几类映射的不动点迭代序列的若干收敛性质
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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其在控制、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在实际运用中起到至关重要的作用。本文主要研究了Ishikawa迭代序列、Mann迭代序列和三步迭代序列,以及介绍了带误差的Ishikawa迭代序列、Mann迭代序列和三步迭代序列的收敛性方面的若干性质,及其在几类映射下的具体结论。
本文主要包括以下三方面内容:第一部分是( L ?α)一致Lipschitz渐近非扩张映射的不动点迭代问题;第二部分是渐近非扩张型映射的不动点迭代问题;第三部分是严格伪压缩映射的不动点迭代逼近问题。
首先,讨论了( L ?α)一致Lipschitz渐近非扩张映射的不动点迭代问题。给出了此映射带误差的Ishikawa迭代序列、Mann迭代序列和三步迭代序列的收敛性条件,并给出了严格证明。用新方法研究了Banach空间中( L ?α)一致Lipschitz渐近非扩张映射不动点的迭代逼近问题,去掉了定义域和值域的有界性假设。
其次,研究了渐近非扩张型映射带误差的三步迭代序列的收敛性问题,在一致凸的Banach空间中给出了带误差的三步迭代序列逼近渐近非扩张型映射不动点的强收敛定理,并在一致渐进正则条件下证明了三步迭代序列强收敛其不动点。其结果把Ishikawa迭代序列和Mann迭代序列推广到三步迭代序列上,改进了一些相关结果。
最后,主要讨论了严格伪压缩映射的不动点迭代逼近问题,给出了在Hilbert空间中的非空有界闭凸集上Lipschitz严格伪压缩映射的带误差的Ishikawa和Mann迭代序列的一些收敛性定理。其结果把非扩张映射推广到Lipschitz严格伪压缩映射上,改进了一些相关结果。
The problem of fixed point is one of the more important problems that people 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 making 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. In this paper, the Ishikawa, Mann and three-step iterative sequences are discussed and some properties of the convergence of the Ishikawa, Mann and three-step iterative processes with errors and their detail conclusion under some mappings are introduced.
The thesis mainly consists of three parts. the first is the convergence problems of fixed point about uniformly ( L-α)- Lipschitz asymptotically nonexpansive mapping; the second is the convergence problem of uniformly L-Lipschitz asymptotically nonexpansive type mappings; the last is the converg-ence theorems of a strictly pseudo-contractive map.
We first discuss the convergence problems of fixed point about uniformly ( L -α)-Lipschitz asymptotically nonexpansive mapping. The results are given by the convergence problem of Ishikawa, Mann and three-step iterative processes with errors, and we prove it. By using a new method, the iterative approximation problems of fixed points for uniformly ( L -α)- Lipschitzasymptotically nonexpansive mapping in Banach spaces were studied, and the bounded assumption of domain and range was dropped.
Second, we study the convergence problem of three-step iterative sequence approximation to fixed points of uniformly L-Lipschitz asymptotically nonexpansive type mappings with error in uniformly convex Banach space, and we prove the convergence theorems of three-step iterative sequence with error approximation asymptotically nonexpansive type mapping, The fixed point properties of Ishikawa and Mann iterative sequences are extended to three-step iterative sequence. and we improve some recent corresponding results.
At last, we get some convergence theorems for Ishikawa and Mann iterative sequences with errors of a strictly pseudo-contractive map in a nonempty bounded closed convex subset of a Hilbert space. The fixed point properties of nonexpansive mapping are extended to strictly pseudo-contractive map. The results improved and extended some recently results.
本文主要包括以下三方面内容:第一部分是( L ?α)一致Lipschitz渐近非扩张映射的不动点迭代问题;第二部分是渐近非扩张型映射的不动点迭代问题;第三部分是严格伪压缩映射的不动点迭代逼近问题。
首先,讨论了( L ?α)一致Lipschitz渐近非扩张映射的不动点迭代问题。给出了此映射带误差的Ishikawa迭代序列、Mann迭代序列和三步迭代序列的收敛性条件,并给出了严格证明。用新方法研究了Banach空间中( L ?α)一致Lipschitz渐近非扩张映射不动点的迭代逼近问题,去掉了定义域和值域的有界性假设。
其次,研究了渐近非扩张型映射带误差的三步迭代序列的收敛性问题,在一致凸的Banach空间中给出了带误差的三步迭代序列逼近渐近非扩张型映射不动点的强收敛定理,并在一致渐进正则条件下证明了三步迭代序列强收敛其不动点。其结果把Ishikawa迭代序列和Mann迭代序列推广到三步迭代序列上,改进了一些相关结果。
最后,主要讨论了严格伪压缩映射的不动点迭代逼近问题,给出了在Hilbert空间中的非空有界闭凸集上Lipschitz严格伪压缩映射的带误差的Ishikawa和Mann迭代序列的一些收敛性定理。其结果把非扩张映射推广到Lipschitz严格伪压缩映射上,改进了一些相关结果。
The problem of fixed point is one of the more important problems that people 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 making 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. In this paper, the Ishikawa, Mann and three-step iterative sequences are discussed and some properties of the convergence of the Ishikawa, Mann and three-step iterative processes with errors and their detail conclusion under some mappings are introduced.
