两类非线性算子的迭代序列的收敛性
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摘要
非线性算子的不动点理论作为非线性泛函分析理论的重要组成部分,它与许多近代数学分支有着紧密的联系,并在处理这些分支中的一些问题中发挥着重要作用。在不动点问题研究的众多方向中,将各种非线性算子类型和构造的收敛的迭代序列应用于优化、控制和微分方程等方面成为研究的主流问题。
本文研究了非线性算子不动点的迭代逼近问题,全文主要包括以下三方面内容。
第一部分,介绍了非线性算子理论的产生背景以及迭代算法的发展情况,为本文的后续研究工作提供了正确方向。
第二部分,研究了p-几乎渐近非扩张型映象Ishikawa迭代序列和Mann迭代序列的收敛性。在一致凸Banach空间中映象是一致渐近正则和一致连续的条件下,应用粘性逼近方法得到了当系数序列满足一定条件时,具随机误差的修正的Ishikawa迭代和Mann迭代序列的强收敛性。
第三部分,引入一类广义p-渐近非扩张型映象,给出了具混合误差的Ishikawa迭代序列强收敛于广义p-渐近非扩张型映象的某一不动点的充要条件,并在实一致凸Banach空间框架下,采用粘性逼近方法得到了广义p-渐近非扩张型映象的两种迭代算法强收敛定理,所得结论推广和改进了张石生、冯先智、向长合等人的相应结果。
The fixed point theory of nonlinear operators, as an important part of nonlinear function analysis, has been closely connected with many branches of modern mathematics and plays an important part for solving problems in these branches. Among many directions of the fixed point researches, it has become the mainstream to realize the application of convergence iterative sequences and various nonlinear operators on optimization, control and derivative equation, etc.
This thesis, studying the problem on approximating to the fixed points of nonlinear operators, mainly contains three parts as following:
In first part we introduce the background of nonlinear operator theory and the development of iterative methods, both of which lead the later parts of the thesis to a right direction.
In second part we study the convergence of modified Ishikawa and Mann iterative sequence with random errors for p-almost asymptotically nonexpansive type of mapping. In uniformly convex Banach spaces, under the condition when Mapping is uniformly asymptotic regular and uniformly continuous, we apply viscosity approximation methods to obtain the convergence of modified Ishikawa and Mann iterative sequences with random errors when the sequence of positive numbers satisfies appropriate conditions.
In third part we bring in a generalized p-asymptotically quasi-nonexpansive type of mapping, gives some necessary and sufficient conditions for the modified Ishikawa iterative sequences with mixed errors to converge strongly to a fixed point of generalized p-asymptotically quasi-nonexpansive type of mapping, and with the application of viscosity approximation methods in uniformly convex Banach spaces, the strong convergence theory of two iterative methods of the generalized p-asymptotically quasi-non-expansive type of mapping are obtained. Our results extend and improve the corresponding results of Chang S S, Feng and Xiang.
本文研究了非线性算子不动点的迭代逼近问题,全文主要包括以下三方面内容。
第一部分,介绍了非线性算子理论的产生背景以及迭代算法的发展情况,为本文的后续研究工作提供了正确方向。
第二部分,研究了p-几乎渐近非扩张型映象Ishikawa迭代序列和Mann迭代序列的收敛性。在一致凸Banach空间中映象是一致渐近正则和一致连续的条件下,应用粘性逼近方法得到了当系数序列满足一定条件时,具随机误差的修正的Ishikawa迭代和Mann迭代序列的强收敛性。
第三部分,引入一类广义p-渐近非扩张型映象,给出了具混合误差的Ishikawa迭代序列强收敛于广义p-渐近非扩张型映象的某一不动点的充要条件,并在实一致凸Banach空间框架下,采用粘性逼近方法得到了广义p-渐近非扩张型映象的两种迭代算法强收敛定理,所得结论推广和改进了张石生、冯先智、向长合等人的相应结果。
The fixed point theory of nonlinear operators, as an important part of nonlinear function analysis, has been closely connected with many branches of modern mathematics and plays an important part for solving problems in these branches. Among many directions of the fixed point researches, it has become the mainstream to realize the application of convergence iterative sequences and various nonlinear operators on optimization, control and derivative equation, etc.
