On a converse to Banach's Fixed Point Theorem
Author:
Márton Elekes
Journal:
Proc. Amer. Math. Soc. 137 (2009), 31393146
MSC (2000):
Primary 54H25, 47H10, 55M20, 03E15, 54H05; Secondary 26A16
Published electronically:
April 30, 2009
MathSciNet review:
2506473
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We say that a metric space possesses the Banach Fixed Point Property (BFPP) if every contraction 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 possessing the BFPP. He also asked if there is even an open example in , and whether there is a `nice' example in . 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 is open or is simultaneously and and has the BFPP, then is closed. Then we show that these results are optimal, as we give an and also a nonclosed example in with the BFPP. We also show that a nonmeasurable set can have the BFPP. Our non examples provide metric spaces with the BFPP that cannot be remetrized by any compatible complete metric. All examples are in addition bounded.
 1.
A.
C. Babu, A converse to a generalized Banach contraction
principle, Publ. Inst. Math. (Beograd) (N.S.) 32(46)
(1982), 5–6. MR 710962
(84g:54055)
 2.
E. Behrends, Problem Session of the 34th Winter School in Abstract Analysis, 2006.
 3.
E. Behrends, private communication, 2006.
 4.
C.
Bessaga, On the converse of the Banach “fixedpoint
principle“, Colloq. Math. 7 (1959),
41–43. MR
0111015 (22 #1882)
 5.
Ljubomir
B. Ćirić, On some mappings in metric spaces and fixed
points, Acad. Roy. Belg. Bull. Cl. Sci. (6) 6 (1995),
no. 16, 81–89. MR 1385507
(97b:54050)
 6.
Herbert
Federer, Geometric measure theory, Die Grundlehren der
mathematischen Wissenschaften, Band 153, SpringerVerlag New York Inc., New
York, 1969. MR
0257325 (41 #1976)
 7.
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
(57 #7564)
 8.
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
(2004e:47090)
 9.
Ludvík
Janoš, A converse of Banach’s
contraction theorem, Proc. Amer. Math. Soc.
18 (1967),
287–289. MR 0208589
(34 #8398), http://dx.doi.org/10.1090/S00029939196702085893
 10.
Ludvík
Janoš, A converse of the generalized Banach’s
contraction theorem, Arch. Math. (Basel) 21 (1970),
69–71. MR
0264628 (41 #9219)
 11.
Alexander
S. Kechris, Classical descriptive set theory, Graduate Texts
in Mathematics, vol. 156, SpringerVerlag, New York, 1995. MR 1321597
(96e:03057)
 12.
W.
A. Kirk, Contraction mappings and extensions, Handbook of
metric fixed point theory, Kluwer Acad. Publ., Dordrecht, 2001,
pp. 1–34. MR 1904272
(2003f:54096)
 13.
K.
Kuratowski, Topology. Vol. I, New edition, revised and
augmented. Translated from the French by J. Jaworowski, Academic Press, New
YorkLondon; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. MR 0217751
(36 #840)
 14.
Philip
R. Meyers, A converse to Banach’s contraction theorem,
J. Res. Nat. Bur. Standards Sect. B 71B (1967),
73–76. MR
0221469 (36 #4521)
 15.
A.
Mukherjea and K.
Pothoven, Real and functional analysis, Plenum Press, New
YorkLondon, 1978. Mathematical Concepts and Methods in Science and
Engineering, Vol. 6. MR 0492145
(58 #11294)
 16.
V.
I. Opoĭcev, A converse of the contraction mapping
principle, Uspehi Mat. Nauk 31 (1976), no. 4
(190), 169–198 (Russian). MR 0420591
(54 #8605)
 17.
John
C. Oxtoby, Measure and category, 2nd ed., Graduate Texts in
Mathematics, vol. 2, SpringerVerlag, New YorkBerlin, 1980. A survey
of the analogies between topological and measure spaces. MR 584443
(81j:28003)
 18.
