Remarks on best approximation in R-trees

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 -