Abstract
This is an examination of the structure of fixed point sets of locally nonexpansive mappings in various geodesic spaces. Among other things, it is shown that if G is a bounded connected open subset of a complete CAT(0) space X and if T:G‾→X is continuous on G‾ and locally nonexpansive on G, then the condition d(u,T(u))<d(x,T(x)) for all u∈G and x∈∂G implies that the fixed point set of T is a nonempty closed convex subset of G. The following theorem is also consequence of one of our main results. Theorem. Let (X,d) be a complete CAT(0) space which has the geodesic extension property and whose Alexandrov curvature is bounded below. Suppose G is a connected open subset of X, and suppose T:G→G is a locally nonexpansive mapping for which Fix(T)‾⊂G and for which int(Fix(T))≠∅. Then Fix(T) is a closed convex subset of G, and moreover the sequence {Tn(x)} converges to a point of Fix(T) for each x∈G.