一致正规结构与Reich的公开问题的解答

曾六川

一致正规结构与Reich的公开问题的解答

曾六川

上海师范大学数学系,上海 200234

基金项目: 高等学校优秀青年教师教学和科研奖励基金资助项目;上海市曙光计划基金资助项目

详细信息

作者简介:
曾六川(1965- ),男,湖南邵东人,教授,博士,博士生导师(E-mail:zenglc@hotmail.com).

中图分类号: O177.91
计量
- 文章访问数: 2277
- HTML全文浏览量: 93
- PDF下载量: 619
- 被引次数: 0
出版历程
- 收稿日期: 2003-08-21
- 修回日期: 2005-03-15
- 刊出日期: 2005-09-15

Uniform Normal Structure and Solutions of Reich’s Open Question

ZENG Lu-chuan

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

摘要

摘要: 在具有一致正规结构且其范数是一致Gateaux可微的Banach空间中，研究了Reich提出的公开问题．在给渐近非扩张映象作更适当的假设下，对Reich的公开问题给出了一个肯定的答复．所得结果在下列方面推广与改进了张石生教授的最新结果：(ⅰ) 去掉了张教授的较强条件“迭代参数列收敛到零”；(ⅱ) 去掉了张教授的较强假设“渐近非扩张映象有不动点”；(ⅲ)也去掉了张教授的较强条件“Banach压缩映象原理生成的序列强收敛”．而且，这些结果也推广与改进了先前由Reich,Shioji,Takahashi,Ueda及Wittmann等多位作者得到的相应结果．
- 渐近非扩张映象 /
- 不动点 /
- 一致正规结构 /
- 一致Gateaux可微范数 /
- 迭代逼近
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 Shi-sheng'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.
- asymptotically nonexpansive mapping /
- fixed point /
- uniform normal structure /
- uniformly Gateaux differentiable norm /
- iterative approximation

HTML全文

参考文献(12)

[1]	Goebel K,Kirk W A.A fixed point theorem for asymptotically nonexpansive mappings[J].Proceedings of the American Mathematical Society,1972,35(1):171—174. doi: 10.1090/S0002-9939-1972-0298500-3
[2]	Edelstein M,O'Brien C R.Nonexpansive mappings, asymptotic regularity and successive approximations[J].Journal of the London Mathematical Society,1978,17:547—554. doi: 10.1112/jlms/s2-17.3.547
[3]	张石生.关于Reich的公开问题[J].应用数学和力学,2003,24(6):572—578.
[4]	Reich S.Some problems and results in fixed point theory[J].Contemporary Mathematics,1983,21:179—187. doi: 10.1090/conm/021/729515
[5]	Reich S.Strong convergence theorems for resolvent of accretive mappings in Banach spaces[J].Journal of Mathematical Analysis and Applications,1980,75:287—292. doi: 10.1016/0022-247X(80)90323-6
[6]	Wittmann R. Approximation of fixed points of nonexpansive mappings[J].Archiv der Mathematik,1992,58:486—491. doi: 10.1007/BF01190119
[7]	Shioji N,Takahashi W.Strong convergence of approximated sequence for nonexpansive mappings[J].Proceedings of the American Mathematical Society,1997,125(12):3641—3645. doi: 10.1090/S0002-9939-97-04033-1
[8]	Takahashi W,Ueda Y.On Reich's strong convergence theorems for resolvents of accretive operators[J].Journal of Mathematical Analysis and Applications,1984,104:546—553. doi: 10.1016/0022-247X(84)90019-2
[9]	Deimling K.Nonlinear Functional Analysis[M].Berlin:Springer-Verlag,1985.
[10]	Lim T C,Xu H K. Fixed point theorems for asymptotically nonexpansive mappings[J].Nonlinear Analysis—Theory Methods & Applications,1994,22(11):1345—1355.
[11]	Chang S S,Cho Y J,Lee B S,et al.Iterative approximations of fixed points and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach spaces[J].Journal of Mathematical Analysis and Applications,1998,224:149—165. doi: 10.1006/jmaa.1998.5993
[12]	Kim T H,Xu H K.Remarks on asymptotically nonexpansive mappings[J].Nonlinear Analysis—Theory Methods & Applications,2000,41:405—415.