Hyperconvexity and Nonexpansive Multifunctions

Abstract

Abstract.It is shown that a ball intersection valued nonexpansive multifunction on a hyperconvex space admits a nonexpansive point valued selection. This implies fixed point theorems for such multifunctions and to certain point valued nonexpansive maps. The result is used to study best approximation and to show the space of all nonexpansive maps of a bounded hyperconvex space is hyperconvex.

Fixed point and selection theorems in hyperconvex spaces

“…This paper focuses on external hyperconvexity, a concept which was also introduced by Aronszajn and Panitchpakdi in their fundamental paper [1]. Our main result, which extends the principal result of Sine [12], yields the fact that a lipschitzian set-valued mapping of a hyperconvex metric space into itself, taking externally hyperconvex values, always has a single valued selection which is lipschitzian for the same constant. This is used to show that the family of all bounded λ-lipschitzian mappings of a hyperconvex space into itself is itself hyperconvex.…”

Metric fixed point theory on hyperconvex spaces: recent progress

“…It turned out that hyperconvex spaces were very adequate to deal with multivalued mappings. More general versions of the next result can be found in [37] (see also [62] or [21]). The same result was proved for admissible valued mappings in [62]; it is an open question posed in [62] whether this theorem remains true if the images are supposed to be hyperconvex instead of admissible sets.…”

“…Admissible subsets of hyperconvex spaces enjoy of a very large number of properties. These properties were very relevant when studying approximation problems and fixed point results; see for instance [21,33,46,61,62] among others. Obviously, the class of admissible sets is closed under arbitrary intersections.…”

“…Selections is also a very important feature of R-trees and hyperconvex spaces [37,62]. This topic was already covered in [21,Section 9].…”

“…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.…”