Nonlinear Rayleigh Quotients and Nonlinear Spectral Theory

Abstract

We give a new and simplified definition of spectrumfor 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 ) .

Keywords

• surjective operators
• measure of noncompactness
• Darbo’s fixed point theorem
• gradient operators
• Ekeland variational principle