Banach空间中的不动点迭代逼近
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摘要
不动点问题一直是人们关注的重点问题之一,有关这方面的研究也取得了显著的成绩。在不动点问题研究的众多方向中,关于构造渐近不动点序列的迭代收敛问题以及在控制、非线性算子和微分方程等方面的理论结合及应用成为研究的主流问题,对这方面问题的研究会在实际运用中起到至关重要的作用。本文主要研究了L-Lipschitz映射的不动点迭代逼近以及不动点迭代序列收敛的等价性问题。
全文共分四部分:
第一章绪论:简单阐述国内外有关不动点理论的发展概况,并介绍本文要讨论的主要内容、背景和意义。
第二章预备知识:主要引入文中用到的一些定义、引理及相关知识。
第三章L-Lipschitz映射的不动点迭代逼近:给出了赋范线性空间中一致L-Lipsc -hitz映射的不动点迭代逼近定理、Banach空间中一致L-Lipschitz映射的(带误差修改的)Ishikawa迭代序列和(带误差修改的)Mann迭代序列的不动点迭代逼近的充要条件和充分条件、修改的三重迭代序列的收敛性以及渐近非扩张映射的不动点迭代逼近等问题的定理。
第四章Ishikawa迭代序列和Mann迭代序列收敛的等价性:给出了带误差的Ishikawa迭代序列和带误差的Mann迭代序列收敛性的等价定理。
The problem of fixed points is one of the emphasis problems of people’s regard and the corresponding research has gained many great achievements.Among many directions of the fixed point researches,it becomes main problem that the convergence problem about constructing approximating fixed point sequences and its application in control ,nonlinear operator and derivative equation etc.The research of this problem will play an important role in its application in reality.The purpose of this paper is to study some problems about iterative approximations of fixed points for L-Lipschitz mappings and the equivalence of iterative sequences convergence of fixed points.
The paper consists of four parts.The main results of this paper are summarized as following:
In the first chapter,we simply narrate the development of the fixed point theory home and abroad in the preface.And the main content that we will discuss,background and significance are introduced.
In the second chapter,some definitions,lemmas and related knowledge used in the article are introduced in the preparation.
In the third chapter,the main theorems about iterative approximations of fixed points for uniformly L-Lipschitz mappings in normed linear spaces,the sufficient and necessary conditions and the sufficient conditions on the convergence of the modified Ishikawa and Mann iterative sequence with errors for uniformly L–Lipschitz mappings in Banach spaces, the convergence of the modified three iterative sequences for uniformly L-Lipschitz mappings in Banach spaces and iterative approximations of fixed points for asymptotically nonexpansive mappings in Banach spaces are described and proved.
In the last chapter,the equivalence theorems about iterative sequences convergence of fixed points are described and proved.
全文共分四部分:
第一章绪论:简单阐述国内外有关不动点理论的发展概况,并介绍本文要讨论的主要内容、背景和意义。
第二章预备知识:主要引入文中用到的一些定义、引理及相关知识。
第三章L-Lipschitz映射的不动点迭代逼近:给出了赋范线性空间中一致L-Lipsc -hitz映射的不动点迭代逼近定理、Banach空间中一致L-Lipschitz映射的(带误差修改的)Ishikawa迭代序列和(带误差修改的)Mann迭代序列的不动点迭代逼近的充要条件和充分条件、修改的三重迭代序列的收敛性以及渐近非扩张映射的不动点迭代逼近等问题的定理。
第四章Ishikawa迭代序列和Mann迭代序列收敛的等价性:给出了带误差的Ishikawa迭代序列和带误差的Mann迭代序列收敛性的等价定理。
The problem of fixed points is one of the emphasis problems of people’s regard and the corresponding research has gained many great achievements.Among many directions of the fixed point researches,it becomes main problem that the convergence problem about constructing approximating fixed point sequences and its application in control ,nonlinear operator and derivative equation etc.The research of this problem will play an important role in its application in reality.The purpose of this paper is to study some problems about iterative approximations of fixed points for L-Lipschitz mappings and the equivalence of iterative sequences convergence of fixed points.
The paper consists of four parts.The main results of this paper are summarized as following:
In the first chapter,we simply narrate the development of the fixed point theory home and abroad in the preface.And the main content that we will discuss,background and significance are introduced.
In the second chapter,some definitions,lemmas and related knowledge used in the article are introduced in the preparation.