This thesis, studying the problem on approximating to the fixed points of nonlinear operators, mainly contains three parts as following:
In first part we introduce the background of nonlinear operator theory and the development of iterative methods, both of which lead the later parts of the thesis to a right direction.
In second part we study the convergence of modified Ishikawa and Mann iterative sequence with random errors for p-almost asymptotically nonexpansive type of mapping. In uniformly convex Banach spaces, under the condition when Mapping is uniformly asymptotic regular and uniformly continuous, we apply viscosity approximation methods to obtain the convergence of modified Ishikawa and Mann iterative sequences with random errors when the sequence of positive numbers satisfies appropriate conditions.
In third part we bring in a generalized p-asymptotically quasi-nonexpansive type of mapping, gives some necessary and sufficient conditions for the modified Ishikawa iterative sequences with mixed errors to converge strongly to a fixed point of generalized p-asymptotically quasi-nonexpansive type of mapping, and with the application of viscosity approximation methods in uniformly convex Banach spaces, the strong convergence theory of two iterative methods of the generalized p-asymptotically quasi-non-expansive type of mapping are obtained. Our results extend and improve the corresponding results of Chang S S, Feng and Xiang.
引文
[1] BANACH S. Sur les Operations Dans les Ensembles Abstraits er Leurs Appli- cations[J]. Fund. Math., 1922, (3): 133-181.
[2] BROUWER L E J. Uber Abbildungen von Mannigfatigkeiten[J]. Math. Ann., 1911, 71(1): 97-115.
[3] SCHAUDER J. Der Fixpunktsatz in Funktionalraumen[J]. Studia Math., 1930, 2(1): 17-22.
[4] GOEBEL K, KIRK A.Fixed Point Theorem for Asymptotically Nonexpansive Mappings[J]. J. Proc. Amer. Soc., 1972, 35(2): 171-174.
[5] XIANG C H.Approximation of Fixed Points of Generalized Asymptotically quasinonexpansive Type Mappings[J].Joumal of Chongqing Normal University. 2005, 4(22): 7-9.
[6] ZENG L C.Iterative Approximation of Fixed Points for Almost Asymptotically Nonexpansive Type Mappings in Banach Spaces[J].Applied Mathematics and Mechanics, 2003, 24(12): 1258-1266.
[7] KIRK W A, SIMS B. Handbook of Metric Fixed Point Theory[M]. Kluwer Academic Publishers, 2001: 1-200.
[8] SHIMIZU, TAKAHASHI W. Strong Convergence Theorem for Asymptotically Nonexpansive Mappings[J]. Nonlinear Anal. TMA., 1996, 26(2): 265-272.
[9] TAKAHASHI W. Viscosity Approximation Methods for Countable Families of Nonexpansive Mappings in Banach Spaces[J]. Nonlinear. Anal. TMA., 2009, 70(2): 719-734.
[10] TAKAHASHI W, TAMURAT. Convergence Theorem for a pair of Nonexpan- sive Mappings[J]. J. Convex Analysis., 1998, 5(1): 45-48.
[11] BOSE S C. Weak Convergence to a Fixed Point of an Asymptotically Nonexpansive Mappings[J]. Proc. Amer. Math. Soc., 1978, 68(1): 305-308.
[12] PASSTY G B Constuction of Fixed Points for Asymptotically Nonexpansive Mappings[J]. Proc. Amer. Math. Soc., 1982, 84(1): 213-216.
[13] CHIDUME C E. Strong and Weak Convergence Theorem for Asymptotically Nonexpansive Mappings[J]. J. Math. Anal. Appl., 2003, 280(1): 364-374.
[14] CHIDUME C E, ALI B. Approximation of Common Fixed Points for Finite families of nonself Asymptotically Nonexpansive Mappings in BanachSpaces[J]. J. Math. Anal. Appl., 2007, 326(1): 960-973.
[15] MOUDAFI A. Vscosity Approximation Methods for Fixed Point Problems[J]. J. Math. Anal. Appl., 2000, 241(1): 46-55.
