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Hyperconvexity and approximate fixed points. (English) Zbl 0694.54033
A metric space (S,d) is called hyperconvex if any collection of closed balls of radius ${r}_{\alpha }$ and center ${s}_{\alpha },\alpha \in A$, which satisfy $d\left({s}_{\alpha },{s}_{\beta }\right)\le {r}_{\alpha }+{r}_{\beta }$ for all $\alpha$, $\beta$ in A, has nonempty intersection. The author begins with a good review and set of references for known results. He defines the $ϵ$-fixed point set of T: $S\to S$ to be ${F}_{ϵ}\left(T\right)=\left\{s:d\left(s,Ts\right)\le ϵ\right\}$, then proves an approximate fixed point theorem: if T: $H\to H$ is nonexpansive and H is hyperconvex, then ${F}_{ϵ}\left(T\right)$ is hyperconvex and, if nonempty, is a nonexpansive retract of H. If ${F}_{ϵ}\left(T\right)$ is also bounded, then the fixed point set $F\left(T\right)={F}_{0}\left(T\right)$ is nonempty as well. The rest of the paper gives interesting results and discussion of the hyperconvex hull, N(S), of S as defined by J. R. Isbell [Comment. Math. Helv. 39, 65- 76 (1964; Zbl 0151.302)]. In particular, if T:N(S)$\to N\left(S\right)$ is nonexpansive, then $S\subset {F}_{ϵ}\left(T\right)$ implies $N\left(S\right)={F}_{ϵ}\left(T\right)$. Also, any realization of $N\left({c}_{0}\right)$ in ${\ell }_{\infty }$ is all of ${\ell }_{\infty }$. This is nice since realizations of N(S) in a hyperconvex space H containing S always exist but, though isometric, are not generally unique. The author’s first “Remark” is unfortunate in that it gives a badly flawed example to discount Remark 3.1 of the Isbell paper cited above. In fact, it is the reviewer’s opinion that minor modification of the author’s lemma proof verifies Isbell’s remark.
Reviewer: D.E.Sanderson

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H09 Mappings defined by “shrinking” properties