A note on Picard iterates of nonexpansive mappings

Eun Suk Kim¹ ; W. A. Kirk²

¹ Department of Applied Mathematics Pukyong National University Pusan 608-737, South Korea
² Department of Mathematics The University of Iowa Iowa City, IA 52242-1419, U.S.A.

Annales Polonici Mathematici, Tome 76 (2001) no. 3, pp. 189-196.

Résumé

Let $X$ be a Banach space, $C$ a closed subset of $X$, and $T:C\rightarrow C$ a nonexpansive mapping. It has recently been shown that if $X$ is reflexive and locally uniformly convex and if the fixed point set $F(T)$ of $T$ has nonempty interior then the Picard iterates of the mapping $T$ always converge to a point of $F(T)$. In this paper it is shown that if $T$ is assumed to be asymptotically regular, this condition can be weakened much further. Finally, some observations are made about the geometric conditions imposed.

Détail

DOI : 10.4064/ap76-3-2

Keywords: banach space closed subset rightarrow nonexpansive mapping has recently shown reflexive locally uniformly convex fixed point set has nonempty interior picard iterates mapping always converge point paper shown assumed asymptotically regular condition weakened much further finally observations made about geometric conditions imposed