[16] SUZUKI T. Strong Convergence of Krasnoselskii and Mann's Type Sequence for One-parameter Nonexpansive Semigroups without Bochner Integerals[J]. J. Math. Anal. Appl., 2005, 305(1): 227-239.
[17] ISA YILDIRIM A. New Iterative Process for Common Fixed Points of Finite Families of Nonself Asymptotically Nonexpansive Mappings[J]. Nonlinear Anal., 2009: 342-357.
[18] XU H K. Viscosity Approximation Methods for Expansive Mappings[J]. J. Math. Anal. Appl., 2004, 298(1): 279-291.
[19] MANN W R. Mean Value Methods in Iteration[J]. Proc. Amer. Math. Soc., 1953, 4(3): 506-510.
[20] ISHIKAWA S. Fixed Point by a New Iteration Method[J]. Proc. Amer. Math. Soc., 1974, 44(1): 147-150.
[21] BAILLON J B. Un Theorem de Type Ergodic Pour les Contractions Non-linear Dans un Espace de Hilbert[J]. C. R. Acad. Sci. Paris, 1975, 280(1): 1511-1514.
[22] REICH S. Some Problems and Results in Fixed Point Theorem[J]. Contem. Math., 1983, 210: 179-187.
[23] SCHU J.Iterative Construction of Fixed Points of Asymptotically Nonexpansive Mappings[J].J.Math.Anal.Appl., 1991, 158: 407-413.
[24] SCHU J. Weak and Strong Convergence of Fixed Points of Asymptotically Nonexpansive Mappings[J].Bull Austral Math.Soc., 1991, 43: 153-159.
[25] TAN K K, Xu H K. Fixed Point Iteration Processes for Asymptotically Nonexpansive Mappings[J]. Proc. Amer. Math. Soc., 1994, 122: 733-739.
[26]胡长松. Banach空间中渐近非扩张型映象的Ishikawa迭代序列的收敛问题[J].数学杂志, 2004, 6(24): 675-679.
[27]张石生. Banach空间中渐近非扩张型映象不动点的迭代逼近问题[J].应用数学和力学, 2001, 22(1): 23-31.
[28]熊明,王绍荣,杨泽恒. Banach空间中几乎渐近非扩张型映象不动点的迭代逼近问题[J].四川大学学报. 2007. 3(44): 485-489.
[29]王缨.渐近拟非扩张映象的带误差的Ishikawa迭代序列[J].西南师范大学学报. 2003, 1(28): 52-54.
[30]王绍荣,熊明.关于渐近拟非扩张型映象不动点的迭代逼近问题[J].纯粹数学与应用数学, 2004, 1(20): 18-23.
[31]王绍荣.渐近拟非扩展型映象不动点的具误差的Ishikawa迭代逼近[J].四川大学学报, 2004, 2(41): 231-235.
[32]冯先智.几乎渐近非扩张型映象不动点具随即误差的迭代逼近[J].宁夏大学学报, 2006, 27(1): 1-5.
[33]冯先智.渐近拟非扩展型映象的Ishikawa迭代序列的强收敛性[J].纯粹数学与应用数学, 2006, 4(22): 477-481.
[34]于延荣,宋义生.渐近非扩张映射不动点的存在与迭代收敛性[J].数学的实践与认识, 2007, 19(37): 144-149.
[35]邓传现,黄发伦.一类新的几乎渐近非扩张型映象不动点的三步迭代逼近问题[J].四川大学学报. 2007, 44(6):1153-1159.
[36]屈静国.几类映射的不动点迭代序列的若干收敛性质[D].中国优秀硕士学位论文全文数据库, 2008, (04).
[37]胡国英,梁天娟.广义渐近拟非扩张型映象不动点的逼近[J].重庆师范大学学报. 2008, 1(25): 10-13.
[38] SONG Y S, CHEN R D. Viscosity Approximation Methods for Nonexpansive Nonself-mappings[J]. J. Math. Anal. Appl., 2006, 321(1): 316-326.
[39] TAKAHASHI W. Nonlinear Functional Analysis Fixed Point Theory and its Applications, Yokohama Publishers,Yokohama, 2000: 243-256.
