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

Author(s): Jacek Jachymski
Journal: Proc. Amer. Math. Soc. 136 (2008), 1359-1373.
MSC (2000): Primary 47H10; Secondary 05C40, 54H25
Posted: December 5, 2007
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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 Lódz, Wólczanska 215, 93-005 Lódz, Poland
Email: jachym@p.lodz.pl

DOI: 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
Posted: 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
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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