Let be a metric space, (resp., ) the family of nonempty closed bounded (resp., the family of nonempty compact) subsets of , and the Hausdorff metric on induced by . Then a map
is called multivalued nonexpansive if
for all . The authors attempt to investigate condions under which a multivalued nonexpansive map may have a fixed point. Since such a map on a complete metric space need not have a fixed point, the authors work in a complete CAT(0) space [cf. W. A. Kirk, in: Proceedings of the international conference on fixed-point theory and its applications, Valencia, Spain, July 13–19, 2003, 113–142 (2004; Zbl 1083.53061)] and assume some “inwardness” requirement on the map. Their main result goes as follows. Let be a nonempty bounded closed convex subset of a complete CAT(0) space and a nonexpansive map. Assume that is weakly inward on . Then has a fixed point.
Another main result is about the existence of a common fixed point of a single-valued nonexpansive map commuting with a multivalued nonexpansive map.