[40] WANG SHAORONG. Some New Strong Convergence Theorem for Ishikawa Iterative Sequence with Errors for Asymptotically Nonexpansive Mappings[J]. Acta Analysis Functionalis Appl, 2004, 6(2): 187-192.
[41] YAO Y H, CHEN R D. Approximating to Fixed Points of Asymptotically Nonexpansive Mappings in Uniformly Convex Banach spaces[J]. Pure and Applied Mathematics, 2006, 22(2): 264-267.
[42] ZHOU Y Y, CHANG S S. Convergence of Implicit Iterative Process for a Family of Asymptotically Nonexpansive Mappings in Banach Spaces[J]. Numer Funct Anal and Optimiz, 2002, 23: 911-921.
[2] BROUWER L E J. Uber Abbildungen von Mannigfatigkeiten[J]. Math. Ann., 1911, 71(1): 97-115.
[3] SCHAUDER J. Der Fixpunktsatz in Funktionalraumen[J]. Studia Math., 1930, 2(1): 17-22.
[4] GOEBEL K, KIRK A.Fixed Point Theorem for Asymptotically Nonexpansive Mappings[J]. J. Proc. Amer. Soc., 1972, 35(2): 171-174.
[5] XIANG C H.Approximation of Fixed Points of Generalized Asymptotically quasinonexpansive Type Mappings[J].Joumal of Chongqing Normal University. 2005, 4(22): 7-9.
[6] ZENG L C.Iterative Approximation of Fixed Points for Almost Asymptotically Nonexpansive Type Mappings in Banach Spaces[J].Applied Mathematics and Mechanics, 2003, 24(12): 1258-1266.
[7] KIRK W A, SIMS B. Handbook of Metric Fixed Point Theory[M]. Kluwer Academic Publishers, 2001: 1-200.
[8] SHIMIZU, TAKAHASHI W. Strong Convergence Theorem for Asymptotically Nonexpansive Mappings[J]. Nonlinear Anal. TMA., 1996, 26(2): 265-272.
[9] TAKAHASHI W. Viscosity Approximation Methods for Countable Families of Nonexpansive Mappings in Banach Spaces[J]. Nonlinear. Anal. TMA., 2009, 70(2): 719-734.
[10] TAKAHASHI W, TAMURAT. Convergence Theorem for a pair of Nonexpan- sive Mappings[J]. J. Convex Analysis., 1998, 5(1): 45-48.
[11] BOSE S C. Weak Convergence to a Fixed Point of an Asymptotically Nonexpansive Mappings[J]. Proc. Amer. Math. Soc., 1978, 68(1): 305-308.
[12] PASSTY G B Constuction of Fixed Points for Asymptotically Nonexpansive Mappings[J]. Proc. Amer. Math. Soc., 1982, 84(1): 213-216.
[13] CHIDUME C E. Strong and Weak Convergence Theorem for Asymptotically Nonexpansive Mappings[J]. J. Math. Anal. Appl., 2003, 280(1): 364-374.
[14] CHIDUME C E, ALI B. Approximation of Common Fixed Points for Finite families of nonself Asymptotically Nonexpansive Mappings in BanachSpaces[J]. J. Math. Anal. Appl., 2007, 326(1): 960-973.
[15] MOUDAFI A. Vscosity Approximation Methods for Fixed Point Problems[J]. J. Math. Anal. Appl., 2000, 241(1): 46-55.
[16] SUZUKI T. Strong Convergence of Krasnoselskii and Mann's Type Sequence for One-parameter Nonexpansive Semigroups without Bochner Integerals[J]. J. Math. Anal. Appl., 2005, 305(1): 227-239.
[17] ISA YILDIRIM A. New Iterative Process for Common Fixed Points of Finite Families of Nonself Asymptotically Nonexpansive Mappings[J]. Nonlinear Anal., 2009: 342-357.
[18] XU H K. Viscosity Approximation Methods for Expansive Mappings[J]. J. Math. Anal. Appl., 2004, 298(1): 279-291.
[19] MANN W R. Mean Value Methods in Iteration[J]. Proc. Amer. Math. Soc., 1953, 4(3): 506-510.
