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.