Abstract
Over the last two decades ultrapower techniques have become major tools for the development and understanding of metric fixed point theory. In this short chapter we develop the Banach space ultrapower and initiate its use in studying the weak fixed point property for nonexpansive mappings. For a more extensive and detailed treatment than is given here the reader is referred to [1] and [21].
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References
Aksoy, A. G. and Khamsi, M. A., Nonstandard methods in fixed point theory, Springer-Verlag, 1990.
Baillon, J. B. and Schöneberg, R., Asymptotic normal structure and fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., 81 (1981), 257–264.
Benyamini, Yoav and Lindenstrauss, Joram, Geometric Nonlinear Functional Analysis, Vol. 1, Amer. Math. Soc., Colloquium Publications 48, Providence Rhode Island, 2000.
Borwein, J. M. and Sims, B., Nonexpansive mappings on Banach lattices and related topics, Houston J. Math., 10 (1984), 339–356.
Dalby, T., Facets of the fixed point theory for nonexpansive mappings, Ph. D. dissertation, Univ. of Newcastle, Australia, 1997.
van Dulst, D. and Pach, A. J., On flatness and some ergodic super-properties of Banach spaces, Indagationes Mathematical, 43 (1981), 153–164.
Elton, J., Lin, P. K., Odell, E. and Szarek, S., Remarks on the fixed point problem for nonexpansive maps, Contemporary Math. 18 (1983), 87–120.
Enflo, P., Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math. 13 (1973), 281–288.
GarcÃa-Falset, J., The fixed point property in B spaces with NUS property, J. Math. Ann. Appl., 215 (1997), 532–542.
Garcia-Falset, J. and Sims, B., Property (M) and the weak fixed point property, Proc. Amer. Math. Soc., 125 (1997), 2891–2896.
Jiménez-Melado, A. and Llorens-Fuster, E., A sufficient condition for the fixed point property, Nonlinear Anal., 20 (1993), 849–853.
Khamsi, Mohamed A. and Kirk, William A, An introduction to metric spaces and fixed point theory, John Wiley, 2000.
Khamsi, M.A. and Turpin, Ph., Fixed points of nonexpansive mappings in Banach lattices, Proc. Amer. Math. Soc., 105 (1989), 102–110.
Lacey, H. Elton, The isometric theory of classical Banach spaces, Springer-Verlag, 1974.
Lin, P. K., Unconditional bases and fixed points of nonexpansive mappings, Pacific J. Math. 116 (1985), 69–76.
Lin, P.K., Stability of the fixed point property of Hilbert spaces, Proc. Amer. Math. Soc., 127 (1999), 3573–3581.
Maurey, B., Points fixes des contractions de certains faiblement compacts de L 1, Seminaire d’Analyse Fonctionnelle, Exposé No. VIII (1980), 18.
Pisier, G., Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), 326350.
Prus, Stanislaw, Multidimensional uniform convexity and uniform smoothness of Banach spaces, in: Recent Advances on metric fixed point theory, Editor: T. Dominguez-Benavides, Universidad de Sevilla, Serie: Ciencias, 48 (1986), 111–136.
Prus, S., Banach spaces which are uniformly noncreasy, Nonlinear Anal., 30 (1987), 2317–2324.
Sims, B., Ultra-techniques in Banach Space theory, Queen’s Papers in Pure and Applied Mathematics, No. 60, Kingston, Canada (1982).
Sims, B., Orthogonality and fixed points of nonexpansive maps, Proc. Centre Math. Anal., Austral. Nat. Uni. 20 (1988), 178–186.
Sims, B., Geometric condition sufficient for the weak and weak fixed point property, Proceedings of the Second International Conference on Fixed Point Theory and Applications, Ed. K. K. Tan, World Scientific Publishers (1992), 278–290.
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Khamsi, M.A., Sims, B. (2001). Ultra-Methods in Metric Fixed Point Theory. In: Kirk, W.A., Sims, B. (eds) Handbook of Metric Fixed Point Theory. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1748-9_6
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DOI: https://doi.org/10.1007/978-94-017-1748-9_6
Publisher Name: Springer, Dordrecht
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