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

We show that ifU is a bounded open set in a complete CAT(0) spaceX, 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. 1