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

 

The contraction principle for mappings on a metric space with a graph


Author: Jacek Jachymski
Journal: Proc. Amer. Math. Soc. 136 (2008), 1359-1373
MSC (2000): Primary 47H10; Secondary 05C40, 54H25
Published electronically: December 5, 2007
MathSciNet review: 2367109
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Abstract | References | Similar Articles | Additional Information

Abstract: We give some generalizations of the Banach Contraction Principle to mappings on a metric space endowed with a graph. This extends and subsumes many recent results of other authors which were obtained for mappings on a partially ordered metric space. As an application, we present a theorem on the convergence of successive approximations for some linear operators on a Banach space. In particular, the last result easily yields the Kelisky-Rivlin theorem on iterates of the Bernstein operators on the space $ C[0,1]$.


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Additional Information

Jacek Jachymski
Affiliation: Institute of Mathematics, Technical University of Łódź, Wólczańska 215, 93-005 Łódź, Poland
Email: jachym@p.lodz.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09110-1
PII: S 0002-9939(07)09110-1
Keywords: Fixed point, Picard operator, partial order, connected graph, Bernstein operator
Received by editor(s): December 12, 2006
Received by editor(s) in revised form: February 13, 2007
Published electronically: December 5, 2007
Additional Notes: $^{*}$ Professor Andrzej Lasota passed away on December 28, 2006.
Dedicated: To the memory of Professor Andrzej Lasota$^{*}$
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.