Fixed points of nonexpansive mappings in spaces of continuous functions
HTML articles powered by AMS MathViewer
- by T. Domínguez Benavides and María A. Japón Pineda PDF
- Proc. Amer. Math. Soc. 133 (2005), 3037-3046 Request permission
Abstract:
Let $K$ be a compact metrizable space and let $C(K)$ be the Banach space of all real continuous functions defined on $K$ with the maximum norm. It is known that $C(K)$ fails to have the weak fixed point property for nonexpansive mappings (w-FPP) when $K$ contains a perfect set. However the space $C(\omega ^{n}+1)$, where $n\in \mathbb {N}$ and $\omega$ is the first infinite ordinal number, enjoys the w-FPP, and so $C(K)$ also satisfies this property if $K^{(\omega )}=\emptyset$. It is unknown if $C(K)$ has the w-FPP when $K$ is a scattered set such that $K^{(\omega )}\not =\emptyset$. In this paper we prove that certain subspaces of $C(K)$, with $K^{(\omega )}\not = \emptyset$, satisfy the w-FPP. To prove this result we introduce the notion of $\omega$-almost weak orthogonality and we prove that an $\omega$-almost weakly orthogonal closed subspace of $C(K)$ enjoys the w-FPP. We show an example of an $\omega$-almost weakly orthogonal subspace of $C(\omega ^{\omega }+1)$ which is not contained in $C(\omega ^{n}+1)$ for any $n\in \mathbb {N}$.References
- Dale E. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc. 82 (1981), no. 3, 423–424. MR 612733, DOI 10.1090/S0002-9939-1981-0612733-0
- C. Bessaga and A. Pełczyński, Spaces of continuous functions. IV. On isomorphical classification of spaces of continuous functions, Studia Math. 19 (1960), 53–62. MR 113132, DOI 10.4064/sm-19-1-53-62
- Jon M. Borwein and Brailey Sims, Nonexpansive mappings on Banach lattices and related topics, Houston J. Math. 10 (1984), no. 3, 339–356. MR 763236
- D. van Dulst, Equivalent norms and the fixed point property for nonexpansive mappings, J. London Math. Soc. (2) 25 (1982), no. 1, 139–144. MR 645871, DOI 10.1112/jlms/s2-25.1.139
- J. Elton, Pei-Kee Lin, E. Odell, and S. Szarek, Remarks on the fixed point problem for nonexpansive maps, Fixed points and nonexpansive mappings (Cincinnati, Ohio, 1982) Contemp. Math., vol. 18, Amer. Math. Soc., Providence, RI, 1983, pp. 87–120. MR 728595, DOI 10.1090/conm/018/728595
- Kazimierz Goebel, On the structure of minimal invariant sets for nonexpansive mappings, Ann. Univ. Mariae Curie-Skłodowska Sect. A 29 (1975), 73–77 (1977) (English, with Russian and Polish summaries). MR 461226
- Kazimierz Goebel and W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, Cambridge, 1990. MR 1074005, DOI 10.1017/CBO9780511526152
- A. Jiménez-Melado and E. Llorens Fuster, A sufficient condition for the fixed point property, Nonlinear Anal. 20 (1993), no. 7, 849–853. MR 1214748, DOI 10.1016/0362-546X(93)90073-2
- Shizuo Kakutani, Concrete representation of abstract $(M)$-spaces. (A characterization of the space of continuous functions.), Ann. of Math. (2) 42 (1941), 994–1024. MR 5778, DOI 10.2307/1968778
- L. A. Karlovitz, On nonexpansive mappings, Proc. Amer. Math. Soc. 55 (1976), no. 2, 321–325. MR 405182, DOI 10.1090/S0002-9939-1976-0405182-X
- William A. Kirk and Brailey Sims (eds.), Handbook of metric fixed point theory, Kluwer Academic Publishers, Dordrecht, 2001. MR 1904271, DOI 10.1007/978-94-017-1748-9
- H. Elton Lacey, The isometric theory of classical Banach spaces, Die Grundlehren der mathematischen Wissenschaften, Band 208, Springer-Verlag, New York-Heidelberg, 1974. MR 0493279, DOI 10.1007/978-3-642-65762-7
- Pei-Kee Lin, Unconditional bases and fixed points of nonexpansive mappings, Pacific J. Math. 116 (1985), no. 1, 69–76. MR 769823, DOI 10.2140/pjm.1985.116.69
- A. Pełczyński and Z. Semadeni, Spaces of continuous functions. III. Spaces $C(\Omega )$ for $\Omega$ without perfect subsets, Studia Math. 18 (1959), 211–222. MR 107806, DOI 10.4064/sm-18-2-211-222
- A. Pełczyński and W. Szlenk, An example of a non-shrinking basis, Rev. Roumaine Math. Pures Appl. 10 (1965), 961–966. MR 203432
- Zbigniew Semadeni, Banach spaces of continuous functions. Vol. I, Monografie Matematyczne, Tom 55, PWN—Polish Scientific Publishers, Warsaw, 1971. MR 0296671
Additional Information
- T. Domínguez Benavides
- Affiliation: Departamento de Análisis Matemático, University of Seville, P.O. Box 1160, 41080-Seville, Spain
- Email: tomasd@us.es
- María A. Japón Pineda
- Affiliation: Departamento de Análisis Matemático, University of Seville, P.O. Box 1160, 41080-Seville, Spain
- Email: japon@us.es
- Received by editor(s): May 30, 2004
- Published electronically: April 20, 2005
- Additional Notes: This research was partially supported by the DGES (research project BMF2000-0344-C02-C01) and the Junta de Andalucia (project 127)
- Communicated by: Jonathan M. Borwein
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 133 (2005), 3037-3046
- MSC (2000): Primary 47H09, 47H10, 46B20, 46B42, 46E05
- DOI: https://doi.org/10.1090/S0002-9939-05-08149-9
- MathSciNet review: 2159783