A New Spectrum for Nonlinear Operators in Banach Spaces

TL;DR

This paper introduces a new concept of spectrum for nonlinear operators in Banach spaces, extending classical spectral theory and exploring its properties and applications in bifurcation analysis.

Contribution

It defines a spectrum for nonlinear operators at a point, generalizing the linear case, and investigates its properties and applications in bifurcation theory.

Findings

sigma(f,p) is always closed

sigma(f,p) equals sigma(f'(p)) when f is C^1

Applications to bifurcation theory are demonstrated

Abstract

Given any continuous self-map f of a Banach space E over K (where K is R or C) and given any point p of E, we define a subset sigma(f,p) of K, called spectrum of f at p, which coincides with the usual spectrum sigma(f) of f in the linear case. More generally, we show that sigma(f,p) is always closed and, when f is C^1, coincides with the spectrum sigma(f'(p)) of the Frechet derivative of f at p. Some applications to bifurcation theory are given and some peculiar examples of spectra are provided.