Let be a hyperconvex metric space, and a given multivalued mapping. The authors consider the following problems in this setting:
1) The best approximation problem: Find the condition on and on a set such that there is satisfying for all
2) The invariant approximation problem consisting in finding conditions for the mapping with invariant set and with implying that where is the metric projection of onto A;
3) The best proximity pair problem: Find conditions on and implying that there is a point such that
These three problems are studied when the multivalued mapping is condensing or nonexpansive, and one obtains sufficient conditions for the existence of the solutions of these problems. For example, if the set is admissible (respectively, bounded externally hyperconvex) and is condensing, continuous in the Hausdorff metric of and the values of are nonempty bounded externally hyperconvex (respectively admissible) subset of then has a point of best approximation (Th. 3.1).