PROXIMAL MAPS, PROX MAPS AND COINCIDENCE POINTS

Given a continuous seminorm p on a Hausdorff locally convex space X and nonempty subsets A, B of X, let Proximal(B;A) (resp. Prox(A,B)) denote the set of p-proximal points of B in A (resp. the set of ordered pairs of p-proximal points of the pair (A,B)). In this article we identify suitable families A,B of nonempty subsets of X and natural topologies on them with a view to study continuity properties of B → Proximal(B;A) and (A,B) → Prox (A, B). This leads us to obtain a best approximation result and a coincidence theorem for multifunctions defined on non-compact sets. © 1990, Taylor & Francis Group, LLC. All rights reserved.