*(English)*Zbl 0957.46033

A metric space $(X,d)$ is said to be hyperconvex if any intersection of closed balls $B({x}_{\alpha};{r}_{\alpha})$ is nonempty whenever $d({x}_{\alpha},{x}_{\beta})\le {r}_{\alpha}+{r}_{\beta}$ for all balls in this collection. The authors are drawing extensively upon an article by *M. A. Khamsi* [J. Math. Anal. Appl. 204, No. 1, 298-306 (1996; Zbl 0869.54045)]. They use the methods of that article to prove fixed point theorems for nonexpansive and condensing maps in hyperconvex spaces.

An interesting feature of the present article is the observation that the theory of hyperconvex spaces may yield fixed point results for nonexpansive mappings in arbitrary Banach spaces: Let $K$ be a closed bounded convex subset of a Banach space and $T:K\to K$ a nonexpansive map with nonempty fixed point set $A$. Assume that there is a $c>0$ such that $\parallel x-Tx\parallel \ge d(x,A)$ for all $x\in K$. Let $f=\frac{1}{2}(I+T)$. Then ${f}^{n}{\left(x\right))}_{n\in \mathbb{N}}$ converges to a fixed point of $T$. Moreover, there is a nonexpansive retraction $r:K\to A$ which commutes with $f$.

##### MSC:

46H10 | Ideals and subalgebras of topological algebras |

47H09 | Mappings defined by “shrinking” properties |

46B20 | Geometry and structure of normed linear spaces |