Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 393470, 4 pages
doi:10.1155/2010/393470
Research Article

A Note on Geodesically Bounded ℝ-Trees

W. A. Kirk

Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA

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.

Abstract

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∈X}=0, then T has a fixed point. The latter result fails if T is only continuous.