Fixed points of locally nonexpansive mappings in geodesic spaces

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))