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Uniform normal structure and solutions of Reich's open question

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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.

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, 200234, Shanghai, P. R. China

    Zeng Liu-chuan (Professor)

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  1. Zeng Liu-chuan
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Correspondence to Zeng Liu-chuan.

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Communicated by Zhang Shi-sheng

Project supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutes of Ministry of Education, China; the Dawn Program Fund of Shanghai

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Liu-chuan, Z. Uniform normal structure and solutions of Reich's open question. Appl Math Mech 26, 1204–1211 (2005). https://doi.org/10.1007/BF02507731

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  • DOI: https://doi.org/10.1007/BF02507731

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2000 Mathematics Subject Classification

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