×

zbMATH — the first resource for mathematics

Some spectral theory for nonlinear operators. (English) Zbl 0876.47042
The purpose of this paper is to study the properties of the nonlinear spectra introduced by R. I. Kachurovskij [Dokl. Akad. Nauk SSSR 188, 274-277 (1969; Zbl 0197.40402)], J. W. Neuberger [Pac. J. Math. 31, 157-159 (1969; Zbl 0182.47203)], and M. Furi, M. Martelli, and A. Vignoli [Ann. Mat. Pura Appl. 118, 229-294 (1978; Zbl 0409.47043)], respectively. The authors also introduce some characteristics for nonlinear operators which give rise to several subdivisions of these spectra. The abstract results are illustrated by numerous examples.

MSC:
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bhatt, S.R.; Bhatt, S.J., On the nonlinear spectrum of furi and vignoli, Math. today, 2, 49-64, (1984) · Zbl 0558.47045
[2] 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
[3] Furi, M.; Vignoli, A., A nonlinear spectral approach to surjectivity in Banach spaces, J. funct. anal., 20, 304-318, (1975) · Zbl 0315.47036
[4] Furi, M.; Vignoli, A., Spectrum for nonlinear maps and bifurcation in the nondifferentiable case, Ann. mat. pura appl., 113, 265-285, (1977) · Zbl 0366.47028
[5] Georg, K.; Martelli, M., On spectral theory for nonlinear operators, J. funct. anal., 24, 140-147, (1977) · Zbl 0345.47048
[6] Kachurovskij, R.I.; Kachurovskij, R.I., Regular points, spectrum and eigenfunctions of nonlinear operators (Russian), Dokl. akad. nauk SSSR, Soviet math. dokl., 10, 1101-1105, (1969), Engl. transl. · Zbl 0197.40402
[7] Maddox, I.J.; Wickstead, A.W., The spectrum of uniformly Lipschitz mappings, (), 101-114 · Zbl 0661.47048
[8] Martin, R.H., Nonlinear operators and differential equations in Banach spaces, (1976), Wiley France
[9] Mininni, M., Coincidence degree and solvability of some nonlinear functional equations in normed spaces: a spectral approach, Nonlinear analysis, 1, 105-122, (1976) · Zbl 0342.47042
[10] Neuberger, J.W., Existence of a spectrum for nonlinear transformations, Pacific J. math., 31, 157-159, (1969) · Zbl 0182.47203
[11] Pejsachowicz, J.; Vignoli, A., On differentiability and surjectivity of α-Lipschitz mappings, Ann. mat. pura appl., 101, 49-63, (1974) · Zbl 0297.47058
[12] Weber, H., Φ-asymptotisches spektrum und surjektivitätssätze vom Fredholm-typ für nichtlineare operatoren mit anwendungen, Math. nachr., 117, 7-35, (1984) · Zbl 0605.47055
[13] Kuratowski, K., Sur LES espaces completes, Fund. math., 15, 301-309, (1930) · JFM 56.1124.04
[14] Darbo, G., Punti uniti in trasformazioni a codominio non compatto, Rend. sem. mat. univ. Padova, 24, 84-92, (1955) · Zbl 0064.35704
[15] Sadovskij, B.N.; Sadovskij, B.N., Limit-compact and condensing operators (Russian), Uspekhi mat. nauk, Russian math. surveys, 27, 1, 85-155, (1972), Engl. transl. · Zbl 0243.47033
[16] Engl. transl.: Birkhäuser, Basel, 1992 · Zbl 0623.47070
[17] Gol’denshtejn, L.S.; Gokhberg, I.; Markus, A.S., Investigation of some properties of bounded linear operators and of the connection with their q-norm (Russian), Uchen. zapiski kishin. GoS. univ., 29, 29-36, (1957)
[18] Granas, A., On a class of nonlinear mappings in Banach spaces, Bull. acad. Pol. sci., 5, 867-870, (1957) · Zbl 0078.11701
[19] Vignoli, A., On α-contractions and surjectivity, Boll. unione mat. ital., 4, 446-455, (1971) · Zbl 0225.47027
[20] Vignoli, A., On quasi-bounded mappings and nonlinear functional equations, Atti accad. naz. lincei rend. cl. sci. fis. mat. natur., 50, 2, 114-117, (1971) · Zbl 0254.47089
[21] Banach, S.; Mazur, S., Über mehrdeutige stetige abbildungen, Studia math., 5, 174-178, (1934) · Zbl 0013.08202
[22] Ambrosetti, A., Proprietà spettrali di certi operatori lineari noncompatti, Rend. sem. mat. univ. Padova, 42, 189-200, (1969)
[23] Nussbaum, R.D., The radius of the essential spectrum, Duke math. J., 37, 473-478, (1970) · Zbl 0216.41602
[24] Engl. transl.: Noordhoff, Leyden, 1976
[25] Fournier, G.; Martelli, M., Eigenvectors for nonlinear maps, Topol. methods nonlin. anal., 2, 203-224, (1993) · Zbl 0812.47059
[26] Dunford, N.; Schwartz, J.T., Linear operators II, (1963), Int. Publ. Moscow
[27] Triebel, H., Höhere analysis, (1972), VEB Dt. Verlag Wiss. Leyden · Zbl 0257.47001
[28] Taylor, A.E., Introduction to functional analysis, (1964), Wiley Berlin
[29] Appell, J., Implicit functions, nonlinear integral equations, and the measure of noncompactness of the superposition operator, J. math. anal. appl., 83, 1, 251-263, (1981) · Zbl 0495.45007
[30] Weissinger, J., Zur theorie und anwendung des iterationsverfahrens, Math. nachr., 8, 193-212, (1952) · Zbl 0046.34105
[31] Petryshyn, W.V., Remarks on condensing and k-set-contractive mappings, J. math. anal. appl., 39, 717-741, (1972) · Zbl 0238.47041
[32] Dörfner, M., A numerical range for nonlinear operators, Zeitschr. anal. anw., 15, 2, 445-456, (1996) · Zbl 0849.47036
[33] Edmunds, D.E.; Webb, J.R.I., Remarks on nonlinear spectral theory, Boll. unione mat. ital., 2B, 377-390, (1983) · Zbl 0539.47042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.