On a converse to Banach’s Fixed Point Theorem
HTML articles powered by AMS MathViewer
- by Márton Elekes PDF
- Proc. Amer. Math. Soc. 137 (2009), 3139-3146 Request permission
Abstract:
We say that a metric space $(X,d)$ possesses the Banach Fixed Point Property (BFPP) if every contraction $f:X\to X$ has a fixed point. The Banach Fixed Point Theorem states that every complete metric space has the BFPP. However, E. Behrends pointed out in 2006 that the converse implication does not hold; that is, the BFPP does not imply completeness; in particular, there is a nonclosed subset of $\mathbb {R}^2$ possessing the BFPP. He also asked if there is even an open example in $\mathbb {R}^n$, and whether there is a ‘nice’ example in $\mathbb {R}$. In this note we answer the first question in the negative, the second one in the affirmative, and determine the simplest such examples in the sense of descriptive set theoretic complexity.
Specifically, first we prove that if $X\subset \mathbb {R}^n$ is open or $X\subset \mathbb {R}$ is simultaneously $F_\sigma$ and $G_\delta$ and $X$ has the BFPP, then $X$ is closed. Then we show that these results are optimal, as we give an $F_\sigma$ and also a $G_\delta$ nonclosed example in $\mathbb {R}$ with the BFPP.
We also show that a nonmeasurable set can have the BFPP. Our non-$G_\delta$ examples provide metric spaces with the BFPP that cannot be remetrized by any compatible complete metric. All examples are in addition bounded.
References
- A. C. Babu, A converse to a generalized Banach contraction principle, Publ. Inst. Math. (Beograd) (N.S.) 32(46) (1982), 5–6. MR 710962
- E. Behrends, Problem Session of the 34th Winter School in Abstract Analysis, 2006.
- E. Behrends, private communication, 2006.
- C. Bessaga, On the converse of the Banach “fixed-point principle“, Colloq. Math. 7 (1959), 41–43. MR 111015, DOI 10.4064/cm-7-1-41-43
- Ljubomir B. Ćirić, On some mappings in metric spaces and fixed points, Acad. Roy. Belg. Bull. Cl. Sci. (6) 6 (1995), no. 1-6, 81–89. MR 1385507
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
- A. A. Ivanov, Fixed points of mappings of metric spaces, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 66 (1976), 5–102, 207 (Russian, with English summary). Studies in topology, II. MR 0467711
- Jacek Jachymski, General solutions of two functional inequalities and converses to contraction theorems, Bull. Polish Acad. Sci. Math. 51 (2003), no. 2, 147–156. MR 1990804
- Ludvík Janoš, A converse of Banach’s contraction theorem, Proc. Amer. Math. Soc. 18 (1967), 287–289. MR 208589, DOI 10.1090/S0002-9939-1967-0208589-3
- Ludvík Janoš, A converse of the generalized Banach’s contraction theorem, Arch. Math. (Basel) 21 (1970), 69–71. MR 264628, DOI 10.1007/BF01220881
- Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
- W. A. Kirk, Contraction mappings and extensions, Handbook of metric fixed point theory, Kluwer Acad. Publ., Dordrecht, 2001, pp. 1–34. MR 1904272
- K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
- Philip R. Meyers, A converse to Banach’s contraction theorem, J. Res. Nat. Bur. Standards Sect. B 71B (1967), 73–76. MR 221469
- A. Mukherjea and K. Pothoven, Real and functional analysis, Mathematical Concepts and Methods in Science and Engineering, Vol. 6, Plenum Press, New York-London, 1978. MR 0492145
- V. I. Opoĭcev, A converse of the contraction mapping principle, Uspehi Mat. Nauk 31 (1976), no. 4 (190), 169–198 (Russian). MR 0420591
- John C. Oxtoby, Measure and category, 2nd ed., Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York-Berlin, 1980. A survey of the analogies between topological and measure spaces. MR 584443
- Ioan A. Rus, Generalized contractions and applications, Cluj University Press, Cluj-Napoca, 2001. MR 1947742
Additional Information
- Márton Elekes
- Affiliation: Rényi Alfréd Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, H-1364 Budapest, Hungary – and – Eötvös Loránd University, Budapest, Hungary
- Email: emarci@renyi.hu
- Received by editor(s): February 2, 2007
- Received by editor(s) in revised form: March 26, 2007
- Published electronically: April 30, 2009
- Additional Notes: The author was partially supported by Hungarian Scientific Foundation grants no. 43620, 49786, 61600, 72655, and the János Bolyai Fellowship.
- Communicated by: Andreas Seeger
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 3139-3146
- MSC (2000): Primary 54H25, 47H10, 55M20, 03E15, 54H05; Secondary 26A16
- DOI: https://doi.org/10.1090/S0002-9939-09-09904-3
- MathSciNet review: 2506473