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 $P_C(x)$ of best $C$-approximant to $x$ contains a $T$-invariant point if $P_C(x)$ 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 $\varepsilon$-approximation in metric spaces thereby generalizing and extending some known results including that of Brosowski, on the subject.

Keywords: $\varepsilon$-approximation; $\varepsilon$-coapproximation; convex metric space; $G$-convex structure; convex set; starshaped set; nonexpansive map and contraction map.

MSC: 41A50, 41A65, 47H10, 54H25

Pages:  165$-$171     

Volume  61 ,  Issue  2 ,  2009