The thesis mainly consists of three parts. the first is the convergence problems of fixed point about uniformly ( L-α)- Lipschitz asymptotically nonexpansive mapping; the second is the convergence problem of uniformly L-Lipschitz asymptotically nonexpansive type mappings; the last is the converg-ence theorems of a strictly pseudo-contractive map.
We first discuss the convergence problems of fixed point about uniformly ( L -α)-Lipschitz asymptotically nonexpansive mapping. The results are given by the convergence problem of Ishikawa, Mann and three-step iterative processes with errors, and we prove it. By using a new method, the iterative approximation problems of fixed points for uniformly ( L -α)- Lipschitzasymptotically nonexpansive mapping in Banach spaces were studied, and the bounded assumption of domain and range was dropped.
Second, we study the convergence problem of three-step iterative sequence approximation to fixed points of uniformly L-Lipschitz asymptotically nonexpansive type mappings with error in uniformly convex Banach space, and we prove the convergence theorems of three-step iterative sequence with error approximation asymptotically nonexpansive type mapping, The fixed point properties of Ishikawa and Mann iterative sequences are extended to three-step iterative sequence. and we improve some recent corresponding results.
At last, we get some convergence theorems for Ishikawa and Mann iterative sequences with errors of a strictly pseudo-contractive map in a nonempty bounded closed convex subset of a Hilbert space. The fixed point properties of nonexpansive mapping are extended to strictly pseudo-contractive map. The results improved and extended some recently results.
引文
[1] SENTER H F, Dotson W G JR. Approximating Fixed Points of Nonexpansive Mappings[J]. Proc. Amer. Math. Soc., 1974, 44: 375-380.
[2] OPIAL Z. Weak Convergence of Successive Approximations for Nonexpansive Mappings[J]. Roll Amer. Math. Soc., 1967, 73: 591-597.
[3]贾如鹏.一致凸Banach空间非扩张映射带误差Ishikawa迭代[J].数学实践与认识,2004,43( 8):158- 161.
[4]邓磊,李胜宏.一致凸Banach空间中非扩张映射的Ishikawa迭代[J].数学年刊,2000,21A(2):159—164.
[5] HALPEM B. Fixed Points of Nonexpanding Maps[J]. Bull Amer. Math Soc., 1967, 73: 957-961.
[6]李志龙,孙经先.有关Amann三解定理的注记[J].山东大学学报(理学版),2003,38(1):20-22.
[7]孙昭洪,何昌.渐近非扩张型映象不动点带误差的迭代逼近[J].数学学报,2004,47(4):811-818.
[8]刘丽莉,师涌江,刘桂霞.Banach空间多值φ-强伪压缩映象不动点的迭代逼近[J].河南师范大学学报(自然科学版),2006,30(1):10-13.
[9]张卷美.一类不动点迭代法的求解[J].河南理工大学学报,2006,25(2):10-13.
[10] GOEBEL K, KIRK W A. A Fixed Point Theorem for Asymptotically Non-expansive Mappings[J]. Proc. Amer. Math. Soc., 1972, 35: 171-174.
[11] SCHU J. Iterative Construction of Fixed Points of Asymptotically Nonexpan-sive Mappings[J]. J. Math. Anal. Appl., 1991, 158: 407-413.
[12] SCHU J. Weak and Strong Convergence of Fixed Points of Asymptotically Nonexpansive Mappings[J]. Bull Austral Math. Soc., 1991, 43: 153-159.
[13] TAN K K, Xu H K. Fixed Point Iteration Processes for Asymptotically Non-expansive Mappings[J]. Proc. Amer. Math. Soc., 1994, 122: 733-739.
[14] XU H K. Inequalities in Banach Spaces with Applications[J]. Nonlinear Anal., 1991, 16: 1127-1138.
