A note on Picard iterates of nonexpansive mappings

Volume 76 / 2001

Eun Suk Kim, W. A. Kirk Annales Polonici Mathematici 76 (2001), 189-196 MSC: Primary 47H10, 46B20. DOI: 10.4064/ap76-3-2


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.


  • Eun Suk KimDepartment of Applied Mathematics
    Pukyong National University
    Pusan 608-737, South Korea
  • W. A. KirkDepartment of Mathematics
    The University of Iowa
    Iowa City, IA 52242-1419, U.S.A.

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