The Brouwer degree associated to classical eigenvalue problems and applications to nonlinear spectral theory

TL;DR

This paper links classical eigenvalue problems with Brouwer degree theory to solve a conjecture in nonlinear spectral theory for finite dimensions, highlighting challenges in extending to infinite dimensions.

Contribution

It establishes a novel connection between eigenvalue problems and Brouwer degree, solving a conjecture in finite-dimensional nonlinear spectral theory.

Findings

Solved a conjecture on global continuation in nonlinear spectral theory

Connected eigenvalue problems with Brouwer degree theory

Identified open problems in infinite-dimensional cases

Abstract

Thanks to a connection between two completely different topics, the classical eigenvalue problem in a finite dimensional real vector space and the Brouwer degree for maps between oriented differentiable real manifolds, we were able to solve, at least in the finite dimensional context, a conjecture regarding global continuation in nonlinear spectral theory that we formulated in some recent papers. The infinite dimensional case seems nontrivial, and is still unsolved.