In the third chapter,the main theorems about iterative approximations of fixed points for uniformly L-Lipschitz mappings in normed linear spaces,the sufficient and necessary conditions and the sufficient conditions on the convergence of the modified Ishikawa and Mann iterative sequence with errors for uniformly L–Lipschitz mappings in Banach spaces, the convergence of the modified three iterative sequences for uniformly L-Lipschitz mappings in Banach spaces and iterative approximations of fixed points for asymptotically nonexpansive mappings in Banach spaces are described and proved.
In the last chapter,the equivalence theorems about iterative sequences convergence of fixed points are described and proved.
引文
[1] Mann W R.Mean value methods in iteration[J].Proc Amer Math Soc,1953,4:506-510
[2] Ishikawa S.Fixed points by a new iteration method[J].Proc Amer Math Soc,1974,44(1): 147-150
[3] Goebel K,Kirk W A.A fixed points theorem for asymoptotically nonexpansive mappings. Proc Amer Math Soc,1972,35(1):171-174
[4] Petryshyn W V,Williamson T E.Strong and weak convergence of the sequence of successiv -e approximations for quasi-nonexpansive mappings.J Math Anal Appl,1973,43:459-497
[5] Kirk W A.Fixed point theorems for non-Lipschitizian mappings of asymptotically nonexpan -sive type.Israel J Math,1974,11:339-346
[6] Schu J.Iterative construction of fixed points of asymoptotically nonexpansive mappings.J Math Anal Appl,1991,158:407-413
[7] Zhou Haiyun.Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilb -ert spaces.J Math Anal Appl,2008,343:546-556
[8] Zhou Haiyun.Convergence theorems of fixed points for k-strict pseudo-contractions in Hilbert spaces.Nonlinear Analysis,2008,69:456-462
[9] Chang S S.On Chidume’s open questions and approximation solutions of multivalued strongly accretive mapping equation in Banach space.J Math Anal Appl,1997,216:94-111.
[10] Gornicki J.Week convergence theorems for asymoptotically nonexpansive mappings in unifo -rmly convex Banach spaces.Comment Math Univ Carolin,1989,301:249-252
[11] Moore C,Nnoli B V.Iterative solution of nonlinear equations involving set-valued uniformly accretive operators,Comput Math Appl,2001,42:131-140
[12] Browder F E.Fixed point theorems for noncompact mappings in Hilbert spaces.Proc Natl Acad Sci USA,1965,53:1272-1276
[13] Lan K Q,Wu J H.Convergence of approximants for demicontinuous pseudo-contractive map -s in Hilbert spaces.Nonlinear Anal Sis,2002,49:737-746
[14] Marino G,Xu H K.Weak and strong convergence theorems for strict pseudo-contraction in Hilbert spaces.J Math Anal Appl,2007,329(1):336-346
[15] Zhang Shih-Sen.Approximation of nearest common fixed point of nonexpansive mappings in Hilbert Spaces.Acta Math Sinica,1999,15(1):1-11
[16]曾六川.一致光滑Banach空间中渐近伪压缩映象不动点的迭代逼近.数学年刊,2005, 26(2):283-290
[17] Yao Yonghong,Chen Rudong,Zhou Haiyun.Interative process for certain nonlinear mapping -s in uniformly smooth Banach spaces.Nonlinear Funct Anal Appl,2005,10(4):651-664
[18] Zhou H Y.Nonexpansive mappings and iterative methods in uniformly convex Banach space -s .Acta Math Sinica,2004,20:829-836
[19] Chidume C E,Naseer Shahzad. Strong convergence of an implicit iteration process for a finit -e family of nonexpansive mappings.Nonlinear Analysis,2005,62:1149-1156
