Asymptotic properties of nonexpansive iterations in reflexive spaces

Let X be a reflexive Banach space and (x(n))(n greater than or equal to 0) a nonexpansive (resp., firmly nonexpansive) sequence in X. It is shown that the set of weak omega-limit points of the sequence (x(n)/n)(n greater than or equal to 1) (resp., (x(n+1) - x(n))(n greater than or equal to 0)) always lies on a convex subset of a sphere centered at the origin of radius d = lim(n-->infinity)\\x(n)/n\\This fact quickly yields previous results of B. Djafari Rouhani as well as recent results of J. S. Jung and J. S. Park. Potential applications are also discussed. (C) 1999 Academic Press.