Bulletin of the
Korean Mathematical Society BKMS

Approximate fixed point property problem for Mann iteration sequence of a nonexpansive mapping has been resolved on a Banach space independent of uniform (strict) convexity by Ishikawa [{\it Fixed points and iteration of a nonexpansive mapping in a Banach space}, Proc. Amer. Math. Soc. {\bf 59} (1976), 65--71]. In this paper, we solve this problem for a class of mappings wider than the class of asymptotically nonexpansive mappings on an arbitrary normed space. Our results generalize and extend several known results.

Keywords: nearly asymptotically nonexpansive mapping, asymptotically nonexpansive mapping, Mann iteration, nearly uniform $k$-contraction mapping, Opial condition

MSC numbers: 47H09, 47H10