A degree associated to linear eigenvalue problems in Hilbert spaces and applications to nonlinear spectral theory

TL;DR

This paper extends the connection between eigenvalue problems and topological degree theory to infinite-dimensional Hilbert spaces, solving a conjecture in nonlinear spectral theory and applying it to a perturbed motion equation.

Contribution

It introduces a new degree concept for linear eigenvalue problems in Hilbert spaces, enabling the proof of a global continuation conjecture in nonlinear spectral theory.

Findings

Solved a conjecture on global continuation in nonlinear spectral theory.

Established a Rabinowitz type global continuation property for solutions.

Extended eigenvalue-degree link to infinite-dimensional Hilbert spaces.

Abstract

We extend to the infinite dimensional context the link between two completely different topics recently highlighted by the authors: the classical eigenvalue problem for real square matrices and the Brouwer degree for maps between oriented finite dimensional real manifolds. Thanks to this extension, we solve a conjecture regarding global continuation in nonlinear spectral theory that we have formulated in a recent article. Our result (the ex conjecture) is applied to prove a Rabinowitz type global continuation property of the solutions to a perturbed motion equation containing an air resistance frictional force.