[20] ISHIKAWA S. Fixed Point by a New Iteration Method[J]. Proc. Amer. Math. Soc., 1974, 44(1): 147-150.
[21] BAILLON J B. Un Theorem de Type Ergodic Pour les Contractions Non-linear Dans un Espace de Hilbert[J]. C. R. Acad. Sci. Paris, 1975, 280(1): 1511-1514.
[22] REICH S. Some Problems and Results in Fixed Point Theorem[J]. Contem. Math., 1983, 210: 179-187.
[23] SCHU J.Iterative Construction of Fixed Points of Asymptotically Nonexpansive Mappings[J].J.Math.Anal.Appl., 1991, 158: 407-413.
[24] SCHU J. Weak and Strong Convergence of Fixed Points of Asymptotically Nonexpansive Mappings[J].Bull Austral Math.Soc., 1991, 43: 153-159.
[25] TAN K K, Xu H K. Fixed Point Iteration Processes for Asymptotically Nonexpansive Mappings[J]. Proc. Amer. Math. Soc., 1994, 122: 733-739.
[26]胡长松. Banach空间中渐近非扩张型映象的Ishikawa迭代序列的收敛问题[J].数学杂志, 2004, 6(24): 675-679.
[27]张石生. Banach空间中渐近非扩张型映象不动点的迭代逼近问题[J].应用数学和力学, 2001, 22(1): 23-31.
[28]熊明,王绍荣,杨泽恒. Banach空间中几乎渐近非扩张型映象不动点的迭代逼近问题[J].四川大学学报. 2007. 3(44): 485-489.
[29]王缨.渐近拟非扩张映象的带误差的Ishikawa迭代序列[J].西南师范大学学报. 2003, 1(28): 52-54.
[30]王绍荣,熊明.关于渐近拟非扩张型映象不动点的迭代逼近问题[J].纯粹数学与应用数学, 2004, 1(20): 18-23.
[31]王绍荣.渐近拟非扩展型映象不动点的具误差的Ishikawa迭代逼近[J].四川大学学报, 2004, 2(41): 231-235.
[32]冯先智.几乎渐近非扩张型映象不动点具随即误差的迭代逼近[J].宁夏大学学报, 2006, 27(1): 1-5.
[33]冯先智.渐近拟非扩展型映象的Ishikawa迭代序列的强收敛性[J].纯粹数学与应用数学, 2006, 4(22): 477-481.
[34]于延荣,宋义生.渐近非扩张映射不动点的存在与迭代收敛性[J].数学的实践与认识, 2007, 19(37): 144-149.
[35]邓传现,黄发伦.一类新的几乎渐近非扩张型映象不动点的三步迭代逼近问题[J].四川大学学报. 2007, 44(6):1153-1159.
[36]屈静国.几类映射的不动点迭代序列的若干收敛性质[D].中国优秀硕士学位论文全文数据库, 2008, (04).
[37]胡国英,梁天娟.广义渐近拟非扩张型映象不动点的逼近[J].重庆师范大学学报. 2008, 1(25): 10-13.
[38] SONG Y S, CHEN R D. Viscosity Approximation Methods for Nonexpansive Nonself-mappings[J]. J. Math. Anal. Appl., 2006, 321(1): 316-326.
[39] TAKAHASHI W. Nonlinear Functional Analysis Fixed Point Theory and its Applications, Yokohama Publishers,Yokohama, 2000: 243-256.
[40] WANG SHAORONG. Some New Strong Convergence Theorem for Ishikawa Iterative Sequence with Errors for Asymptotically Nonexpansive Mappings[J]. Acta Analysis Functionalis Appl, 2004, 6(2): 187-192.
[41] YAO Y H, CHEN R D. Approximating to Fixed Points of Asymptotically Nonexpansive Mappings in Uniformly Convex Banach spaces[J]. Pure and Applied Mathematics, 2006, 22(2): 264-267.
[42] ZHOU Y Y, CHANG S S. Convergence of Implicit Iterative Process for a Family of Asymptotically Nonexpansive Mappings in Banach Spaces[J]. Numer Funct Anal and Optimiz, 2002, 23: 911-921.