A new approach to relatively nonexpansive mappings
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- by Rafa Espínola PDF
- Proc. Amer. Math. Soc. 136 (2008), 1987-1995 Request permission
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 $\|Tx-Ty\|\le \|x-y\|$ 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.References
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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
- 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
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 1987-1995
- MSC (2000): Primary 47H10
- DOI: https://doi.org/10.1090/S0002-9939-08-09323-4
- MathSciNet review: 2383505