摘要
非自映射不动点的迭代逼近问题已成为近年来学术界研究的活跃课题。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,并在实际运用中起到至关重要的作用。本文主要研究了Banach空间上的几类非扩张映射下迭代序列的收敛性问题。
首先,我们讲述了迭代序列的发展概况。通过引用大量前人的定义和定理,使我们对迭代序列的发展史有了一定程度的认识。同时,对于不动点的发展史和Banach空间相关知识也有一定程度的了解。
其次,我们主要研究Hilbert空间中均衡问题和不动点问题的迭代解。我们先提出均衡问题,给出与定理相关的定义,同时给出Hilbert空间的一些特性。接着给出一个关于均衡问题的迭代,讨论了Hilbert空间中在严格伪压缩映射与渐近伪压缩映射下的该迭代的弱收敛性和强收敛性问题。
最后,在Banach空间中,我们讨论了一类渐近非扩张? -伪压缩映射下的三重迭代序列的收敛性和Lipschitz映射下三种迭代(改进的Mann迭代,改进的Ishikawa迭代和改进的三重迭代)收敛性的等价性;然后讨论具有一致Ga?teaux可微范数的一致凸Banach空间中迭代序列的强收敛问题。
In recent years, the problem of iterative approximation of fixed point for non-self mappings becomes a very popular subject. Among many directions of the fixed point researches, it becomes a main problem that the convergence of making approximating fixed point sequence and its application in control, nonlinear operator and derivative equation etc. Its research will play an important role in its application in reality. In this paper, we mainly studied fixed point iterative sequences of some kinds of non-self mappings in Banach spaces.
First, the history of iterative scheme is represented. Lots of theories and definitions are quoted. We can know something about the iterative scheme development. As the same time, we show the fixed point development and the theorems related with Banach spaces.
Second, it is primarily studied the equilibrium problem and iterative scheme about the fixed point in Hilbert spaces. In this paper, we introduce an equilibrium problem and definitions related with this paper firstly. In addition, it is proved some qualities in Hilbert spaces. Secondly, we propose an iterative scheme of an equilibrium problem and establish some weak and strong convergence theorems of the sequences generated by our proposed scheme of a strict pseudo-contraction mapping and an asymptotically pseudo-contraction mapping in Hilbert spaces.
Finally, we discuss convergence problem of three step iterations for certain asymptotically nonexpansive ? -pseudo-contractive mapping in a real Banach space. Then it is provided that the convergences of three kinds of iterations for a Lipschitz operator are equivalent. Finally, the strong convergence of the scheme to a fixed point is shown in a Banach space with uniformly Ga?teaux differenti-able norm.
引文
[1] KIRK W, SIMS B. Handbook of Metric Fixed Point Theory[M]. Boston: Kluwer Academic Publishers, 2001: 1-200.
[2] MANN W R. Mean Value Method in Iteration[J]. Proc. Amer. Math. Soc., 1953, 4(6): 506-510.
[3] ISHIKAWA S. Fixed Points by a New Iterations Method[J]. Proc. Amer. Math. Soc., 1974, 44(1): 147-150.
[4] LIU L S. Ishikawa and Mann Iterative Process with Errors for Nonlinear Str-ongly Accretive Mappings in Banach Spaces[J]. J. Math. Anal. Appl., 1995, 194(1): 114-125.
[5] CHIDUME C E. Global Iteration Schemes for Strongly Pseudo-contractive Maps[J]. Proc. Amer. Math. Soc., 1998, 126(9): 2641-2649.
[6] CHIDUME C E, MOORE C. Fixed Point Iteration for Pseudo-contractive Maps[J]. Proc. Amer. Math. Soc., 1999, 127(4): 1163-1170.
[7] ZHANG S S. Some Results for Asymptoticallu Pseudo-contractive Mappings and Asymptoticallu Nonexpansive Mappings[J]. Amer. Math. Soc., 2000, 129(3): 845-853.
[8] ZHANG S S, GU F, ZHANG X L. Convergence Problem of Iterative Appro-ximation for Pseudo-contractive Type Mappings[J]. Journal of Sichuan Uni-versity, 2000, 6: 795-802.
[9] ZHANG S S. Some Convergence Problem of Iterative Sequences for Accre-tive and Pseudo-contractive Type Mappings in Banach Spaces[J]. Applied Mathematics and Mechanics, 2002, (23): 359-370.
[10] ZAMFIRESCU T. Fix Point Theorems in Metric Spaces[J]. Arch. Math., 1972, 23 (1): 292-298.
[11] RHOADES B E. Fixed Point Iterations Using Infinite Matrices[J]. Trans. Amer. Math. Soc., 1974, 196: 161-176.
[12] BERINDE V. On the Convergence of the Ishikawa Iteration in the Class of Quasi-contractive Operators[J]. Acta Mathematica Universitatis Commenianae, 2004, 1: 119-126.
[13] RAFIQ A. A Convergence Theorem for Mann Fixed Point Iteration Proce-dure[J]. Applied Mathemaics E-notes, 2006, 6: 289-293.
[14] AL-THAGAFI M A, NASEER SHAHZAD. Generalized I-nonexpansive Selfmaps and Invariant Approximations[J]. Acta Mathematica Sinica, 2008, 24(5): 867-876.
[15] HUANG JIANFENG, WANG YUANHENG. Strong Convergence of Appro-ximated Sequences for Nonexpansive Mappings in Banach Space[J]. Appl. Math. J. Chinese Univ. Ser., 2007, 22(3): 311-315.
