Summary: 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.