Remarks on the preservation and breaking of translational symmetry for a class of ODEs

Abstract

In this paper, we provide both a preservation and breaking of symmetry theorem for $2\pi$-periodic problems of the form \begin{align*} \begin{cases} -u''(t) + g(u(t)) = f(t)\cr u(0) - u(2π) = u'(0) - u'(2π) = 0 \end{cases} \end{align*} where $g:\mathbb{R}\to \mathbb{R}$ is a given $C^1$ function and $f:[0,2\pi ]\to \mathbb{R}$ is continuous. We provide a preservation of symmetry result that is analogous to one given by Willem (Willem, 1989) and a generalization of the theorem given by Costa-Fang (Costa and Fang, 2019). Both of these theorems use group actions that are not normally considered in the literature.