Open Access

Approximating fixed points of total asymptotically nonexpansive mappings

Fixed Point Theory and Applications20062006:10673

https://doi.org/10.1155/FPTA/2006/10673

Received: 10 March 2005

Accepted: 28 August 2005

Published: 2 March 2006

Abstract

We introduce a new class of asymptotically nonexpansive mappings and study approximating methods for finding their fixed points. We deal with the Krasnosel'skii-Mann-type iterative process. The strong and weak convergence results for self-mappings in normed spaces are presented. We also consider the asymptotically weakly contractive mappings.

[1234567891011121314151617181920212223242526272829303132333435363738]

Authors’ Affiliations

(1)
Department of Mathematics, The Technion-Israel Institute of Technology
(2)
The Abdus Salam International Centre for Theoretical Physics

References

  1. Alber YaI: On the solution of equations and variational inequalities with maximal monotone operators. Soviet Mathematics Doklady 1979,20(4):871–876.MathSciNetGoogle Scholar
  2. Alber YaI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Appl. Math.. Volume 178. Edited by: Kartsatos AG. Dekker, New York; 1996:15–50.Google Scholar
  3. Alber YaI, Guerre-Delabriere S: Problems of fixed point theory in Hilbert and Banach spaces. Functional Differential Equations. Israel Seminar 1994, 2: 5–10 (1995).MathSciNetMATHGoogle Scholar
  4. Alber YaI, Guerre-Delabriere S: Principle of weakly contractive maps in Hilbert spaces. In New Results in Operator Theory and Its Applications, Oper. Theory Adv. Appl.. Volume 98. Birkhäuser, Basel; 1997:7–22.View ArticleGoogle Scholar
  5. Alber YaI, Guerre-Delabriere S: On the projection methods for fixed point problems. Analysis. International Mathematical Journal of Analysis and its Applications 2001,21(1):17–39.MathSciNetMATHGoogle Scholar
  6. Alber YaI, Iusem AN: Extension of subgradient techniques for nonsmooth optimization in Banach spaces. Set-Valued Analysis. An International Journal Devoted to the Theory of Multifunctions and its Applications 2001,9(4):315–335. 10.1023/A:1012665832688MathSciNetMATHGoogle Scholar
  7. Alber YaI, Iusem AN, Solodov MV: Minimization of nonsmooth convex functionals in Banach spaces. Journal of Convex Analysis 1997,4(2):235–255.MathSciNetMATHGoogle Scholar
  8. Alber YaI, Reich S: An iterative method for solving a class of nonlinear operator equations in Banach spaces. Panamerican Mathematical Journal 1994,4(2):39–54.MathSciNetMATHGoogle Scholar
  9. Alber YaI, Reich S, Yao J-C: Iterative methods for solving fixed-point problems with nonself-mappings in Banach spaces. Abstract and Applied Analysis 2003,2003(4):193–216. 10.1155/S1085337503203018MathSciNetView ArticleMATHGoogle Scholar
  10. Bose SC: Weak convergence to the fixed point of an asymptotically nonexpansive map. Proceedings of the American Mathematical Society 1978,68(3):305–308. 10.1090/S0002-9939-1978-0493543-4MathSciNetView ArticleMATHGoogle Scholar
  11. Bruck RE, Kuczumow T, Reich S: Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property. Colloquium Mathematicum 1993,65(2):169–179.MathSciNetMATHGoogle Scholar
  12. Chang SS, Cho YJ, Zhou H: Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings. Journal of the Korean Mathematical Society 2001,38(6):1245–1260.MathSciNetMATHGoogle Scholar
  13. Chidume CE: Nonexpansive mappings, generalizations and iterative algorithms. to appear in Nonlinear AnalGoogle Scholar
  14. Chidume CE, Ofoedu EU, Zegeye H: Strong and weak convergence theorems for asymptotically nonexpansive mappings. Journal of Mathematical Analysis and Applications 2003,280(2):364–374. 10.1016/S0022-247X(03)00061-1MathSciNetView ArticleMATHGoogle Scholar
  15. Chidume CE, Shahzad N, Zegeye H: Strong convergence theorems for nonexpansive mappings in arbitrary Banach spaces. to appear in Nonlinear AnalGoogle Scholar
  16. Diestel J: Geometry of Banach Spaces—Selected Topics, Lecture Notes in Mathematics. Volume 485. Springer, Berlin; 1975:xi+282.Google Scholar
  17. Figiel T: On the moduli of convexity and smoothness. Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica 1976,56(2):121–155.MathSciNetMATHGoogle Scholar
  18. Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proceedings of the American Mathematical Society 1972, 35: 171–174. 10.1090/S0002-9939-1972-0298500-3MathSciNetView ArticleMATHGoogle Scholar
