A proof of the Generalized Banach Contraction Conjecture

Author:
Alexander D. Arvanitakis

Journal:
Proc. Amer. Math. Soc. **131** (2003), 3647-3656

MSC (2000):
Primary 05C55, 47H10

Published electronically:
February 26, 2003

MathSciNet review:
1998170

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Abstract: We introduce the notion of -continuity, which generalizes both continuity and the hypothesis in the Generalized Banach Contraction Conjecture, and prove that any -continuous self-map on a scattered compact space, has an invariant finite set. We use the results and the techniques to prove the Generalized Banach Contraction Conjecture.

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A generalization of the Banach Contraction Principle.

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Additional Information

**Alexander D. Arvanitakis**

Affiliation:
MPLA, Department of Mathematics, University of Athens, 15784 Panepistimiopolis, Athens, Greece

Email:
aarvan@cc.uoa.gr

DOI:
https://doi.org/10.1090/S0002-9939-03-06937-5

Keywords:
Fixed point,
GBCC,
Banach Contraction Principle,
scattered compact

Received by editor(s):
May 22, 2002

Received by editor(s) in revised form:
July 9, 2002

Published electronically:
February 26, 2003

Communicated by:
John R. Stembridge

Article copyright:
© Copyright 2003
American Mathematical Society