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A new approach to relatively nonexpansive mappings


Author: Rafa Espínola
Journal: Proc. Amer. Math. Soc. 136 (2008), 1987-1995
MSC (2000): Primary 47H10
DOI: https://doi.org/10.1090/S0002-9939-08-09323-4
Published electronically: February 15, 2008
MathSciNet review: 2383505
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Abstract: In this paper we study the nonexpansivity of the so-called relatively nonexpansive mappings. A relatively nonexpansive mapping with respect to a pair of subsets $ (A,B)$ of a Banach space $ X$ is a mapping defined from $ A\cup B$ into $ X$ such that $ \Vert Tx-Ty\Vert\le \Vert x-y\Vert$ for $ x\in A$ and $ y\in B$. These mappings were recently considered in a paper by Eldred et al. (Proximinal normal structure and relatively nonexpansive mappings, Studia Math. 171 (3) (2005), 283-293) to obtain a generalization of Kirk's Fixed Point Theorem. In this work we show that, for certain proximinal pairs $ (A,B)$, there exists a natural semimetric for which any relatively nonexpansive mapping with respect to $ (A,B)$ is nonexpansive. This fact will be used to improve one of the two main results from the aforementioned paper by Eldred et al. At that time we will also obtain several consequences regarding the strong continuity properties of relatively nonexpansive mappings and the relation between the two main results from the same work.


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Additional Information

Rafa Espínola
Affiliation: Departamento de Análisis Matemático, Universidad de Sevilla, P.O. Box 1160, 41080-Sevilla, Spain
Email: espinola@us.es

DOI: https://doi.org/10.1090/S0002-9939-08-09323-4
Received by editor(s): July 18, 2006
Published electronically: February 15, 2008
Additional Notes: The author was partially supported by the Ministry of Science and Technology of Spain, Grant MTM 2006-13997-CO2-01 and La Junta de Antalucía project FQM-127.
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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