Banach空间中的Mann迭代和Ishikawa迭代
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摘要
本文主要研究了定义在Banach空间上的Mann迭代与Ishikawa迭代,以及在这些迭代下的几类映射的收敛性问题。文章首先介绍了在紧空间下的上述映射的收敛性问题,由于紧空间具有任何无穷序列都有收敛子列的性质,从而使问题容易得到证明,接着本文又继续讨论了在非紧空间中上述映射的收敛性问题,由于去掉了紧性条件,因此我们引入了条件I,从而使结论得以证明。
首先,我们讲述了不动点理论的发展概况。通过引用大量前人的定义和定理,使我们对不动点的发展史有了一定程度的认识。
其次,我们主要研究在紧空间下的非扩张映射的Mann迭代和Ishikawa迭代的收敛性问题。首先给出与定理相关的定义,如非扩张映射,点到集合的距离,Mann和Ishikawa的具体迭代方式,Hausdoff距离和本文中用到的一些引理。接着给出了紧空间下的非扩张映射的Mann迭代和Ishikawa迭代的收敛性。
最后,研究了在非紧空间下的映射的迭代收敛性问题。由于紧性条件的减弱,我们加入了条件I,也使收敛性结论成立。本章首先给出了具体的Mann迭代方式,继而分别给出非扩张映射,亚非扩张映射和广义非扩张映射在该种迭代下收敛到相应映射不动点的结论。
It is primarily studied in Banach space that some kinds of mappings converge to these mappings’fixed point in sense of Mann iterations and Ishikawa iterations. First, the mappings’convergence problems in compac-tive space is extended in this paper In this paper. Because there is at least one subsequence in any numerous sequences in compactive space, the question mentioned above can be proved easily. In addition, it continues to discuss the mappings’convergence problems in non-compactive space. Since the condition of compact is not available, condition I is introduced. The result of the above is still correct.
First, the history of fixed points is represented. Lots of theories and defi-nitions are quoted. We can know something about the fixed point development.
Second, it is primarily studied nonexpansive mappings’convergence pro-blems in sense of Mann iteration or Ishikawa iteration. At first, it can be known some definitions related with this paper. Such as nonexpansive mapping, Mann iteration and Ishikawa iteration, Hausdoff metric and some related theorems. In addition, it is proved that in compactive space, the mapping in sense of Mann iteration or Ishikawa iteration converges to the fixed point of the mapping.
Finally, it is proved that these mappings converge in non-compactive space. Since the compactive space is not available, the result is still true by assumption of condition I. A new Mann iteraton is introduced. Then this paper gives us the results that the mapping converges to its fixed point which can be nonexpansive mapping, quasi-nonexpansive mapping or generalized nonexpansive mapping.
首先,我们讲述了不动点理论的发展概况。通过引用大量前人的定义和定理,使我们对不动点的发展史有了一定程度的认识。
其次,我们主要研究在紧空间下的非扩张映射的Mann迭代和Ishikawa迭代的收敛性问题。首先给出与定理相关的定义,如非扩张映射,点到集合的距离,Mann和Ishikawa的具体迭代方式,Hausdoff距离和本文中用到的一些引理。接着给出了紧空间下的非扩张映射的Mann迭代和Ishikawa迭代的收敛性。
最后,研究了在非紧空间下的映射的迭代收敛性问题。由于紧性条件的减弱,我们加入了条件I,也使收敛性结论成立。本章首先给出了具体的Mann迭代方式,继而分别给出非扩张映射,亚非扩张映射和广义非扩张映射在该种迭代下收敛到相应映射不动点的结论。
It is primarily studied in Banach space that some kinds of mappings converge to these mappings’fixed point in sense of Mann iterations and Ishikawa iterations. First, the mappings’convergence problems in compac-tive space is extended in this paper In this paper. Because there is at least one subsequence in any numerous sequences in compactive space, the question mentioned above can be proved easily. In addition, it continues to discuss the mappings’convergence problems in non-compactive space. Since the condition of compact is not available, condition I is introduced. The result of the above is still correct.
First, the history of fixed points is represented. Lots of theories and defi-nitions are quoted. We can know something about the fixed point development.
Second, it is primarily studied nonexpansive mappings’convergence pro-blems in sense of Mann iteration or Ishikawa iteration. At first, it can be known some definitions related with this paper. Such as nonexpansive mapping, Mann iteration and Ishikawa iteration, Hausdoff metric and some related theorems. In addition, it is proved that in compactive space, the mapping in sense of Mann iteration or Ishikawa iteration converges to the fixed point of the mapping.
Finally, it is proved that these mappings converge in non-compactive space. Since the compactive space is not available, the result is still true by assumption of condition I. A new Mann iteraton is introduced. Then this paper gives us the results that the mapping converges to its fixed point which can be nonexpansive mapping, quasi-nonexpansive mapping or generalized nonexpansive mapping.
