Fixed Point Theory and Applications
Volume 2004 (2004), Issue 4, Pages 309-316
doi:10.1155/S1687182004406081

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

Abstract

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 p∈U such that x∉[p,f(x)) for all x∈∂U. It is also shown that if K is a geodesically bounded closed convex subset of a complete ℝ-tree with int(K)≠∅, and if f:K→X is a continuous mapping for which x∉[p,f(x)) for some p∈int(K) and all x∈∂K, then f has a fixed point. It is also noted that a geodesically bounded complete ℝ-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.