[15] RHOADES B E. Fixed Point Iterations for Certain Nonlinear Mappings[J]. J. Math. Anal. Appl., 1994, 183: 118-120.
[16] TAN K K, XU H K. The Nonlinear Ergodic Theorem for Asymptotically Non-expansive Mappings in Banach Spaces[J]. Proc. Amer. Math. Soc., 1992, 114: 399-404.
[17] TAKAHASHI W, KIM G E. Approximating Fixed Points of Nonexpansive Mappings in Banach Spaces[J]. Math. Japonica, 1998, 48: 1-9.
[18]曾六川.逼近Banach空间中渐近非扩张映射的不动点[J].数学物理学报,2003,23:31-37.
[19] XU H K. Existence and Convergence for Fixed Points of Mappings of Asymp-totically Nonexpansive Type[J], Nonlinear Anal. TMA, 1991, 18(12): 1139-1146.
[20] BOSE S C. Weak Convergence to the Fixed Point of an Asymptotically Non-expansive Map[J], Proc. Amer. Math. Soc., 1978, 88(3): 305-308.
[21] ZENG L C. A Note on Approximating Fixed Points of Nonexpansive Mapping by the Ishikawa Iterative Processes[J]. J. Math. Anal. Appl., 1998, 226: 245-250.
[22] NOOR M A. Three-step Iterative Algorithms for Multivalued Quasi Varitional Inclusions[J]. J Math Anal. Appl., 2001, 225: 589-604.
[23] NOOR M A. Some Preditor-corrector Algorithms for Multivalude Variational Inequalities[J]. J Optim. Theory Appl., 2001, 108(3): 659-670.
[24] NOOR M A. Three-step Iterations for Nonlinear Accretive Operator Equa-tions[J]. Math Anal Appl., 2002, 274: 59-68.
[25] NOOR M A. New Approximation Schemes for General Variational Inequalities, [J]. J. Math. Anal. App1., 2000, 251: 217-229.
[26] GLOWINSKI R, TALLEC P LE. Augemented Lagrangian and Operator-splitting Methods in Nonlinear Mechanics[M]. SM: Philadelphia, 1989: 20-40.
[27] HAUBRUGE S, NGUYEN V H, STRODIOT J J. Convergence Analysis and Applications of the Glowinslci-LeTallec Splitting Method for Finding a Zero of the Sum of Two Maximal Monotone. Operators[J]. J. Optim. Theory App1., 1998, 97: 45-673.
[28]张石生.Banach空间中渐近非扩张映象不动点的迭代逼近问题[J].应用数学学报,2001,24(2):236-241.
[29]曾六川.关于Banach空间中Lipschitz映象对的公共不动点的存在性[J].应用数学和力学,2003,24(3):305-314.
[30]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[J].数学年刊,2005,26A(2):283-290.
[31] LIU Q H. Iterative Sequences for Asymptotically Quasi-nonexpansive Mapping with Error Members[J]. J. Math. Anal. Appl., 2001, 259: 18-24.
[32] ZHAO H S. Iterative Approximation of Fixed Points for Asymptotically Non-expansive Type Mapping with Error Member[J]. ACTC math 2004, 47(4): 811-818.
[33] CHANG S S. Approximation of Fixed Point of Nonexpansive Mappings[J]. Arch Math, 1992, 58: 486- 491.
[34] XU H K. Inequalities in Banach Spaces with Applications[J]. nonlinear Anal TMA, 1991, 16(2): 1127-1137.
[35] OSILIKE M O. Implicit Iteration Process for Common Fixed Point sofa Finite Family of Strictly Pseudo-contractive Maps[J]. J Math Anal. Appl. 2004, 294(1): 73-81
[36] OSILIKE M O. Udomene A Demiclosed Principle and Convergence Results for Strictly Pseudo-contractive Mappings of Brower-Petryshyn Type[J]. J. Math Anal Appl., 2001, 256(2): 431-445.
[37] REICH S. Weak Convergence Theorems for Nonexpansive Mapping in Banach Spaces[J]. Math Anal Appl., 1979, 67(2): 274-281.
[38] GENEL A, LINDENSTRASS J. An Example Concerning Fixed Points[J]. Israel J Math, 1975, 22(1): 81-86.
[39] CHANG S S. Some Problems and Results in the Study of Nonlinear Ana-lysis[J ], Nonlinear Anal., 1997, 30: 4197-4208.
[40] CHANG S S, Cho, Y J Zhou. Demi-closed Principle and Weak Convergence Problems for Asymptotically Nonexpansive Mappings[J]. Koeran Math. Soc., 1989, 38(6): 1245-1260.
