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Fixed points and positive eigenvalues for nonlinear operators. (English) Zbl 1090.47041

The authors present some results related to fixed points and eigenvalues. They prove, in particular, a new fixed point theorem for nonexpansive mappings in Banach spaces, which is inspired by the result proved recently in J.–P.Penot [Proc.Am.Math.Soc.131, 2371–2377 (2003; Zbl 1035.47043)] and by the notion of the scalar derivative. The results related to eigenvalues are based on this new fixed point theorem and on a fixed point theorem given by the authors in [J.Math.Anal.Appl.290, No. 2, 452–468 (2004; Zbl 1066.47056)] as a variant of Altman’s fixed point theorem. This paper can be considered as a complementary part of the cited paper.

MSC:

47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
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References:

[1] Akhmerov, R. R.; Kamenskii, M. I.; Potapova, A. S.; Rodkina, A. E.; Sadovskii, B. N., Measure of Noncompactness and Condensing Operators (1992), Birkhäuser: Birkhäuser Basel · Zbl 0748.47045
[2] Banas, J.; Goebel, K., Measure of Noncompactness in Banach Spaces (1980), Dekker: Dekker New York · Zbl 0441.47056
[3] Ciorănescu, I., Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems (1990), Kluwer Academic
[4] Feng, W., A new spectral theory for nonlinear operators and its applications, Abstr. Appl. Anal., 1-2, 163-183 (1997) · Zbl 0952.47047
[5] Hedenmalm, P. J.H., Spectral properties of invariant subspaces in the Bergman space, J. Funct. Anal., 116, 441-448 (1993) · Zbl 0814.46039
[6] Isac, G., On some generalization of Karamardian’s theorem on the complementarity problem, Boll. Un. Mat. Ital. B, 7, 323-332 (1998) · Zbl 0653.90076
[7] G. Isac, S.Z. Németh, Scalar derivatives and asymptotic scalar derivatives. An Altman type fixed point theorem on cones and some applications, J. Math. Anal. Appl., in press; G. Isac, S.Z. Németh, Scalar derivatives and asymptotic scalar derivatives. An Altman type fixed point theorem on cones and some applications, J. Math. Anal. Appl., in press · Zbl 1066.47056
[8] Giles, J. R., Classes of semi-inner-product spaces, Trans. Amer. Math. Soc., 129, 436-446 (1967) · Zbl 0157.20103
[9] Lumer, G., Semi-inner-product spaces, Trans. Amer. Math. Soc., 100, 29-43 (1961) · Zbl 0102.32701
[10] Nussbaum, R. D., The fixed point index for local condensing maps, Ann. Mat. Pura Appl., 84, 217-258 (1971) · Zbl 0226.47031
[11] Penot, J. P., A fixed point theorem for asymptotically contractive mappings, Proc. Amer. Math. Soc., 131, 2371-2377 (2003) · Zbl 1035.47043
[12] Richter, S., Invariant subspaces in Banach spaces of analytic functions, Trans. Amer. Math. Soc., 304, 585-616 (1987) · Zbl 0646.47023
[13] Sadovsky, B. N., Limit compact and condensing operators, Russian Math. Surveys, 27, 85-155 (1972) · Zbl 0243.47033
[14] Takahashi, W., Nonlinear Functional Analysis (Fixed Point Theory and its Applications) (2000), Yokohoma Publishers · Zbl 0997.47002
[15] Weber, V. H., \(φ\)-asymptotisches Spectrum und Surjektivitätssätze vom Fredholm Typ für nichtlineare Operatoren mit Anwendungen, Math. Nachr., 117, 7-35 (1984) · Zbl 0605.47055
[16] Zeidler, E., Nonlinear Functional Analysis and its Applications, Part 1: Fixed Point Theorems (1986), Springer-Verlag: Springer-Verlag New York
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