A note on Picard iterates of nonexpansive mappings
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.