A new spectrum for continuous 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 $\sigma(f,p)$ of $\K$, called \emph{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 $C\sp 1$, coincides with the spectrum $\sigma(f'(p))$ of the Fr\'echet derivative of $f$ at $p$. Some applications to bifurcation theory are given and some peculiar examples of spectra are provided.