[16] XIAOLONG QIN, YONGFU SU, MEIJUAN SHANG. Approximating Com-mon Fixed Points of Asymptotically Nonexpansive Mappings by Composite Algorithm in Banach Spaces[J]. Central European Journal of Mathematics, 2007, 5(2): 345-357.
[17] NADEZHKINA N, TAKAHASHI W. Weak Convergence Theorem by an Extragradient Method for Nonexpansive Mappings and Monotone Map-pings[J]. Journal of Optimization Theory and Applications, 2006, 128(1): 191-201.
[18] TAKANORI IBARAKI, WATARU TAKAHASHI. Weak and Strong Conver-gence Theorems for New Resolvents of Maximal Monotone Operators in Banach Spaces[J]. Adv. Math. Econ, 2007, 10: 51-67.
[19] TAKAHASHI W, TOYODA M. Weak Convergence Theorems for Nonex-pansive Mappings and Monotone Mappings[J]. Journal of Optimization Theory and Applications, 2003, 118(2): 417-428.
[20] FUMIAKI KOHSAKA, WATARU TAKAHASHI. Fixed Point Theorems for a Class of Nonlinear Mappings Related to Maximal Monotone Operators in Banach Spaces[J]. Arch. Math., 2008, 91: 166-177.
[21] YONGFU SU, MEIJUAN SHANG, XIAOLONG QIN. A General Iterative Scheme for Nonexpansive Mappings and Inverse-strongly Monotone Map-pings[J]. Journal of Applied Mathematics and Computing, 2008, 28(1-2): 283-294.
[22] SHENGHUA WANG, BAOHUA GUO. Viscosity Approximation Methods for Nonexpansive Mappings and Inverse-strongly Monotone Mappings in Hilbert Spaces[J]. Journal of Applied Mathematics and Computing, 2008, 28(1-2): 351-365.
[23] ZHANG S S, HUANG F L. The Theory of Fixed Points for Mean Nonex-pansive in Banach Space[J]. Journal of Sichuan University, 1975, 2: 67-78.
[24] ZHAO H B. Existence Theorems of Fixed Point of Average Nonexpansive Mappings in Banach Space[J]. Acta Mathematics Sinica, 1979, 22(2): 459-470.
[25] GU Z H, LI Y J. Approximation Methods for Common Fixed Points of Mean Nonexpansive Mapping in Banach Spaces[J]. Fixed Point Theory Appl., 2008: 1-7.
[26] RHOADES B E, SOLTUZ S M. The Equivalence between the Convergences of Ishikawa and Mann Iterations for an Asymptotically Pseudo-contractive Map[J]. Math. Anal. Appl., 2003, 283: 681-688.
[27] RHOADES B E, SOLTUZ S M. The Equivalence of Mann Iteration and Ishikawa Iteration for Nonlipschitzian Oprators[J]. Int. J. Math. Math. Sci., 2003, 2003: 2645-2652.
[28] ZHANG S S, CHO Y J, KIM J K. The Equivalence Between the Convergence of Mdified Picard, Modified Mann, and Modified Ishikawa Iterations[J]. Mathematical and Computer Modelling, 2003, 37(9-10): 985-991.
[29] SOLTUZ S M. The Equivalence of Picard, Mann and Ishikawa Iteration Dealing with Quasi-contractive Operators[J]. Mathematical Communications, 2005, 10: 81-88.
[30] GIUSEPPE MARINO, HONGKUN XU. Weak and Strong Convergence Theorems for Strict Pseudo-contractions in Hilbert Spaces[J]. Math. Anal. Appl., 2007, 329: 336-346.
[31] TAKAHASHI S, TAKAHASHI W. Viscosity Approximation Methods for Equilibrium Problems and Fixed Point Problems in Hilbert Spaces[J]. Math. Anal. Appl., 2007, 331:506-515.
[32] COMBETTES P L, HIRSTOAGA S A. Equilibrium Programming in Hilbert Spaces [J]. Nonlinear Convex Anal., 2005, 6: 117-136.
[33] XU YUGANG. Ishikawa and Mann Iterative Process with Errors for Nonlinear Strongly Accretive Operator Equations[J]. Math. Anal. Appl., 1998, 224:91-101.
[34] WEN G X. Fixed Point Iteration for Local Strictly Pseudocontractive Mapping [J]. Proc. Amer. Math. Soc., 1991, 113:727-731.
[35] LIU QIHOU. Iterative Sequences for Asymptotically Quasi-nonexpansive Mapping with Error Members[J]. Math. Anal. Appl., 2001, 259:18-24.
[36] SUZUKI T. Strong Convergence of Krasnoselskii and Mann’s Type Sequences for One-parameter Nonexpansive[J]. Math. Anal. Appl., 2005, 305:227-239.
[37] XU H K. Iterative Algorithms for Nonlinear Operators[J]. London Math. Soc., 2002, 240-256.
[38] CENG L C, HOMIDAM S AL, ANSARI Q H, YAO J C. An Iterative Scheme for Equilibrium Problems and Fixed Point Problems of Strict Pseudo-Contraction Mappings[J]. Joural of Computational and Applied Mathematics, 2009, 223:967-974.
[39] CHANG SHIHSEN. A New Method for Solving Equilibrium Problem Fixed Point Problem and Variational Inequality Problem with Application to Optimization[J]. Nonlinear Analysis, 2009, 70:3307-3319.
[40] ZHANG QINGBANG. Approximation of Fixed Points of Nonexpansive Mapping in Banach Spaces[J]. Mathematical and Computer Modelling, 2009, 49:1173-1179.
