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A comparison of different spectra for nonlinear operators. (English) Zbl 0956.47035
The authors express the opinion that it is only slightly exaggerated to say that the spectral theory for linear operators is one of the most important topics of functional analysis and operator theory. As a matter of fact, much information on a linear operator is “hidden” in it’s spectrum, and thus knowing the spectrum means knowing a large part of the properties of the operator. In view of the importance of spectral theory for linear operators, it is not surprising at all that various attempts have been made to define and study spectra also for nonlinear operators.
In this paper the authors discuss spectra for various classes of nonlinear operators and compare their properties from the viewpoint of the above requirements. The classes they are interested in are Fréchet differentiable operators, Lipschitz continuous operators, general continuous operators, special continuous operators, \(k\)-epi continuous, and linearly bounded operators.

47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
46T20 Continuous and differentiable maps in nonlinear functional analysis
46G05 Derivatives of functions in infinite-dimensional spaces
Full Text: DOI
[1] Appell, J.; Dörfner, M., Some spectral theory for nonlinear operators, Nonlinear anal. TMA, 28, 12, 1955-1976, (1997) · Zbl 0876.47042
[2] Banach, S.; Mazur, S., Über mehrdeutige stetige abbildungen, Studia math., 5, 174-178, (1934) · Zbl 0013.08202
[3] Darbo, G., Punti uniti in trasformazioni a codominio non compatto, Rend. sem. mat. univ. Padova, 24, 84-92, (1955) · Zbl 0064.35704
[4] Dörfner, M., Spektraltheorie für nichtlineare operatoren, Ph.D. thesis, (1997), University of Würzburg
[5] Edmunds, D.E.; Webb, J.R.L., Remarks on nonlinear spectral theory, Boll. unione mat. ital., 2-B, 377-390, (1983) · Zbl 0539.47042
[6] Feng, W., A new spectral theory for nonlinear operators and its applications, Abstracts appl. anal., 2, 163-183, (1997) · Zbl 0952.47047
[7] Furi, M.; Martelli, M.; Vignoli, A., Stably solvable operators in Banach spaces, Atti accad. naz. lincei rend. cl. sci. fis. mat. nat., 60, 21-26, (1976) · Zbl 0361.47024
[8] Furi, M.; Martelli, M.; Vignoli, A., Contributions to the spectral theory for nonlinear operators in Banach spaces, Ann. mat. pura appl., 118, 229-294, (1978) · Zbl 0409.47043
[9] Furi, M.; Vignoli, A., On a property of the unit sphere in a linear normed space, Bull. acad. Pol. sci., 18, 333-334, (1970) · Zbl 0194.43501
[10] Furi, M.; Vignoli, A., A nonlinear spectral approach to surjectivity in Banach spaces, J. funct. anal., 20, 304-318, (1975) · Zbl 0315.47036
[11] Georg, K.; Martelli, M., On spectral theory for nonlinear operators, J. funct. anal., 24, 140-147, (1977) · Zbl 0345.47048
[12] Granas, A., On a class of nonlinear mappings in Banach spaces, Bull. acad. Pol. sci., 5, 867-870, (1957) · Zbl 0078.11701
[13] Kachurovskij, R.I., Regular points, spectrum and eigenfunctions of nonlinear operators, Dokl. akad. nauk SSSR, 188, 274-277, (1969), (Engl. transl.: Soviet Math. Dokl. 10 (1969) 1101-1105) [in Russian] · Zbl 0197.40402
[14] Krasnosel’skij, M.A., Topological methods in the theory of nonlinear integral equations, (1956), Gostekhizdat Moscow, (Engl. transl.: Macmillan, New York, 1964) (in Russian) · Zbl 0072.09702
[15] Maddox, I.J.; Wickstead, A.W., The spectrum of uniformly Lipschitz mappings, Proc. royal irish acad., 89-A, 101-114, (1989) · Zbl 0661.47048
[16] Martin, R.H., Nonlinear operator and differential equations in Banach spaces, (1976), Wiley New York
[17] Nemytskij, V.V., Some problems concerning the structure of the spectrum of completely continuous nonlinear operators, Dokl. akad. nauk SSSR, 80, 2, 161-164, (1951), (in Russian)
[18] Nemytskij, V.V., Structure of the spectrum of completely continuous nonlinear operators, Mat. sbornik, 33, 3, 545-558, (1953), (in Russian) · Zbl 0052.34801
[19] Neuberger, J.W., Existence of a spectrum for nonlinear transformations, Pacific J. math., 31, 157-159, (1969) · Zbl 0182.47203
[20] Rhodius, A., Der numerische wertebereich und die Lösbarkeit linearer und nichtlinearer operatorgleichungen, Math. nachr., 79, 343-360, (1977) · Zbl 0371.47053
[21] Sadovskij, B.N., On a fixed point principle, Funkt. anal. prilozh., 1, 2, 74-76, (1967), (in Russian) · Zbl 0165.49102
[22] Sadovskij, B.N., Limit-compact and condensing operators, Uspekhi mat. nauk, 27, 81-146, (1972), (Engl.transl.: Russian Math. Surveys 27 (1972) 85-155) (in Russian) · Zbl 0232.47067
[23] P. Santucci, M. Väth, On the definition of nonlinear eigenvalues, submitted for publication.
[24] Schauder, J., Der fixpunktsatz in funktionalräumen, Studia math., 2, 171-180, (1930) · JFM 56.0355.01
[25] Tarafdar, E.U.; Thompson, H.B., On the solvability of nonlinear noncompact operator equations, J. austral. math. soc., 43, 103-114, (1987) · Zbl 0623.47072
[26] Vignoli, A., On α-contractions and surjectivity, Boll. unione mat. ital., 4-A, 446-455, (1971) · Zbl 0225.47027
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