The problem of extending uniformly continuous mappings between metric spaces was proposed by Aronszajn and Panitchpakdi in 1956, with respect to Hyperconvex. The structure offered by the hyperconvexity of the metric to space was evident from the very beginning. Due to the richness of this, hyperconvectic metric locations, notably in the late 1880's by pioneering works due to Baillon, Sine and Soardi, were developed to establish a very deep and exhaustive Fixed Point Theory. This theory refers both to single and multivalued mappings and to the best outcomes. In the last decade of metric fixed point theory on hyperconvex metric spaces we offer an exposition of progress in this article. Therefore, we cover primarily observations where the mapping requirements are metric. We shall recall effects of non-expansive proximinal retractions and their effect on the principle of best approximation and best pairs for proximity. Finally, few reflections and new findings are shown on the expansion of compact maps.
Key words: hyperconvex metric spaces, proximinal retractions
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