ON ε-APPROXIMATION AND FIXED POINTS OF NONEXPANSIVE MAPPINGS IN METRIC SPACES

Abstract. Using fixed point theory, B. Brosowski [2] 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 Brosowski’s result have appeared. We also obtain some results on invariant points of a nonexpansive mapping for the set of ε-approximation in metric spaces thereby generalizing and extending some known results including that of Brosowski, on the subject. Using fixed point theory, the theorem of Meinardus [6] on invariant approxi-mation was generalized by Brosowski [2] who 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, various generalizations of Brosowski’s result have appeared (see e.g. [5]). In the present work we also obtain some results on invariant points of a nonexpansive mapping T on the set of ε-approximation in metric spaces. Our results contain some of th