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ISSN 1088-6826(e) ISSN 0002-9939(p)

Equivalent conditions involving common fixed points for maps on the unit interval

Author(s): Jacek R. Jachymski
Journal: Proc. Amer. Math. Soc. 124 (1996), 3229-3233.
MSC (1991): Primary 54H25; Secondary 54E45
MathSciNet review: 1328352
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Abstract: Let $g$ be a continuous self-map of the unit interval $I$ . Equivalent conditions are given to ensure that $g$ has a common fixed point with every continuous map $f:I\mapsto I$ that commutes with $g$ on a suitable subset of $I$ . This extends a recent result of Gerald Jungck.

References:

1.: W. M. BOYCE, $\Gamma$ -compact maps on an interval and fixed points, Trans. Amer. Math. Soc. 160 (1971), 87-102. MR 43:6374
2.: J. CANO, Common fixed points for a class of commuting mappings on an interval, Proc. Amer. Math. Soc. 86 (1982), 336-338. MR 83h:54054
3.: S. C. CHU and R. D. MOYER, On continuous functions, commuting functions, and fixed points, Fund. Math. 59 (1966), 90-95. MR 34:805
4.: E. M. COVEN and G. A. HEDLUND, Continuous maps of the interval whose periodic points form a closed set, Proc. Amer. Math. Soc. 79 (1980), 127-133. MR 81a:54042
5.: R. ENGELKING, General topology, Polish Scientific Publishers, Warszawa, 1977. MR 58:18316b
6.: J. P. HUNEKE, Two counterexamples to a conjecture on commuting continuous functions of the closed unit interval, Abstract no. 67T-2311, Notices Amer. Math. Soc. 14 (1967), 284.
7.: G. JUNGCK, Common fixed points for compatible maps on the unit interval, Proc. Amer. Math. Soc. 115 (1992), 495-499. MR 92m:54076

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Additional Information:

Jacek R. Jachymski
Affiliation: Institute of Mathematics, Technical University of Łódź, Żwirki 36, 90-924 Łódź, Poland
Email: jachymsk@lodz1.p.lodz.pl

DOI: 10.1090/S0002-9939-96-03397-7
PII: S 0002-9939(96)03397-7
Keywords: Common fixed points, equicontinuity, commuting maps, iterates, unit interval
Received by editor(s): September 9, 1993
Received by editor(s) in revised form: March 29, 1995
Communicated by: James E. West
Copyright of article: Copyright 1996, American Mathematical Society

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