Best Approximation, Fixed Points and Invariant Approximation in Linear Metric Spaces

Abstract

If G is a nonempty subset of a metric space (X, d) and x ∈ X, a point g₀ ∈ G is called a best approximation to x in G if d(x; g₀) = dist(x;G) ≡
inf{d(x, g) : g ∈ G}. The set of all best approximations to x in G is denoted by P_G(x). The set-valued map P_G is called a best approximation map. This paper shows how the best approximation map helps in finding fixed points of certain mappings in linear metric spaces. The results proved in the paper generalize and extend several known results on the subject.

Key words and phrases. Best approximation, proximinal set, Chebyshev set, metric projection, approximatively compact set, uniformly convex linear metric space, convex continuous map.

2000 Mathematics Subject Classification. 41A50, 41A65, 47H10, 54H25