无限族非扩张非自射映象公共不动点的迭代逼近与Cesàro均值迭代收敛性

数学物理学报 ›› 2010, Vol. 30 ›› Issue (6): 1666-1676.

无限族非扩张非自射映象公共不动点的迭代逼近与Cesàro均值迭代收敛性

饶若峰

宜宾学院数学系四川宜宾 644007

收稿日期:2008-08-27 修回日期:2009-12-31 出版日期:2010-12-25 发布日期:2010-12-25
基金资助:
四川省教育厅(青年)自然科学基金(08ZB002)资助

Iterative Approximation to Common |Fixed Points of An Infinite Family of Non-self Non-expansive Maps and Strong Convergence of Cesàro Mean Iterations

RAO Ruo-Feng

Department of Mathematics, Yibin University, Sichuan Yibin 644007

Received:2008-08-27 Revised:2009-12-31 Online:2010-12-25 Published:2010-12-25
Supported by:
四川省教育厅(青年)自然科学基金(08ZB002)资助

摘要/Abstract

摘要：

该文在严格凸的自反Banach空间引进并研究了关于无限族非扩张非自射映象的迭代算法, 获一强收敛结果. 应用这个结果又
获得了Cesàro型均值迭代算法的强收敛性结果. 所获主要结果将2007年Zhang-Lee-Chan获得的主要结果从Hilbert空间推广到Banach空间, 将自射映象推广到非自射. 也将2007年R. Wangkeeree获得的主要结果从一个非扩张映象Cesàro均值迭代算法推广到两个. 该文还得到关于非扩张映象不动点集的一全新结果.

关键词: Sunny非扩张收缩映象, 公共不动点, Cesàro均值迭代

Abstract:

The author introduces and studies an iteration with infinitely many non-expansive nonself mappings in the framework of reflexive and strictly convex Banach spaces. A new strong convergence theorem is obtained, which immediately extends the main result obtained by Zhang-Lee-Chan in 2007 from non-expanive self-mapping to non-expanive non-self-mapping, from Hilbert spaces to Banach spaces. In addition, other main results of this paper extend the main result obtained by R. Wangkeeree in 2007 from Cesàro mean iteration with one non-expansive mapping to Cesàro mean iteration with two ones. Finally, an interesting new result about the set of fixed points of non-expansive mappings is given.

Key words: Sunny non-expansive retraction, Common fixed point, Cesàro mean iterations

中图分类号:

47H09

饶若峰. 无限族非扩张非自射映象公共不动点的迭代逼近与Cesàro均值迭代收敛性[J]. 数学物理学报, 2010, 30(6): 1666-1676.

RAO Ruo-Feng. Iterative Approximation to Common |Fixed Points of An Infinite Family of Non-self Non-expansive Maps and Strong Convergence of Cesàro Mean Iterations[J]. Acta mathematica scientia,Series A, 2010, 30(6): 1666-1676.

参考文献

[1] Chang S S, Yao J C, Kim J K, Yang L. Iterative approximation to convex feasibility problems in Banach space. Fixed Point Theory and Applications, doi: 10.1155/2007/46797

[2] Zhang S S, Lee H W J, Chan C K. Approximation of nearest common fixed point of nonexpansive mappings in Hilbert spaces. Acta Mathematica Sinica (English Series), 2007, 23(10): 1889--1896

[3] Goebel K, Reich S. Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Monographs and Textbooks in Pure and Applied Mathematics, Vol 83. New York: Marcel Dekker, 1984

[4] Goebel K, Kark W A. Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics, 28. Cambridge, UK: Cambridge University Press, 1990

[5] Browder F E. Convergence theorems for sequences of nonlinear operators in Banach spaces. Math Z, 1967, 100: 201--225

[6] Bruck R E. Nonexpansive projections on subsets of Banach spaces. Pacific J Math, 1973, 47: 341--355

[7] 饶若峰. 严格渐进伪压缩映象之修正型Mann迭代算法的强收敛性. 数学进展, 2010, 39(3): 283--288

[8] Reich S. Asymptotic behavior of contractions in Banach spaces. Journal of Mathematical Analysis and Applications, 1973, 44(1): 57--70

[9] Kopeck\'{a} E, Reich S. Nonexplosive retractions in Banach spaces. Erwin Schrodinger Institute, Preprint No.1787, 2006

[10] Liu L S. Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive mappings in Banach space. J Math Anal Appl, 1995, 194: 114--125

[11] Chang S S. On Chidume's open questions and approximation solutions of multivalued strongly accretive mappings equations in Banach spaces. J Math Anal Appl, 1997, 216: 94--111

[12] Chang S S, Cho Y J, Zhou H Y. Iterative Methods for Nonlinear Operator Equations in Banach Spaces. New York: Nova Science Publishers, 2002

[13] Chang S S. Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces. J Math Anal Appl, 2006, 323: 1402--1416

[14] Shimizu T, Takahashi W. Strong convergence to common fixed points of families of nonexpansive mappings. J Math Anal Appl, 1997, 211: 71--83

[15] Brezis H, Browder F E. Nonlinear ergodic theorems. Bull Amer Math Soc, 1976, 82: 959--961

[16] Bruck R E. A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces. Israel J Math, 1979, 32(2/3): 107--116

[17] Matsushita S, Takahashi W. Strong convergence theorems for nonexpansive nonself-mappings without boundary conditions. Nonlinear Anal, 2008, 68: 412--419

[18] Wangkeeree R. Strong convergence of Cesàro mean iterations for nonexpansive nonself-mappings in Banach spaces. Fixed Point Theory and Applications, doi: 10.1155/2007/59262

[19] Matsushita S, Kuroiwa D. Strong convergence of averaging iterations of nonexpansive nonself-mappings. J Math Anal Appl, 2004, 294(1): 206--214

[20] 饶若峰. 涉及无限族非扩张映象{T_n}^∞_n=1的迭代算法x_n+1 =α_n+1f (x_n) +(1 − α_n+1)T_n+1x_n. 数学研究与评论, 2009, 29(4): 639--648

[21] 饶若峰. 渐近非扩张映象具误差的合成隐迭代序列的弱收敛和强收敛定理. 数学年刊, 2008, 29A(4): 461--470

[22] 饶若峰. 带误差的合成隐迭代新算法. 数学物理学报, 2009, 29A(3): 823--831

[23] Bruck R E. Properties of fixed -point sets of nonexpansive mappings in Banach spaces. Trans Amer Math Soc, 1973, 179: 251--262