$\varepsilon$-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 anormed linear space $X$, $C$ a $T$-invariant subset of $X$ and $x$ a$T$-invariant point, then the set $P_C(x)$ of best $C$-approximantto $x$ contains a $T$-invariant point if $P_C(x)$ is non-empty,compact and convex. Subsequently, many generalizations of theBrosowski's result have appeared. We also obtain some results oninvariant points of a nonexpansive mapping for the set of$\varepsilon$-approximation in metric spaces thereby generalizingand extending some known results including that of Brosowski, on the subject.