We give a survey on spectra for various classes of nonlinear operators, with a particular emphasis on a comparison of their advantages and drawbacks. Here the most useful spectra are the asymptotic spectrum by M. Furi, M. Martelli and A. Vignoli (1978), the global spectrum by W. Feng (1997), and the local spectrum (called “phantom”) by P. Santucci and M. Väth (2000). In the last part we discuss these spectra for homogeneous operators (of any degree), and derive a discreteness result and a nonlinear Fredholm alternative for such operators. This may be applied to an eigenvalue problem for the -Laplace operator which arises in various fields of applied mathematics, mechanics, and physics.
@article{702482,
title = {A la recherche du spectre perdu: An invitation to nonlinear spectral theory},
booktitle = {Nonlinear Analysis, Function Spaces and Applications},
series = {GDML\_Books},
publisher = {Czech Academy of Sciences, Mathematical Institute},
address = {Praha},
year = {2003},
pages = {1-20},
url = {http://dml.mathdoc.fr/item/702482}
}
Appell, Jürgen. A la recherche du spectre perdu: An invitation to nonlinear spectral theory, dans Nonlinear Analysis, Function Spaces and Applications, GDML_Books, (2003), pp. 1-20. http://gdmltest.u-ga.fr/item/702482/
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