Remarks on best approximation in R-trees

An R-tree is a geodesic space for which there is a unique arc joining any two of its points, and this arc is a metric segment. If X is a closed convex subset of an R-tree Y; and if T : X ! 2 Y is a multivalued mapping, then a point z for which 0 < dist(z;T (z)) = inf x2X dist(x;T (z)) is called a point of best approximation. It is shown here that if T is an "- semicontinuous mapping whose values are nonempty closed convex subsets of Y; and if T has at least two distinct points of best approximation, then T must have a fixed point. We also obtain a common best approximation theorem for a commuting pair of mappings t : X ! Y and T : X ! 2 Y where t is single-valued continuous and T is "-semicontinuous.