Fixed Point Theory and Applications
Volume 2004 (2004), Issue 4, Pages 309-316
Fixed point theorems in spaces and -trees
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 is a bounded open set in a complete space , and if is nonexpansive,
then always has a fixed point if there exists such that for all . It is also shown that if is a geodesically bounded closed convex subset
of a complete -tree with , and if is a continuous mapping for which
for some and all , then 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.