Hyperconvexity and approximate fixed points

Introduction to Hyperconvex Spaces

“…The set of ε-fixed points of T is denoted by Fix ε (T ). Sine [55] obtained the following wonderful result: Theorem 6.4. Let H be a bounded hyperconvex metric space and T : H → H a nonexpansive map.…”

“…Sine [55] began the study of the retraction property in hyperconvex spaces in a little more detailed way than what we have stated so far. His results are crucial in investigating nonexpansive mappings defined on hyperconvex metric spaces.…”

Metric fixed point theory on hyperconvex spaces: recent progress

“…These results can be summarized in the next one. A more general and very celebrated version of this result on ε-fixed points was later proved by Sine [63] or [36,Section 4.6].…”

“…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.…”

“…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.…”