Uniform normal structure and solutions of Reich's open question

Abstract

The open question raised by Reich is studied in a Banach space with uniform normal structure, whose norm is uniformly Gateaux differentiable. Under more suitable assumptions imposed on an asymptotically nonexpansive mapping, an affirmative answer to Reich's open question is given. The results presented extend and improve Zhang Shisheng's recent ones in the following aspects: (i) Zhang's stronger condition that the sequence of iterative parameters converges to zero is removed; (ii) Zhang's stronger assumption that the asymptotically nonexpansive mapping has a fixed point is removed; (iii) Zhang's stronger condition that the sequence generated by the Banach Contraction Principle is strongly convergent is also removed. Moreover, these also extend and improve the corresponding ones obtained previously by several authors including Reich, Shioji, Takahashi, Ueda and Wittmann.