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

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- by Jacek Jachymski
- Proc. Amer. Math. Soc.
**136**(2008), 1359-1373 - DOI: https://doi.org/10.1090/S0002-9939-07-09110-1
- Published electronically: 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 $C[0,1]$.## References

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## Bibliographic Information

**Jacek Jachymski**- Affiliation: Institute of Mathematics, Technical University of Łódź, Wólczańska 215, 93-005 Łódź, Poland
- Email: jachym@p.lodz.pl
- 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.
- Communicated by: Joseph A. Ball
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2367109

Dedicated: To the memory of Professor Andrzej Lasota$^*$