[41]刘丽莉,师涌江,刘桂霞.Banach空间多值φ-强伪压缩映象不动点的迭代逼近[J].河南师范大学学报(自然科学版),2006,30(1):10-13.
[42]王彬,徐家斌.Browder不动点定理的推广[J].内江师范学院学报,2006,21(2):20-21.
[43]黄小平,李雪松.一致拟Lipschitzian映象的迭代逼近[J].电子科技大学学报,2006,35(4):10-13.
[44]丁争尚.Banach空间不放大映射的迭代法[J].商洛学院学报,2006,200(4):21-22.
[45]许静波.Banach空间中的不动点[J].吉林师范大学学报(自然科学版),2006,30(1):24-26.
[46]廖正琦.Hilbert空间集值非扩张映象的耦合不动点[J].重庆交通学院学报,2005,24(3):161-165.
[47]刘桂霞,姚力.Banach空间中两类压缩映射的迭代逼近[J].大学数学,2006,22(6):48-52.
[48]冯先智.几乎渐近非扩张型映象不动点具随机误差的迭代逼近[J].宁夏大学学报(自然科学版),2006,27(1):1-5.
[2] OPIAL Z. Weak Convergence of Successive Approximations for Nonexpansive Mappings[J]. Roll Amer. Math. Soc., 1967, 73: 591-597.
[3]贾如鹏.一致凸Banach空间非扩张映射带误差Ishikawa迭代[J].数学实践与认识,2004,43( 8):158- 161.
[4]邓磊,李胜宏.一致凸Banach空间中非扩张映射的Ishikawa迭代[J].数学年刊,2000,21A(2):159—164.
[5] HALPEM B. Fixed Points of Nonexpanding Maps[J]. Bull Amer. Math Soc., 1967, 73: 957-961.
[6]李志龙,孙经先.有关Amann三解定理的注记[J].山东大学学报(理学版),2003,38(1):20-22.
[7]孙昭洪,何昌.渐近非扩张型映象不动点带误差的迭代逼近[J].数学学报,2004,47(4):811-818.
[8]刘丽莉,师涌江,刘桂霞.Banach空间多值φ-强伪压缩映象不动点的迭代逼近[J].河南师范大学学报(自然科学版),2006,30(1):10-13.
[9]张卷美.一类不动点迭代法的求解[J].河南理工大学学报,2006,25(2):10-13.
[10] GOEBEL K, KIRK W A. A Fixed Point Theorem for Asymptotically Non-expansive Mappings[J]. Proc. Amer. Math. Soc., 1972, 35: 171-174.
[11] SCHU J. Iterative Construction of Fixed Points of Asymptotically Nonexpan-sive Mappings[J]. J. Math. Anal. Appl., 1991, 158: 407-413.
[12] SCHU J. Weak and Strong Convergence of Fixed Points of Asymptotically Nonexpansive Mappings[J]. Bull Austral Math. Soc., 1991, 43: 153-159.
[13] TAN K K, Xu H K. Fixed Point Iteration Processes for Asymptotically Non-expansive Mappings[J]. Proc. Amer. Math. Soc., 1994, 122: 733-739.
[14] XU H K. Inequalities in Banach Spaces with Applications[J]. Nonlinear Anal., 1991, 16: 1127-1138.
[15] RHOADES B E. Fixed Point Iterations for Certain Nonlinear Mappings[J]. J. Math. Anal. Appl., 1994, 183: 118-120.
[16] TAN K K, XU H K. The Nonlinear Ergodic Theorem for Asymptotically Non-expansive Mappings in Banach Spaces[J]. Proc. Amer. Math. Soc., 1992, 114: 399-404.
[17] TAKAHASHI W, KIM G E. Approximating Fixed Points of Nonexpansive Mappings in Banach Spaces[J]. Math. Japonica, 1998, 48: 1-9.
[18]曾六川.逼近Banach空间中渐近非扩张映射的不动点[J].数学物理学报,2003,23:31-37.
[19] XU H K. Existence and Convergence for Fixed Points of Mappings of Asymp-totically Nonexpansive Type[J], Nonlinear Anal. TMA, 1991, 18(12): 1139-1146.
[20] BOSE S C. Weak Convergence to the Fixed Point of an Asymptotically Non-expansive Map[J], Proc. Amer. Math. Soc., 1978, 88(3): 305-308.
[21] ZENG L C. A Note on Approximating Fixed Points of Nonexpansive Mapping by the Ishikawa Iterative Processes[J]. J. Math. Anal. Appl., 1998, 226: 245-250.
