An antimaximum principle for periodic solutions of a forced oscillator

Abstract

Consider the equation of the linear oscillator $u"+u=h(\theta)$, where the forcing term $h:\mathbb R\to\mathbb R$ is $2\pi$-periodic and positive. We show that the existence of a periodic solution implies the existence of a positive solution. To this aim we establish connections between this problem and some separation questions of convex analysis.