A new spectrum for nonlinear operators in Banach spaces

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 (f, p) of K, called the ‘spectrum of f at p’, which coincides with the usual spectrum (f) of f in the linear case. More generally, we show that (f, p) is always closed and, when f is C1, coincides with the spectrum (f0(p)) of the Fr´echet derivative of f at p. Some applications to bifurcation theory are given and some peculiar examples of spectra are provided.