A new spectral theory for nonlinear operators and its applications. (English) Zbl 0952.47047

A new definition of the spectrum for nonlinear operators is given by using \((p,k)\)-epi mapping theory. The new spectrum \(\Sigma (f)\) of a nonlinear operator \(f\) is closed, bounded, upper semicontinuous and contains all the eigenvalues. More precise connection with eigenvalues is obtained for the case of positively homogeneous operators. The new spectrum is compared with the spectrum introduced sooner for Lipschitz continuous operators (\(\Sigma_{\text{lip}} (f)\)) and with that introduced for nonlinear operators by Furi, Martelli, Vignoli (\(\Sigma_{fmv} (f)\)). It is proved that in general \(\Sigma_{fmv} (f) \subset \Sigma (f) \subset \Sigma_{\text{lip}} (f) \). Examples are shown and, particularly, it is shown that all spectra mentioned can be empty. Some applications of the theory are given. A nontrivial existence result for a global Cauchy problem is obtained, generalizations of the Birkoff-Kellogg theorem and of the Hopf theorem on spheres are proved.
Reviewer: M.Kučera (Praha)


47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
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