Fixed points of locally nonexpansive mappings in geodesic spaces

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.