Metric fixed point theory on hyperconvex spaces: recent progress

Abstract

In this survey we present an exposition of the development during the last decade of metric fixed point theory on hyperconvex metric spaces. Therefore we mainly cover results where the conditions on the mappings are metric. We will recall results about proximinal nonexpansive retractions and their impact into the theory of best approximation and best proximity pairs. A central role in this survey will be also played by some recent developments on R-trees. Finally, some considerations and new results on the extension of compact mappings will be shown. Mathematics Subject Classification

Diversities, hyperconvexity and fixed points

“…A very well known fact, first indepently discovered by R. Sine [16] and P. Soardi [20] and then revisited by J.B. Baillon in [2], is that bounded hyperconvex metric spaces have the fixed point property for nonexpansive mappings. In fact a complete fixed point theory has been developed on hyperconvex spaces since then, the interested reader may check the surveys [6,7]. For a more general treatment on metric fixed point theory the reader may check [9] or for a really exhaustive and more recent monograph [13].…”

“…Hyperconvex metric spaces had been introduced some years earlier by N. Aronszajn and P. Panitchpakdi in [1] as metric spaces which are absolute nonexpansive retracts. Since then a lot has been written on hyperconvex metric spaces, the reader may find a gentle introduction to most of this information in the recent surveys [6,7] where hyperconvexity and its connections to existence of fixed points for nonexpansive mappings are explained. These surveys do not deal however with the connection of metric tight spans with phylogenetic problems.…”

“…In the way to give answers to this problem we will need to show new properties and provide examples regarding diversities and induced metric spaces. The work is organized as follows: in Section 2 we recall main facts and definitions from [3] which are relevant to our discussion as well as main facts on hyperconvex metric spaces which can be found in a more detailed way in any of [6,7]. Section 2 is closed with a new fact on the problem of extending nonexpansive mappings from an induced metric space to the diversity.…”

Uniformly Lipschitzian group actions on hyperconvex spaces

“…Dress in [7] as the tight span in the context of optimal networks and phylogenetic analysis. For a deeper discussion of hyperconvex spaces we refer the reader to [9,10,11].…”

Best Proximity Points for Relatively -Continuous Mappings in Banach and Hyperconvex Spaces

“…If is an admissible set, then ( ) is also an admissible set [11]. For recent progress in hyperconvex metric spaces, we refer the reader to [12].…”