Ioan
A. Rus, Generalized contractions and applications, Cluj
University Press, ClujNapoca, 2001. MR 1947742
(2004f:54043)
 1.
 A. C. Babu, A converse to a generalised Banach contraction principle, Publ. Inst. Math. (Beograd) (N.S.) 32(46) (1982), 56. MR 710962 (84g:54055)
 2.
 E. Behrends, Problem Session of the 34th Winter School in Abstract Analysis, 2006.
 3.
 E. Behrends, private communication, 2006.
 4.
 C. Bessaga, On the converse of the Banach ``fixedpoint principle'', Colloq. Math. 7 (1959), 4143. MR 0111015 (22:1882)
 5.
 L. B. Ćirić, On some mappings in metric spaces and fixed points, Acad. Roy. Belg. Bull. Cl. Sci. (6) 6 (1995), no. 16, 8189. MR 1385507 (97b:54050)
 6.
 H. Federer, Geometric Measure Theory. SpringerVerlag, New York, 1969. MR 0257325 (41:1976)
 7.
 A. A. Ivanov, Fixed points of mappings of metric spaces, Studies in topology, II. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 66 (1976), 5102, 207. MR 0467711 (57:7564)
 8.
 J. Jachymski, General solutions of two functional inequalities and converses to contraction theorems, Bull. Polish Acad. Sci. Math. 51 (2003), no. 2, 147156. MR 1990804 (2004e:47090)
 9.
 L. Janoš, A converse of Banach's contraction theorem, Proc. Amer. Math. Soc. 18 (1967), 287289. MR 0208589 (34:8398)
 10.
 L. Janoš, A converse of the generalised Banach's contraction theorem, Arch. Math. (Basel) 21 (1970), 6971. MR 0264628 (41:9219)
 11.
 A. S. Kechris, Classical Descriptive Set Theory. Graduate Texts in Math., vol. 156, SpringerVerlag, New York, 1995. MR 1321597 (96e:03057)
 12.
 W. A. Kirk, Contraction mappings and extensions, Handbook of Metric Fixed Point Theory, 134, Kluwer Acad. Publ., Dordrecht, 2001. MR 1904272 (2003f:54096)
 13.
 K. Kuratowski, Topology. Academic Press, New YorkLondon, 1966. MR 0217751 (36:840)
 14.
 P. R. Meyers, A converse to Banach's contraction theorem, J. Res. Nat. Bur. Standards Sect. B 71B (1967), 7376. MR 0221469 (36:4521)
 15.
 A. Mukherjea and K. Pothoven, Real and Functional Analysis. Mathematical Concepts and Methods in Science and Engineering, Vol. 6. Plenum Press, New YorkLondon, 1978. MR 0492145 (58:11294)
 16.
 V. I. Opoĭcev, A converse of the contraction mapping principle, Uspehi Mat. Nauk 31 (1976), no. 4 (190), 169198. MR 0420591 (54:8605)
 17.
 J. C. Oxtoby: Measure and Category. A Survey of the Analogies between Topological and Measure Spaces. Second edition. Graduate Texts in Mathematics, No. 2, SpringerVerlag, New YorkBerlin, 1980. MR 584443 (81j:28003)
 18.
 I. A. Rus: Generalised Contractions and Applications. Cluj University Press, ClujNapoca, 2001. MR 1947742 (2004f:54043)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
54H25,
47H10,
55M20,
03E15,
54H05,
26A16
Retrieve articles in all journals
with MSC (2000):
54H25,
47H10,
55M20,
03E15,
54H05,
26A16
Additional Information
Márton Elekes
Affiliation:
Rényi Alfréd Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, H1364 Budapest, Hungary – and – Eötvös Loránd University, Budapest, Hungary
Email:
emarci@renyi.hu
DOI:
http://dx.doi.org/10.1090/S0002993909099043
PII:
S 00029939(09)099043
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
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
