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一致正规结构与Reich的公开问题的解答

曾六川

曾六川. 一致正规结构与Reich的公开问题的解答[J]. 应用数学和力学, 2005, 26(9): 1097-1104.
引用本文: 曾六川. 一致正规结构与Reich的公开问题的解答[J]. 应用数学和力学, 2005, 26(9): 1097-1104.
ZENG Lu-chuan. Uniform Normal Structure and Solutions of Reich’s Open Question[J]. Applied Mathematics and Mechanics, 2005, 26(9): 1097-1104.
Citation: ZENG Lu-chuan. Uniform Normal Structure and Solutions of Reich’s Open Question[J]. Applied Mathematics and Mechanics, 2005, 26(9): 1097-1104.

一致正规结构与Reich的公开问题的解答

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

Uniform Normal Structure and Solutions of Reich’s Open Question

  • 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.
  • [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.
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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

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