An antimaximum principle for periodic solutions of a forced oscillator

TL;DR

This paper proves that for a linear periodic oscillator with positive forcing, the existence of a periodic solution guarantees a positive solution, linking the problem to convex analysis separation questions.

Contribution

It establishes a novel connection between periodic solutions of linear oscillators and convex analysis separation, providing new insights into solution positivity.

Findings

Existence of periodic solution implies existence of positive solution.

Links between oscillator solutions and convex analysis separation.

Provides theoretical foundation for positivity in forced oscillators.

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.