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 2π-periodic problems of the form −u′′(t)+g(u(t))=f(t)u(0)−u(2π)=u′(0)−u′(2π)=0where g:R→R is a given C1 function and f:[0,2π]→R is continuous. We provide a preservation of symmetry result that is analogous to one given by (Willem, 1989) and a generalization of the theorem given by (Costa and Fang, 2019). Both of these theorems consider the group action of translation — which corresponds to periodicity.