# Remarks on best approximation in R-trees

• Volume: 63, Issue: 1, page 133-138
• ISSN: 2083-7402

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## Abstract

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An R-tree is a geodesic space for which there is a unique arc joining any two of its points, and this arc is a metric segment. If X is a closed convex subset of an R-tree Y, and if T: X → 2Y is a multivalued mapping, then a point z for which [...] is called a point of best approximation. It is shown here that if T is an ε-semicontinuous mapping whose values are nonempty closed convex subsets of Y, and if T has at least two distinct points of best approximation, then T must have a fixed point. We also obtain a common best approximation theorem for a commuting pair of mappings t: X → Y and T: X → 2Y where t is single-valued continuous and T is ε-semicontinuous.

## How to cite

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William Kirk, and Bancha Panyanak. "Remarks on best approximation in R-trees." Annales UMCS, Mathematica 63.1 (2009): 133-138. <http://eudml.org/doc/267770>.

@article{WilliamKirk2009,
abstract = {An R-tree is a geodesic space for which there is a unique arc joining any two of its points, and this arc is a metric segment. If X is a closed convex subset of an R-tree Y, and if T: X → 2Y is a multivalued mapping, then a point z for which [...] is called a point of best approximation. It is shown here that if T is an ε-semicontinuous mapping whose values are nonempty closed convex subsets of Y, and if T has at least two distinct points of best approximation, then T must have a fixed point. We also obtain a common best approximation theorem for a commuting pair of mappings t: X → Y and T: X → 2Y where t is single-valued continuous and T is ε-semicontinuous.},
author = {William Kirk, Bancha Panyanak},
journal = {Annales UMCS, Mathematica},
keywords = {Best approximation; R-tree; fixed points; semicontinuity; best approximation; -tree},
language = {eng},
number = {1},
pages = {133-138},
title = {Remarks on best approximation in R-trees},
url = {http://eudml.org/doc/267770},
volume = {63},
year = {2009},
}

TY - JOUR
AU - William Kirk
AU - Bancha Panyanak
TI - Remarks on best approximation in R-trees
JO - Annales UMCS, Mathematica
PY - 2009
VL - 63
IS - 1
SP - 133
EP - 138
AB - An R-tree is a geodesic space for which there is a unique arc joining any two of its points, and this arc is a metric segment. If X is a closed convex subset of an R-tree Y, and if T: X → 2Y is a multivalued mapping, then a point z for which [...] is called a point of best approximation. It is shown here that if T is an ε-semicontinuous mapping whose values are nonempty closed convex subsets of Y, and if T has at least two distinct points of best approximation, then T must have a fixed point. We also obtain a common best approximation theorem for a commuting pair of mappings t: X → Y and T: X → 2Y where t is single-valued continuous and T is ε-semicontinuous.
LA - eng
KW - Best approximation; R-tree; fixed points; semicontinuity; best approximation; -tree
UR - http://eudml.org/doc/267770
ER -

## References

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1. Fan, K., Extensions of two fixed point theorems of F. E. Browder, Math. Zeit. 112 (1969), 234-240.[Crossref] Zbl0185.39503
2. Kirk, W. A., Fixed point theorems in CAT(0) spaces and R-trees, Fixed Point Theory Appl. 2004:4 (2004), 309-316. Zbl1089.54020
3. Kirk, W. A., Panyanak, B., Best approximation in R-trees, Numer. Funct. Anal. Optimiz. 28 (2007), 681-690; Erratum: Numer. Funct. Anal. Optimiz. 30 (2009), 403.[Crossref] Zbl1132.54025
4. Lin, T., Proximity maps, best approximations and fixed points, Approx. Theory Appl. (N. S.) 16, no. 4 (2000), 1-16. Zbl0989.47052
5. Markin, J. T., Fixed points, selections and best approximation for multivalued mappings in R-trees, Nonlinear Anal. 67 (2007), 2712-2716.[WoS] Zbl1128.47052
6. Piątek, B., Best approximation of coincidence points in metric trees, Ann. Univ. Mariae Curie-Skłodowska Sect. A 62 (2008), 113-121. Zbl1182.54055
7. Shahzad, N., Markin, J., Invariant approximations for commuting mappings in CAT(0) and hyperconvex spaces, J. Math. Anal. Appl. 337 (2008), 1457-1464. Zbl1137.47043

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