g(t, z) + r(t, z)$ 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.