We show that if U 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… CONTINUE READING