[20] Guo Weiping,Cho Y J.On the strong convergence of the implicit iterative process with errors for a finite family of asymoptotically nonexpansive mappings.Appl Math Lett,2008,21:1046-1052
[21] Liu Qihou.Iterative Sequences for Asymptotically Quasi-nonexpansive Mappings with Error Member. J Math Anal Appl,2001,259:18-24
[22]朱玲娣.渐近伪压缩映象的迭代序列强收敛的充要条件.绍兴文理学院学报,2002,22 (4):4-9
[23] Chidume C E,Zegeye H.Approximate fixed point sequences and convergence theorems for asymptotically pseudocontractive mappings.J Math Anal Appl,2003,278:354-366
[24]曾六川.Banach空间中几乎渐近非扩张型映象的不动点的迭代逼近.应用数学和力学,2003,24(12):1258-1266
[25]曾六川.Banach空间中带误差的修改的Ishikawa迭代程序.数学学报,2004,47(2):219- 228
[26]曾六川.关于渐近伪压缩型映象的不动点的迭代构造.系统科学与数学,2004,24(2): 261-270
[27]王朝,刘理蔚.渐近伪压缩映象的Ishikawa迭代序列强收敛的充要条件.应用泛函分析学报,2006,8(3):252-258
[28] Zhou Haiyun.Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractions in Banach spaces.Nonlinear Analysis,2008,68:2977-2983
[29]向长合.渐近伪压缩映象不动点的迭代逼近.西南师范大学学报,2007,32(5):6-9
[30]向长合.一致L-Lipschitz的渐近拟伪压缩型映象迭代收敛的充要条件.系统科学与数学,2008,28(4):447-455
[31] Chang Shih-sen,Cho Yeol Je,Jong Kyu kim.Some results for uniformly L-Lipschitzian ma -ppings in Banach spaces.Appl Math Lett(2008),doi:10.1016/j.aml.2008.02.16
[32] Chang S S,Tan K K,Lee H W J,et al.On the convergence of implicit iteration process wit -h error for a finite family of asymptotically nonexpansive mappings.J Math Anal Appl,2006,313:2 -283
[33] Chang S S, Lee H W J,Chi Kin Chan.On Reich’strong convergence theorem for asy-moptot -ically nonexpansive mappings in Banach spaces.Nonlinear Analysis(2006),doi:10.1016/j.na.2006.03.025
[34]姚永红,陈汝栋.渐近伪压缩和渐近非扩张映像不动点的迭代逼近问题.应用泛函分析学报,2007,9(3):233-237
[35]罗元松.一类迭代序列的收敛性.四川大学学报(自然科学版),2006,43(6):1187-1191
[36] GU Feng,DU Xiufeng.Iterative approximations of fixed points for asymoptotically pseudo-contractive mappings in normed linear spaces.应用泛函分析学报,2003,5(2):125-131
[37]谷峰.赋范空间中渐进一致φ-伪压缩型映象不动点的迭代逼近.数学的实践与认识, 2006,26(3):282-287
[38]唐玉超,刘理蔚.赋范线性空间中渐近伪压缩映象的不动点迭代逼近.应用数学学报,2007,5:810-815
[39] Soltuz S M.The convergence of Mann iteration with errors is equivalent to the convergence of Ishikawa iteration with errors[J].Lectures Mathematicas,2004,25:5-13
[40] Soltuz S M.The equivalence of Picard,Mann and Ishikawa iterations dealing with quasi-constractive operators[J].Math Comm,2005,(10):81-88
[41]苏巨国,唐玉超.Mann迭代和带误差的Ishikawa迭代的等价性.上饶师范学院学报,2005,25(6):18-21
[42]野金花,崔云安.一致伪压缩映射的具误差的Ishikawa和Mann迭代序列收敛性的等价性问题.黑龙江大学自然科学学报,2006,23(2):242-244
[43]薛志群,范瑞琴.关于非线性算子不动点收敛定理等价性的探讨.石家庄铁道学院学报,2007,20(4):54-56
[44]王亚宁.关于一类非线性映射迭代序列收敛的等价性.北华航天工业学院学报,2009,19(4):38-40
[45] Tan K K.Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process.J Math Anal Appl,1993,178:301-308
[46] Zeng L C.A note on approximating fixed points of nonexpansive mappings by the Ishikawa iteration process.J Math Anal Appl,1998,226:245-250
[47] Xu H K.Existence and convergence for fixed points of mapping of asymoptotically nonexp -ansive type.Nonlinear Anal TMA,1991,16(12):1139-1146
[48] Sun Z H.Strong convergence of an implicit iteration process for a finite family of asymoptotically quasi-nonexpansive mappings.J Math Anal Appl,2003,286:351-358
[49] Ofoedu E U.Strong convergence theorem for uniformly L-Lipschitzian asymptotically pseudocontractive mapping in real Banach space.J Math Anal Appl,2006,321: 722-728
