Fixed Point Theory and Applications
Volume 2004 (2004), Issue 4, Pages 309-316

Fixed point theorems in CAT(0) spaces and -trees

W. A. Kirk

Department of Mathematics, The University of Iowa, Iowa City 52242-1419, IA, USA

Received 10 June 2004

Copyright © 2004 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.


We show that if U is a bounded open set in a complete CAT(0) space X, and if f:U¯X is nonexpansive, then f always has a fixed point if there exists pU such that x[p,f(x)) for all xU. It is also shown that if K is a geodesically bounded closed convex subset of a complete -tree with int(K), and if f:KX is a continuous mapping for which x[p,f(x)) for some pint(K) and all xK, then f has a fixed point. It is also noted that a geodesically bounded complete -tree has the fixed point property for continuous mappings. These latter results are used to obtain variants of the classical fixed edge theorem in graph theory.