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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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 $J$-continuity, which generalizes both continuity and the hypothesis in the Generalized Banach Contraction Conjecture, and prove that any $J$-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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Additional Information

Alexander D. Arvanitakis
Affiliation: MPLA, Department of Mathematics, University of Athens, 15784 Panepistimiopolis, Athens, Greece
Email: aarvan@cc.uoa.gr

DOI: http://dx.doi.org/10.1090/S0002-9939-03-06937-5
PII: S 0002-9939(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