引文
[1] SHIZUO KAKUTANI. Topological Properties of the Unit Sphere of a Hilbert Spaces[J]. Proc. Imp. Acad., 1943, 19(6): 269-271.
[2] KIRK W, SIMS B. Handbook of Metric Fixed Point Theory[M]. Boston: Kluwer Academic Publishers, 2001: 1-200.
[3] MANN W R. Mean Value Method in Iteration[J]. Proc. Amer. Math. Soc., 1953, 4(6): 506-510.
[4] ISHIKAWA S. Fixed Points by a New Iterations Method[J]. Proc. Amer. Math. Soc., 1974, 44(1): 147-150.
[5] 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.
[6] CHIDUME C E. Global Iteration Schemes for Strongly Pseudo-contractive Maps[J]. Proc. Amer. Math. Soc., 1998, 126(9): 2641-2649.
[7] CHIDUME C E, MOORE C. Fixed Point Iteration for Pseudo-contractive Maps[J]. Proc. Amer. Math. Soc., 1999, 127(4): 1163-1170.
[8] ZHANG S S. Some Results for Asymptoticallu Pseudo-contractive Mappings and Asymptoticallu Nonexpansive Mappings[J]. Amer. Math. Soc., 2000, 129(3): 845-853.
[9] 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.
[10] 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.
[11] ZAMFIRESCU T. Fix Point Theorems in Metric Spaces[J]. Arch. Math., 1972, 23 (1): 292-298.
[12] RHOADES B E. Fixed Point Iterations Using Infinite Matrices[J]. Trans. Amer. Math. Soc., 1974, 196: 161-176.
[13] 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.
[14] RAFIQ A. A Convergence Theorem for Mann Fixed Point Iteration Proce-dure[J]. Applied Mathemaics E-notes, 2006, 6: 289-293.
[15] AL-THAGAFI M A, NASEER SHAHZAD. Generalized I-nonexpansive Selfmaps and Invariant Approximations[J]. Acta Mathematica Sinica, 2008, 24(5): 867-876.
[16] 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.
[17] 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.
[18] 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.
[19] 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.
[20] 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.
[21] 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.
[22] 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.
[23] 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.
[24] 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.
[25] ZHAO H B. Existence Theorems of Fixed Point of Average Nonexpansive Mappings in Banach Space[J]. Acta Mathematics Sinica, 1979, 22(2): 459-470.
[26] 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.
[27] RHOADES B E, SOLTUZ S M. The Equivalence between the Convergences of Ishikawa and Mann Iterations for an Asymptotically Pseudo-contractive Map[J]. J. Math. Anal. Appl., 2003, 283: 681-688.
[28] 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.
[29] 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.
[30] SOLTUZ S M. The Equivalence of Picard, Mann and Ishikawa Iteration Dealing with Quasi-contractive Operators[J]. Mathematical Communications, 2005, 10: 81-88.
[31] XUE Z. Remarks of Equivalence of Picard, Mann and Ishikawa Iteration in Normed Spaces[J]. Fixed Point Theory Appl., 2007: 1-5.
[32] BERINDE V. On the Converges Faster than Mann Iteration for a Class of Quasi-contractive Operators[J]. Fixed Point Theory Appl., 2004, 2: 97-105.
[33] BABU G V R, VARA PRASAD K N V V. Mann Iteration Converges Faster than Ishikawa Iteration for the Class of Zamfirescu Operators[J]. Fixed Point Theory Appl., 2006: 1-6.
[34] POPESCU O. Picard Iteration Converges Faster than Mann Iteration for a Class of Quasi-contractive Operators[J]. Mathematical Communications, 2007, 12: 195-202.
[35] KIRK W A. A Fixed Point Theorem for Mappings Which Do not Increase Distance[J]. Amer. Math. Monthly, 1965, 72: 1004-1006.
[36] ISHIKAWA S. Fixed Points by a New Iteration Method[J]. Proc. Amer. Math.Soc., 1974, 44: 147-150.
[37] LIM T C. A Fixed Point Theorem for Multivalued Nonexpansive Mappings in a Uniformly Convex Banach Space[J]. Bull. Amer. Math. Soc., 1974, 80: 1123-1126.
[38] SASTRY K P R, BABU G V R. Convergence of Ishikawa Iterates for a Multi-valued Mapping with a Fixed Point[J]. Czechoslovak Math. J, 2005, 130(55): 817-826.
[39] XU H K. Inequalities in Banach Spaces with Applications[J]. Nonlinear Anal., 1991, 16 (12): 1127-1138.
[40] DOTSON JR W G. On the Manniterative Process[J]. Trans. Amer. Math. Soc., 1970, 149: 65-73.
