A new approach to relatively nonexpansive mappings

Author:
Rafa Espínola

Journal:
Proc. Amer. Math. Soc. **136** (2008), 1987-1995

MSC (2000):
Primary 47H10

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 of a Banach space is a mapping defined from into such that for and . 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 , there exists a natural semimetric for which any relatively nonexpansive mapping with respect to 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:
http://dx.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.