Research Article A Note on Geodesically Bounded R-Tree | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 393470 4 pages doi 2010 393470 Research Article A Note on Geodesically Bounded R-Trees W. A. Kirk Department of Mathematics University of Iowa Iowa City IA 52242 USA Correspondence should be addressed to W. A. Kirk kirk@ Received 4 March 2010 Accepted 10 May 2010 Academic Editor Mohamed Amine Khamsi Copyright 2010 W. A. Kirk. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. It is proved that a complete geodesically bounded R-tree is the closed convex hull of the set of its extreme points. It is also noted that if X is a closed convex geodesically bounded subset of a complete R-tree Y and if a nonexpansive mapping T X Y satisfies inf d x T x x e X 0 then T has a fixed point. The latter result fails if T is only continuous. 1. Introduction Recall that for a metric space X d a geodesic path or metric segment joining x and y in X is a mapping c of a closed interval 0 into X such that c 0 x c l y and d c t c f f - f for each t f e 0 l . Thus c is an isometry and d x y l. An R-tree or metric tree is a metric space X such that i there is a unique geodesic path denoted by x y joining each pair of points x y e X ii if y x n x z x then y x u x z y z . From i and ii it is easy to deduce that iii if x y z e X then x y n x z x w for some w e X. The concept of an R-tree goes back to a 1977 article of Tits 1 . Complete R-trees posses fascinating geometric and topological properties. Standard examples of R-trees include the radial and river metrics on R2. For the radial metric consider all rays emanating from the origin in R2. Define the radial distance dr between x y e R2 to be the usual distance if they are on the same ray otherwise take dr x y d x 0 d 0 y . 2 Fixed Point Theory and Applications Here d .
