*(English)*Zbl 1086.47019

Let $(X,D)$ be a metric space, $CB\left(X\right)$ (resp., $K\left(X\right)$) the family of nonempty closed bounded (resp., the family of nonempty compact) subsets of $X$, and $H$ the Hausdorff metric on $CB\left(X\right)$ induced by $d$. Then a map

is called multivalued nonexpansive if

for all $x,y\in X$. 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 $E$ be a nonempty bounded closed convex subset of a complete CAT(0) space $X$ and $T:X\to K\left(X\right)$ a nonexpansive map. Assume that $T$ is weakly inward on $E$. Then $T$ 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.

##### MSC:

47H10 | Fixed point theorems for nonlinear operators on topological linear spaces |

47H09 | Mappings defined by “shrinking” properties |

54H25 | Fixed-point and coincidence theorems in topological spaces |

51K10 | Synthetic differential geometry |

05C05 | Trees |

53C70 | Direct methods ($G$-spaces of Busemann, etc.) |

58C30 | Fixed point theorems on manifolds |