an:06121002
Zbl 1273.54052
Esp??nola, Rafa; Lorenzo, Pepa
Metric fixed point theory on hyperconvex spaces: recent progress
EN
Arab. J. Math. 1, No. 4, 439-463 (2012).
00310937
2012
j
54H25 54E40 54-02
hyperconvex metric spaces; proximal nonexpansive retracts; best approximation; best proximity pairs
A metric space \(M\) is said to be hyperconvex if given any family \(\{x_{\alpha}\}\) of points of \(M\) and any family \(\{r_{\alpha}\}\) of real numbers satisfying \( d(x_{\alpha}, x_{\beta}) \leq r_{\alpha}+r_{\beta}\), then \(\bigcap_{\alpha} B(x_{\alpha},r_{\alpha}) \neq \varnothing\). In this survey paper, the authors review the development of metric fixed point theory on hyperconvex metric spaces. In Section 4, they discuss the problem of characterizing proximal nonexpansive retracts of hyperconvex spaces and its connections to several problems in best approximation and best proximity pairs. In Section 5, recent developments on \(\mathbb{R}\) trees and metric fixed point theory are treated. In the last section, some recent advances on the theory of extensions of H??lder maps and their relation to extensions of uniformly continuous mappings under \(\chi_0\) hyperconvex conditions are presented. Some new results on the extension of compact mappings are also given.
D. S. Diwan (Bhilai)