[41] SENTER H F, DOTSON W G. Approximating Fixed Points of Nonexpansive Mappings[J]. Proc. Amer. Math. Soc., 1974, 44(2): 375-380.
[2] KIRK W, SIMS B. Handbook of Metric Fixed Point Theory[M]. Boston: Kluwer Academic Publishers, 2001: 1-200.
[3] MANN W R. Mean Value Method in Iteration[J]. Proc. Amer. Math. Soc., 1953, 4(6): 506-510.
[4] ISHIKAWA S. Fixed Points by a New Iterations Method[J]. Proc. Amer. Math. Soc., 1974, 44(1): 147-150.
[5] 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.
[6] CHIDUME C E. Global Iteration Schemes for Strongly Pseudo-contractive Maps[J]. Proc. Amer. Math. Soc., 1998, 126(9): 2641-2649.
[7] CHIDUME C E, MOORE C. Fixed Point Iteration for Pseudo-contractive Maps[J]. Proc. Amer. Math. Soc., 1999, 127(4): 1163-1170.
[8] ZHANG S S. Some Results for Asymptoticallu Pseudo-contractive Mappings and Asymptoticallu Nonexpansive Mappings[J]. Amer. Math. Soc., 2000, 129(3): 845-853.
[9] 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.
[10] 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.
[11] ZAMFIRESCU T. Fix Point Theorems in Metric Spaces[J]. Arch. Math., 1972, 23 (1): 292-298.
[12] RHOADES B E. Fixed Point Iterations Using Infinite Matrices[J]. Trans. Amer. Math. Soc., 1974, 196: 161-176.
[13] 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.
[14] RAFIQ A. A Convergence Theorem for Mann Fixed Point Iteration Proce-dure[J]. Applied Mathemaics E-notes, 2006, 6: 289-293.
[15] AL-THAGAFI M A, NASEER SHAHZAD. Generalized I-nonexpansive Selfmaps and Invariant Approximations[J]. Acta Mathematica Sinica, 2008, 24(5): 867-876.
[16] 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.
[17] 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.
[18] 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.
[19] 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.
[20] 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.
[21] 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.
[22] 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.
[23] 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.
[24] 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.
[25] ZHAO H B. Existence Theorems of Fixed Point of Average Nonexpansive Mappings in Banach Space[J]. Acta Mathematics Sinica, 1979, 22(2): 459-470.
[26] 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.
[27] RHOADES B E, SOLTUZ S M. The Equivalence between the Convergences of Ishikawa and Mann Iterations for an Asymptotically Pseudo-contractive Map[J]. J. Math. Anal. Appl., 2003, 283: 681-688.
[28] 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.
[29] 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.
[30] SOLTUZ S M. The Equivalence of Picard, Mann and Ishikawa Iteration Dealing with Quasi-contractive Operators[J]. Mathematical Communications, 2005, 10: 81-88.
[31] XUE Z. Remarks of Equivalence of Picard, Mann and Ishikawa Iteration in Normed Spaces[J]. Fixed Point Theory Appl., 2007: 1-5.
[32] BERINDE V. On the Converges Faster than Mann Iteration for a Class of Quasi-contractive Operators[J]. Fixed Point Theory Appl., 2004, 2: 97-105.
[33] BABU G V R, VARA PRASAD K N V V. Mann Iteration Converges Faster than Ishikawa Iteration for the Class of Zamfirescu Operators[J]. Fixed Point Theory Appl., 2006: 1-6.
[34] POPESCU O. Picard Iteration Converges Faster than Mann Iteration for a Class of Quasi-contractive Operators[J]. Mathematical Communications, 2007, 12: 195-202.
[35] KIRK W A. A Fixed Point Theorem for Mappings Which Do not Increase Distance[J]. Amer. Math. Monthly, 1965, 72: 1004-1006.
[36] ISHIKAWA S. Fixed Points by a New Iteration Method[J]. Proc. Amer. Math.Soc., 1974, 44: 147-150.
[37] LIM T C. A Fixed Point Theorem for Multivalued Nonexpansive Mappings in a Uniformly Convex Banach Space[J]. Bull. Amer. Math. Soc., 1974, 80: 1123-1126.
[38] SASTRY K P R, BABU G V R. Convergence of Ishikawa Iterates for a Multi-valued Mapping with a Fixed Point[J]. Czechoslovak Math. J, 2005, 130(55): 817-826.
[39] XU H K. Inequalities in Banach Spaces with Applications[J]. Nonlinear Anal., 1991, 16 (12): 1127-1138.
[40] DOTSON JR W G. On the Manniterative Process[J]. Trans. Amer. Math. Soc., 1970, 149: 65-73.
[41] SENTER H F, DOTSON W G. Approximating Fixed Points of Nonexpansive Mappings[J]. Proc. Amer. Math. Soc., 1974, 44(2): 375-380.