A global bifurcation theorem for critical values of $C^1$ maps in Banach spaces

TL;DR

This paper establishes a global bifurcation theorem for critical values of $C^1$ maps in Banach spaces using topological methods, and extends results to $C^2$ maps via spectral flow, advancing bifurcation theory.

Contribution

It introduces a topological approach to global bifurcation for $C^1$ maps and extends results to $C^2$ maps using spectral flow, providing new theoretical insights.

Findings

Global bifurcation theorem for $C^1$ maps in Banach spaces

Extension of bifurcation results to $C^2$ maps using spectral flow

Topological approach based on homotopy equivalence

Abstract

We present a global bifurcation result for critical values of $C^1$ maps in Banach spaces. The approach is topological based on homotopy equivalence of pairs of topological spaces. For $C^2$ maps, we prove a particular global bifurcation result, based on the notion of spectral flow.