Remarks on approximate fixed points

A mapping ff from a topological space XX to a metric space (M,d)(M,d) is said to be rr-continuous (r>0r>0) if for any x∈Xx∈X there exists a neighborhood UxUx of xx such that diam(f(Ux))≤r. In 1961 Victor Klee proved that an rr-continuous self-mapping of a compact convex subset of a Banach space always has a γγ-invariant point for any γ>rγ>r. Recently H.X. Phu extended Klee’s result to mappings that are ‘almost’ rr-continuous. In this note we extend Phu’s result to compact hyperconvex spaces and to geodesically bounded complete RR-trees. We also extend Klee’s stability result to certain geodesic spaces.