A generalization of a theorem of Janos and Edelstein

Author:
Sehie Park

Journal:
Proc. Amer. Math. Soc. **66** (1977), 344-346

MSC:
Primary 54C25

MathSciNet review:
0454907

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Abstract: A generalization of Edelstein's version of a theorem of Janos and its converse are obtained:

Theorem. *Let X be a compact metrizable topological space, and f, g be continuous self-maps of X such that* *and f is bijective. Then g is injective and* , *where* , *iff, given* , *a homeomorphism h of X into* *exists such that*

*for all*.

**[1]**Michael Edelstein,*A short proof of a theorem of L. Janos*, Proc. Amer. Math. Soc.**20**(1969), 509–510. MR**0234426**, 10.1090/S0002-9939-1969-0234426-9**[2]**L. Janos,*Converse of the Banach theorem in the case of one-to-one contracting mapping*, Notices Amer. Math. Soc.**11**(1964), 686. Abstract #64T-469.**[3]**-,*Homothetic property of contractive one-to-one mappings*, Notices Amer. Math. Soc.**13**(1966), 818. Abstract #638-11.**[4]**-,*One-to-one contractive mappings on compact space*, Notices Amer. Math. Soc.**14**(1967), 133. Abstract #67T-21.**[5]**Gerald Jungck,*Commuting mappings and fixed points*, Amer. Math. Monthly**83**(1976), no. 4, 261–263. MR**0400196**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1977-0454907-7

Article copyright:
© Copyright 1977
American Mathematical Society