Asymptotic Properties of Nonexpansive Iterations in Reflexive Spaces

Let X be a reflexive Banach space and xn.nG0 a nonexpansive resp., firmly nonexpansive. sequence in X. It is shown that the set of weak v-limit points of the sequence xnrn.nG1 resp., xnq1yxn.nG0.always lies on a con¨ex subset of a sphere centered at the origin of radius dslim nª`5xnrn5. 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.