Introduction to Hyperconvex Spaces

Metric fixed point theory on hyperconvex spaces: recent progress

“…In the monograph Handbook of Metric Fixed Point Theory edited by Kirk and Sims, and published in 2001 by Kluwer Academic Publishers, we can find the chapter Introduction to hyperconvex spaces [21] by Espínola and Khamsi. This survey was mainly devoted to the study of metric hyperconvexity and its impact on metric fixed point theory which began in late seventies of the twentieth century [60,64].…”

“…In this survey we will address some of these facts with a special attention to those involving metric conditions on the mappings. Therefore, it is not our intention to recall in detail basic and fundamental facts on hyperconvex metric spaces, which were already very generously explained along the first five sections of [21], but rather to present an update of some of the topics covered by the other last six sections of [21].…”

“…After the fundamental work of Sine on hyperconvexity and fixed points, which may be compiled in the series of works [35,50,[60][61][62][63], many authors continued stating new facts during the past decade of the twentieth century. The aforementioned chapter [21] in the Handbook of Metric Fixed Point Theory was devoted to collect many of those results. Since then new interesting facts have been obtained about fixed points and best approximation theory in hyperconvex metric spaces.…”

“…The connection between R-tress and hyperconvexity became evident for most fixed point theorists after the work [40] from 1998 by Kirk; however, it was needed to wait for some time until fixed point results on R-trees began to be studied [22] in a more systematic way. This explains why one of the most fruitful topics in the field nowadays, which is fixed point results on R-trees, was not covered in [21].…”

“…4 we deal with nonexpansive proximinal retracts, best approximation, and best proximity pairs. In [21] a lot of attention was given to the study of nonexpansive proximinal retracts. By then the problem of characterizing such subsets of hyperconvex spaces was still open.…”