[50] Morales C H,Jun J S.Convergence of paths for pseudocontractive mappings in Banach spa -ces.Proc Amer Math Soc,2000,128:3411-3419
[51] Chang S S.Some results for asymoptotically pseudo-contractive mappings and asymoptotic -ally nonexpansive mappings.Proc Amer Math Soc,2001,129(3):845-853
[52]野金花.不动点的若干性质[D]:[硕士学位论文].哈尔滨:哈尔滨理工大学,2006
[2] Ishikawa S.Fixed points by a new iteration method[J].Proc Amer Math Soc,1974,44(1): 147-150
[3] Goebel K,Kirk W A.A fixed points theorem for asymoptotically nonexpansive mappings. Proc Amer Math Soc,1972,35(1):171-174
[4] Petryshyn W V,Williamson T E.Strong and weak convergence of the sequence of successiv -e approximations for quasi-nonexpansive mappings.J Math Anal Appl,1973,43:459-497
[5] Kirk W A.Fixed point theorems for non-Lipschitizian mappings of asymptotically nonexpan -sive type.Israel J Math,1974,11:339-346
[6] Schu J.Iterative construction of fixed points of asymoptotically nonexpansive mappings.J Math Anal Appl,1991,158:407-413
[7] Zhou Haiyun.Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilb -ert spaces.J Math Anal Appl,2008,343:546-556
[8] Zhou Haiyun.Convergence theorems of fixed points for k-strict pseudo-contractions in Hilbert spaces.Nonlinear Analysis,2008,69:456-462
[9] Chang S S.On Chidume’s open questions and approximation solutions of multivalued strongly accretive mapping equation in Banach space.J Math Anal Appl,1997,216:94-111.
[10] Gornicki J.Week convergence theorems for asymoptotically nonexpansive mappings in unifo -rmly convex Banach spaces.Comment Math Univ Carolin,1989,301:249-252
[11] Moore C,Nnoli B V.Iterative solution of nonlinear equations involving set-valued uniformly accretive operators,Comput Math Appl,2001,42:131-140
[12] Browder F E.Fixed point theorems for noncompact mappings in Hilbert spaces.Proc Natl Acad Sci USA,1965,53:1272-1276
[13] Lan K Q,Wu J H.Convergence of approximants for demicontinuous pseudo-contractive map -s in Hilbert spaces.Nonlinear Anal Sis,2002,49:737-746
[14] Marino G,Xu H K.Weak and strong convergence theorems for strict pseudo-contraction in Hilbert spaces.J Math Anal Appl,2007,329(1):336-346
[15] Zhang Shih-Sen.Approximation of nearest common fixed point of nonexpansive mappings in Hilbert Spaces.Acta Math Sinica,1999,15(1):1-11
[16]曾六川.一致光滑Banach空间中渐近伪压缩映象不动点的迭代逼近.数学年刊,2005, 26(2):283-290
[17] Yao Yonghong,Chen Rudong,Zhou Haiyun.Interative process for certain nonlinear mapping -s in uniformly smooth Banach spaces.Nonlinear Funct Anal Appl,2005,10(4):651-664
[18] Zhou H Y.Nonexpansive mappings and iterative methods in uniformly convex Banach space -s .Acta Math Sinica,2004,20:829-836
[19] Chidume C E,Naseer Shahzad. Strong convergence of an implicit iteration process for a finit -e family of nonexpansive mappings.Nonlinear Analysis,2005,62:1149-1156
[20] Guo Weiping,Cho Y J.On the strong convergence of the implicit iterative process with errors for a finite family of asymoptotically nonexpansive mappings.Appl Math Lett,2008,21:1046-1052
[21] Liu Qihou.Iterative Sequences for Asymptotically Quasi-nonexpansive Mappings with Error Member. J Math Anal Appl,2001,259:18-24
[22]朱玲娣.渐近伪压缩映象的迭代序列强收敛的充要条件.绍兴文理学院学报,2002,22 (4):4-9