  19. Górnicki J: Nonlinear ergodic theorems for asymptotically nonexpansive mappings in Banach spaces satisfying Opial's condition. Journal of Mathematical Analysis and Applications 1991,161(2):440–446. 10.1016/0022-247X(91)90343-XMathSciNetView ArticleMATHGoogle Scholar
  20. Ishikawa S: Fixed points by a new iteration method. Proceedings of the American Mathematical Society 1974, 44: 147–150. 10.1090/S0002-9939-1974-0336469-5MathSciNetView ArticleMATHGoogle Scholar
  21. Ishikawa S: Fixed points and iteration of a nonexpansive mapping in a Banach space. Proceedings of the American Mathematical Society 1976,59(1):65–71. 10.1090/S0002-9939-1976-0412909-XMathSciNetView ArticleMATHGoogle Scholar
  22. Kim GE, Kim TH: Mann and Ishikawa iterations with errors for non-Lipschitzian mappings in Banach spaces. Computers & Mathematics with Applications. An International Journal 2001,42(12):1565–1570. 10.1016/S0898-1221(01)00262-0MathSciNetView ArticleMATHGoogle Scholar
  23. Kirk WA: Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type. Israel Journal of Mathematics 1974, 17: 339–346. 10.1007/BF02757136MathSciNetView ArticleMATHGoogle Scholar
  24. Lindenstrauss J, Tzafriri L: Classical Banach Spaces. II. Function Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. Volume 97. Springer, Berlin; 1979:x+243.Google Scholar
  25. Mann W: Mean value methods in iteration. Proceedings of the American Mathematical Society 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3MathSciNetView ArticleMATHGoogle Scholar
  26. Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bulletin of the Australian Mathematical Society 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleMATHGoogle Scholar
  27. Osilike MO, Aniagbosor SC: Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings. Mathematical and Computer Modelling 2000,32(10):1181–1191. 10.1016/S0895-7177(00)00199-0MathSciNetView ArticleMATHGoogle Scholar
  28. Passty GB: Construction of fixed points for asymptotically nonexpansive mappings. Proceedings of the American Mathematical Society 1982,84(2):212–216. 10.1090/S0002-9939-1982-0637171-7MathSciNetView ArticleMATHGoogle Scholar
  29. Reich S: Weak convergence theorems for nonexpansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications 1979,67(2):274–276. 10.1016/0022-247X(79)90024-6MathSciNetView ArticleMATHGoogle Scholar
  30. Rhoades BE: Fixed point iterations for certain nonlinear mappings. Journal of Mathematical Analysis and Applications 1994,183(1):118–120. 10.1006/jmaa.1994.1135MathSciNetView ArticleMATHGoogle Scholar
  31. Schu J: Iterative construction of fixed points of asymptotically nonexpansive mappings. Journal of Mathematical Analysis and Applications 1991,158(2):407–413. 10.1016/0022-247X(91)90245-UMathSciNetView ArticleMATHGoogle Scholar
  32. Schu J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bulletin of the Australian Mathematical Society 1991,43(1):153–159. 10.1017/S0004972700028884MathSciNetView ArticleMATHGoogle Scholar
  33. Senter HF, Dotson WG Jr.: Approximating fixed points of nonexpansive mappings. Proceedings of the American Mathematical Society 1974, 44: 375–380. 10.1090/S0002-9939-1974-0346608-8MathSciNetView ArticleMATHGoogle Scholar
  34. Tan K-K, Xu HK: A nonlinear ergodic theorem for asymptotically nonexpansive mappings. Bulletin of the Australian Mathematical Society 1992,45(1):25–36. 10.1017/S0004972700036972MathSciNetView ArticleMATHGoogle Scholar
  35. Tan K-K, Xu HK: The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces. Proceedings of the American Mathematical Society 1992,114(2):399–404. 10.1090/S0002-9939-1992-1068133-2MathSciNetView ArticleMATHGoogle Scholar
  36. Tan K-K, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. Journal of Mathematical Analysis and Applications 1993,178(2):301–308. 10.1006/jmaa.1993.1309MathSciNetView ArticleMATHGoogle Scholar
  37. Tan K-K, Xu HK: Fixed point iteration processes for asymptotically nonexpansive mappings. Proceedings of the American Mathematical Society 1994,122(3):733–739. 10.1090/S0002-9939-1994-1203993-5MathSciNetView ArticleMATHGoogle Scholar
  38. Xu HK: Existence and convergence for fixed points of mappings of asymptotically nonexpansive type. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 1991,16(12):1139–1146. 10.1016/0362-546X(91)90201-BMathSciNetView ArticleMATHGoogle Scholar

Copyright

© Alber et al. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.