[22] NOOR M A. Three-step Iterative Algorithms for Multivalued Quasi Varitional Inclusions[J]. J Math Anal. Appl., 2001, 225: 589-604.
[23] NOOR M A. Some Preditor-corrector Algorithms for Multivalude Variational Inequalities[J]. J Optim. Theory Appl., 2001, 108(3): 659-670.
[24] NOOR M A. Three-step Iterations for Nonlinear Accretive Operator Equa-tions[J]. Math Anal Appl., 2002, 274: 59-68.
[25] NOOR M A. New Approximation Schemes for General Variational Inequalities, [J]. J. Math. Anal. App1., 2000, 251: 217-229.
[26] GLOWINSKI R, TALLEC P LE. Augemented Lagrangian and Operator-splitting Methods in Nonlinear Mechanics[M]. SM: Philadelphia, 1989: 20-40.
[27] HAUBRUGE S, NGUYEN V H, STRODIOT J J. Convergence Analysis and Applications of the Glowinslci-LeTallec Splitting Method for Finding a Zero of the Sum of Two Maximal Monotone. Operators[J]. J. Optim. Theory App1., 1998, 97: 45-673.
[28]张石生.Banach空间中渐近非扩张映象不动点的迭代逼近问题[J].应用数学学报,2001,24(2):236-241.
[29]曾六川.关于Banach空间中Lipschitz映象对的公共不动点的存在性[J].应用数学和力学,2003,24(3):305-314.
[30]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[J].数学年刊,2005,26A(2):283-290.
[31] LIU Q H. Iterative Sequences for Asymptotically Quasi-nonexpansive Mapping with Error Members[J]. J. Math. Anal. Appl., 2001, 259: 18-24.
[32] ZHAO H S. Iterative Approximation of Fixed Points for Asymptotically Non-expansive Type Mapping with Error Member[J]. ACTC math 2004, 47(4): 811-818.
[33] CHANG S S. Approximation of Fixed Point of Nonexpansive Mappings[J]. Arch Math, 1992, 58: 486- 491.
[34] XU H K. Inequalities in Banach Spaces with Applications[J]. nonlinear Anal TMA, 1991, 16(2): 1127-1137.
[35] OSILIKE M O. Implicit Iteration Process for Common Fixed Point sofa Finite Family of Strictly Pseudo-contractive Maps[J]. J Math Anal. Appl. 2004, 294(1): 73-81
[36] OSILIKE M O. Udomene A Demiclosed Principle and Convergence Results for Strictly Pseudo-contractive Mappings of Brower-Petryshyn Type[J]. J. Math Anal Appl., 2001, 256(2): 431-445.
[37] REICH S. Weak Convergence Theorems for Nonexpansive Mapping in Banach Spaces[J]. Math Anal Appl., 1979, 67(2): 274-281.
[38] GENEL A, LINDENSTRASS J. An Example Concerning Fixed Points[J]. Israel J Math, 1975, 22(1): 81-86.
[39] CHANG S S. Some Problems and Results in the Study of Nonlinear Ana-lysis[J ], Nonlinear Anal., 1997, 30: 4197-4208.
[40] CHANG S S, Cho, Y J Zhou. Demi-closed Principle and Weak Convergence Problems for Asymptotically Nonexpansive Mappings[J]. Koeran Math. Soc., 1989, 38(6): 1245-1260.
[41]刘丽莉,师涌江,刘桂霞.Banach空间多值φ-强伪压缩映象不动点的迭代逼近[J].河南师范大学学报(自然科学版),2006,30(1):10-13.
[42]王彬,徐家斌.Browder不动点定理的推广[J].内江师范学院学报,2006,21(2):20-21.
[43]黄小平,李雪松.一致拟Lipschitzian映象的迭代逼近[J].电子科技大学学报,2006,35(4):10-13.
[44]丁争尚.Banach空间不放大映射的迭代法[J].商洛学院学报,2006,200(4):21-22.
[45]许静波.Banach空间中的不动点[J].吉林师范大学学报(自然科学版),2006,30(1):24-26.
[46]廖正琦.Hilbert空间集值非扩张映象的耦合不动点[J].重庆交通学院学报,2005,24(3):161-165.
[47]刘桂霞,姚力.Banach空间中两类压缩映射的迭代逼近[J].大学数学,2006,22(6):48-52.
[48]冯先智.几乎渐近非扩张型映象不动点具随机误差的迭代逼近[J].宁夏大学学报(自然科学版),2006,27(1):1-5.