[23] Chidume C E,Zegeye H.Approximate fixed point sequences and convergence theorems for asymptotically pseudocontractive mappings.J Math Anal Appl,2003,278:354-366
[24]曾六川.Banach空间中几乎渐近非扩张型映象的不动点的迭代逼近.应用数学和力学,2003,24(12):1258-1266
[25]曾六川.Banach空间中带误差的修改的Ishikawa迭代程序.数学学报,2004,47(2):219- 228
[26]曾六川.关于渐近伪压缩型映象的不动点的迭代构造.系统科学与数学,2004,24(2): 261-270
[27]王朝,刘理蔚.渐近伪压缩映象的Ishikawa迭代序列强收敛的充要条件.应用泛函分析学报,2006,8(3):252-258
[28] Zhou Haiyun.Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractions in Banach spaces.Nonlinear Analysis,2008,68:2977-2983
[29]向长合.渐近伪压缩映象不动点的迭代逼近.西南师范大学学报,2007,32(5):6-9
[30]向长合.一致L-Lipschitz的渐近拟伪压缩型映象迭代收敛的充要条件.系统科学与数学,2008,28(4):447-455
[31] Chang Shih-sen,Cho Yeol Je,Jong Kyu kim.Some results for uniformly L-Lipschitzian ma -ppings in Banach spaces.Appl Math Lett(2008),doi:10.1016/j.aml.2008.02.16
[32] Chang S S,Tan K K,Lee H W J,et al.On the convergence of implicit iteration process wit -h error for a finite family of asymptotically nonexpansive mappings.J Math Anal Appl,2006,313:2 -283
[33] Chang S S, Lee H W J,Chi Kin Chan.On Reich’strong convergence theorem for asy-moptot -ically nonexpansive mappings in Banach spaces.Nonlinear Analysis(2006),doi:10.1016/j.na.2006.03.025
[34]姚永红,陈汝栋.渐近伪压缩和渐近非扩张映像不动点的迭代逼近问题.应用泛函分析学报,2007,9(3):233-237
[35]罗元松.一类迭代序列的收敛性.四川大学学报(自然科学版),2006,43(6):1187-1191
[36] GU Feng,DU Xiufeng.Iterative approximations of fixed points for asymoptotically pseudo-contractive mappings in normed linear spaces.应用泛函分析学报,2003,5(2):125-131
[37]谷峰.赋范空间中渐进一致φ-伪压缩型映象不动点的迭代逼近.数学的实践与认识, 2006,26(3):282-287
[38]唐玉超,刘理蔚.赋范线性空间中渐近伪压缩映象的不动点迭代逼近.应用数学学报,2007,5:810-815
[39] Soltuz S M.The convergence of Mann iteration with errors is equivalent to the convergence of Ishikawa iteration with errors[J].Lectures Mathematicas,2004,25:5-13
[40] Soltuz S M.The equivalence of Picard,Mann and Ishikawa iterations dealing with quasi-constractive operators[J].Math Comm,2005,(10):81-88
[41]苏巨国,唐玉超.Mann迭代和带误差的Ishikawa迭代的等价性.上饶师范学院学报,2005,25(6):18-21
[42]野金花,崔云安.一致伪压缩映射的具误差的Ishikawa和Mann迭代序列收敛性的等价性问题.黑龙江大学自然科学学报,2006,23(2):242-244
[43]薛志群,范瑞琴.关于非线性算子不动点收敛定理等价性的探讨.石家庄铁道学院学报,2007,20(4):54-56
[44]王亚宁.关于一类非线性映射迭代序列收敛的等价性.北华航天工业学院学报,2009,19(4):38-40
[45] Tan K K.Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process.J Math Anal Appl,1993,178:301-308
[46] Zeng L C.A note on approximating fixed points of nonexpansive mappings by the Ishikawa iteration process.J Math Anal Appl,1998,226:245-250
[47] Xu H K.Existence and convergence for fixed points of mapping of asymoptotically nonexp -ansive type.Nonlinear Anal TMA,1991,16(12):1139-1146
[48] Sun Z H.Strong convergence of an implicit iteration process for a finite family of asymoptotically quasi-nonexpansive mappings.J Math Anal Appl,2003,286:351-358
[49] Ofoedu E U.Strong convergence theorem for uniformly L-Lipschitzian asymptotically pseudocontractive mapping in real Banach space.J Math Anal Appl,2006,321: 722-728
[50] Morales C H,Jun J S.Convergence of paths for pseudocontractive mappings in Banach spa -ces.Proc Amer Math Soc,2000,128:3411-3419
[51] Chang S S.Some results for asymoptotically pseudo-contractive mappings and asymoptotic -ally nonexpansive mappings.Proc Amer Math Soc,2001,129(3):845-853
[52]野金花.不动点的若干性质[D]:[硕士学位论文].哈尔滨:哈尔滨理工大学,2006