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
DOI:
https://doi.org/10.1090/S0002-9939-07-09110-1
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
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.
