Fixed points of quasi-nonexpansive mappings and best approximation

Abstract

Using fixed point theory, B.Brosowski [Mathematica (Cluj) 11 (1969), 195-220] proved that if T is a nonexpansive linear operator on a normed linear space X, C a T-invariant subset of X and x a T-invariant point, then the set PC(x) of best C-approximant to x contains a T-invariant point if PC(x) is non-empty, compact and convex. Subsequently, many generalizations of the Brosowskis result have appeared. In this paper, we also prove some extensions of the results of Brosowski and others for quasi-nonexpansive mappings when the underlying spaces are metric linear spaces or convex metric spaces.