Nonlinear rayleigh quotients and nonlinear spectral theory

We give a new and simplified definition of spectrum for a nonlinear operator F acting in a real Banach space X, and study some of its features in terms of (qualitative and) quantitative properties of F such as the measure of noncompactness, α(F), of F. Then, using as a main tool the Ekeland Variational Principle, we focus our attention on the spectral properties of F when F is a gradient operator in a real Hilbert space, and in particular on the role played by its Rayleigh quotient R(F) and by the best lower and upper bounds, m(F) and M(F), of R(F).