Skip to main content
Fixed Point Theory and Algorithms for Sciences and Engineering
Download PDF
  • Research Article
  • Open access
  • Published:

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

Fixed Point Theory and Applications volume 2007, Article number: 021972 (2007) Cite this 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.

[12345678910111213141516171819202122232425262728]

References

  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)90375-3

    Article  MATH  MathSciNet  Google Scholar 

  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. 10.1080/00207169008803865

    Article  MATH  Google Scholar 

  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-Q

    Article  MATH  MathSciNet  Google Scholar 

  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.1998.6026

    Article  MATH  MathSciNet  Google Scholar 

  5. Flåm SD, Zowe J: Relaxed outer projections, weighted averages and convex feasibility. BIT 1990,30(2):289–300. 10.1007/BF02017349

    Article  MATH  MathSciNet  Google Scholar 

  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.

    MATH  MathSciNet  Google Scholar 

  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-W

    Article  MATH  MathSciNet  Google Scholar 

  8. Kitahara S, Takahashi W: Image recovery by convex combinations of sunny nonexpansive retractions. Topological Methods in Nonlinear Analysis 1993,2(2):333–342.

    MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.1996.3093

    Article  MATH  MathSciNet  Google Scholar 

  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.2001.151

    Article  MATH  MathSciNet  Google Scholar 

  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/S1687182004310089

    Article  MATH  MathSciNet  Google Scholar 

  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. 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.2005.02.007

    Article  MATH  MathSciNet  Google Scholar 

  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. 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/S105262340139611X

    Article  MathSciNet  Google Scholar 

  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.

    Article  Google Scholar 

  18. Censor Y, Reich S: Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization. Optimization 1996,37(4):323–339. 10.1080/02331939608844225

    Article  MATH  MathSciNet  Google Scholar 

  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/S1085337504309036

    Article  MATH  MathSciNet  Google Scholar 

  20. Cioranescu I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Mathematics and Its Applications. Volume 62. Kluwer Academic, Dordrecht, The Netherlands; 1990.

    Book  Google Scholar 

  21. Diestel J: Geometry of Banach Spaces—Selected Topics, Lecture Notes in Mathematics. Volume 485. Springer, Berlin, Germany; 1975.

    Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. Xu HK: Inequalities in Banach spaces with applications. Nonlinear Analysis 1991,16(12):1127–1138. 10.1016/0362-546X(91)90200-K

    Article  MATH  MathSciNet  Google Scholar 

  25. Zălinescu C: On uniformly convex functions. Journal of Mathematical Analysis and Applications 1983,95(2):344–374. 10.1016/0022-247X(83)90112-9

    Article  MATH  MathSciNet  Google Scholar 

  26. Zălinescu C: Convex Analysis in General Vector Spaces. World Scientific, River Edge, NJ, USA; 2002.

    Book  MATH  Google Scholar 

  27. Takahashi W: Convex Analysis and Approximation of Fixed Points, Mathematical Analysis Series. Volume 2. Yokohama Publishers, Yokohama, Japan; 2000.

    Google Scholar 

  28. Takahashi W: Nonlinear Functional Analysis. Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama, Japan; 2000.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Information Environment, Tokyo Denki University, Muzai Gakuendai, Inzai, Chiba, 270-1382, Japan

    Fumiaki Kohsaka

  2. Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Oh-Okayama, Meguro-Ku, Tokyo, 152-8552, Japan

    Wataru Takahashi

Authors
  1. Fumiaki Kohsaka
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wataru Takahashi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fumiaki Kohsaka.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Kohsaka, F., Takahashi, W. Block Iterative Methods for a Finite Family of Relatively Nonexpansive Mappings in Banach Spaces. Fixed Point Theory Appl 2007, 021972 (2007). https://doi.org/10.1155/2007/21972

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/2007/21972

Keywords