| UNIFORM NORMAL STRUCTURE AND SOLUTIONS OF REICH’S OPEN QUESTION |
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| 引用本文： | 曾六川.UNIFORM NORMAL STRUCTURE AND SOLUTIONS OF REICH’S OPEN QUESTION[J].应用数学和力学(英文版),2005,26(9):1204-1211. |
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| 作者姓名： | 曾六川 |
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| 作者单位： | [1]Department of Mathematics, Shanghai Normal University, Shanghai 200234, P. R. China |
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| 基金项目： | Project supported by the Teaching and Research Award Fund for 0utstanding Young Teachers in Higher Education Institutes of Ministry of Education, China; the Dawn Program Fund of Shanghai |
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| 摘 要： | 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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| 关 键 词： | Banach空间 标准结构 开放空间 收敛性 参数 |
| 文章编号： | 0253-4827（2005）09-1204-08 |
| 收稿时间： | 2003-08-21 |
| 修稿时间： | 2005-03-15 |
Uniform normal structure and solutions of Reich's open question |
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| Zeng Liu-chuan.Uniform normal structure and solutions of Reich's open question[J].Applied Mathematics and Mechanics(English Edition),2005,26(9):1204-1211. |
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| Authors: | Zeng Liu-chuan |
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| Institution: | Department of Mathematics, Shanghai Normal University, Shanghai 200234, P. R. China |
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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.
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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| Keywords: | asymptotically nonexpansive mapping fixed point uniform normal structure uniformly CJateaux differentiable norm iterative approximation |
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