Advertisement

Fixed Point Theory and Applications

, 2007:021972 | Cite as

Block Iterative Methods for a Finite Family of Relatively Nonexpansive Mappings in Banach Spaces

  • Fumiaki Kohsaka
  • Wataru Takahashi
Open Access
Research Article

Abstract

Using the convex combination based on Bregman distances due to Censor and Reich, we define an operator from a given family of relatively nonexpansive mappings in a Banach space. We first prove that the fixed-point set of this operator is identical to the set of all common fixed points of the mappings. Next, using this operator, we construct an iterative sequence to approximate common fixed points of the family. We finally apply our results to a convex feasibility problem in Banach spaces.

Keywords

Banach Space Iterative Method Differential Geometry Nonexpansive Mapping Convex Combination 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Aharoni R, Censor Y: Block-iterative projection methods for parallel computation of solutions to convex feasibility problems. Linear Algebra and Its Applications 1989, 120: 165–175. 10.1016/0024-3795(89)90375-3MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Butnariu D, Censor Y: On the behavior of a block-iterative projection method for solving convex feasibility problems. International Journal of Computer Mathematics 1990,34(1–2):79–94. 10.1080/00207169008803865MATHCrossRefGoogle Scholar
  3. 3.
    Butnariu D, Censor Y: Strong convergence of almost simultaneous block-iterative projection methods in Hilbert spaces. Journal of Computational and Applied Mathematics 1994,53(1):33–42. 10.1016/0377-0427(92)00123-QMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Cohen N, Kutscher T: On spherical convergence, convexity, and block iterative projection algorithms in Hilbert space. Journal of Mathematical Analysis and Applications 1998,226(2):271–291. 10.1006/jmaa.1998.6026MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Flåm SD, Zowe J: Relaxed outer projections, weighted averages and convex feasibility. BIT 1990,30(2):289–300. 10.1007/BF02017349MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Kikkawa M, Takahashi W: Approximating fixed points of nonexpansive mappings by the block iterative method in Banach spaces. International Journal of Computational and Numerical Analysis and Applications 2004,5(1):59–66.MATHMathSciNetGoogle Scholar
  7. 7.
    Crombez G: Image recovery by convex combinations of projections. Journal of Mathematical Analysis and Applications 1991,155(2):413–419. 10.1016/0022-247X(91)90010-WMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Kitahara S, Takahashi W: Image recovery by convex combinations of sunny nonexpansive retractions. Topological Methods in Nonlinear Analysis 1993,2(2):333–342.MATHMathSciNetGoogle Scholar
  9. 9.
    Takahashi W: Iterative methods for approximation of fixed points and their applications. Journal of the Operations Research Society of Japan 2000,43(1):87–108.MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Takahashi W, Tamura T: Limit theorems of operators by convex combinations of nonexpansive retractions in Banach spaces. Journal of Approximation Theory 1997,91(3):386–397. 10.1006/jath.1996.3093MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Butnariu D, Reich S, Zaslavski AJ: Asymptotic behavior of relatively nonexpansive operators in Banach spaces. Journal of Applied Analysis 2001,7(2):151–174. 10.1515/JAA.2001.151MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Matsushita S, Takahashi W: Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces. Fixed Point Theory and Applications 2004,2004(1):37–47. 10.1155/S1687182004310089MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Matsushita S, Takahashi W: An iterative algorithm for relatively nonexpansive mappings by a hybrid method and applications. In Nonlinear Analysis and Convex Analysis. Edited by: Takahashi W, Tanaka T. Yokohama Publishers, Yokohama, Japan; 2004:305–313.Google Scholar
  14. 14.
    Matsushita S, Takahashi W: A strong convergence theorem for relatively nonexpansive mappings in a Banach space. Journal of Approximation Theory 2005,134(2):257–266. 10.1016/j.jat.2005.02.007MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Alber YI: 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. Markel Dekker, New York, NY, USA; 1996:15–50.Google Scholar
  16. 16.
    Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM Journal on Optimization 2002,13(3):938–945. 10.1137/S105262340139611XMathSciNetCrossRefGoogle Scholar
  17. 17.
    Bregman LM: A relaxation method of finding a common point of convex sets and its application to the solution of problems in convex programming. USSR Computational Mathematics and Mathematical Physics 1967, 7: 200–217.CrossRefGoogle Scholar
  18. 18.
    Censor Y, Reich S: Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization. Optimization 1996,37(4):323–339. 10.1080/02331939608844225MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Kohsaka F, Takahashi W: Strong convergence of an iterative sequence for maximal monotone operators in a Banach space. Abstract and Applied Analysis 2004,2004(3):239–249. 10.1155/S1085337504309036MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Cioranescu I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Mathematics and Its Applications. Volume 62. Kluwer Academic, Dordrecht, The Netherlands; 1990.CrossRefGoogle Scholar
  21. 21.
    Diestel J: Geometry of Banach Spaces—Selected Topics, Lecture Notes in Mathematics. Volume 485. Springer, Berlin, Germany; 1975.Google Scholar
  22. 22.
    Gossez J-P, Lami Dozo E: Some geometric properties related to the fixed point theory for nonexpansive mappings. Pacific Journal of Mathematics 1972, 40: 565–573.MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Reich S: A weak convergence theorem for the alternating method with Bregman distances. 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. Markel Dekker, New York, NY, USA; 1996:313–318.Google Scholar
  24. 24.
    Xu HK: Inequalities in Banach spaces with applications. Nonlinear Analysis 1991,16(12):1127–1138. 10.1016/0362-546X(91)90200-KMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Zălinescu C: On uniformly convex functions. Journal of Mathematical Analysis and Applications 1983,95(2):344–374. 10.1016/0022-247X(83)90112-9MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Zălinescu C: Convex Analysis in General Vector Spaces. World Scientific, River Edge, NJ, USA; 2002.MATHCrossRefGoogle Scholar
  27. 27.
    Takahashi W: Convex Analysis and Approximation of Fixed Points, Mathematical Analysis Series. Volume 2. Yokohama Publishers, Yokohama, Japan; 2000.Google Scholar
  28. 28.
    Takahashi W: Nonlinear Functional Analysis. Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama, Japan; 2000.MATHGoogle Scholar

Copyright information

© F. Kohsaka and W. Takahashi. 2007

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.

Authors and Affiliations

  1. 1.Department of Information EnvironmentTokyo Denki University, Muzai GakuendaiInzai, ChibaJapan
  2. 2.Department of Mathematical and Computing SciencesTokyo Institute of TechnologyTokyoJapan

Personalised recommendations