Skip to main content
Fixed Point Theory and Algorithms for Sciences and Engineering
Download PDF

Weak and strong convergence of finite family with errors of nonexpansive nonself-mappings

Fixed Point Theory and Applications volume 2006, Article number: 81493 (2006) Cite this article

Abstract

We are concerned with the study of a multistep iterative scheme with errors involving a finite family of nonexpansive nonself-mappings. We approximate the common fixed points of a finite family of nonexpansive nonself-mappings by weak and strong convergence of the scheme in a uniformly convex Banach space. Our results extend and improve some recent results, Shahzad (2005) and many others.

[12345678910111213141516171819]

References

  1. Browder FE: Semicontractive and semiaccretive nonlinear mappings in Banach spaces. Bulletin of the American Mathematical Society 1968, 74: 660–665. 10.1090/S0002-9904-1968-11983-4

    Article  MathSciNet  MATH  Google Scholar 

  2. García Falset J, Kaczor W, Kuczumow T, Reich S: Weak convergence theorems for asymptotically nonexpansive mappings and semigroups. Nonlinear Analysis 2001,43(3):377–401. 10.1016/S0362-546X(99)00200-X

    Article  MathSciNet  MATH  Google Scholar 

  3. Ishikawa S: Fixed points by a new iteration method. Proceedings of the American Mathematical Society 1974,44(1):147–150. 10.1090/S0002-9939-1974-0336469-5

    Article  MathSciNet  MATH  Google Scholar 

  4. Jung JS, Kim SS: Strong convergence theorems for nonexpansive nonself-mappings in Banach spaces. Nonlinear Analysis 1998,33(3):321–329. 10.1016/S0362-546X(97)00526-9

    Article  MathSciNet  MATH  Google Scholar 

  5. Kaczor W: Weak convergence of almost orbits of asymptotically nonexpansive commutative semigroups. Journal of Mathematical Analysis and Applications 2002,272(2):565–574. 10.1016/S0022-247X(02)00175-0

    Article  MathSciNet  MATH  Google Scholar 

  6. Khan SH, Fukhar-ud-din H: Weak and strong convergence of a scheme with errors for two nonexpansive mappings. Nonlinear Analysis 2005,61(8):1295–1301. 10.1016/j.na.2005.01.081

    Article  MathSciNet  MATH  Google Scholar 

  7. Mann WR: Mean value methods in iteration. Proceedings of the American Mathematical Society 1953,4(3):506–510. 10.1090/S0002-9939-1953-0054846-3

    Article  MathSciNet  MATH  Google Scholar 

  8. Matsushita S-Y, Kuroiwa D: Strong convergence of averaging iterations of nonexpansive nonself-mappings. Journal of Mathematical Analysis and Applications 2004,294(1):206–214. 10.1016/j.jmaa.2004.02.010

    Article  MathSciNet  MATH  Google Scholar 

  9. Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bulletin of the American Mathematical Society 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0

    Article  MathSciNet  MATH  Google Scholar 

  10. Rhoades BE: Fixed point iterations for certain nonlinear mappings. Journal of Mathematical Analysis and Applications 1994,183(1):118–120. 10.1006/jmaa.1994.1135

    Article  MathSciNet  MATH  Google Scholar 

  11. 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-U

    Article  MathSciNet  MATH  Google Scholar 

  12. Shahzad N: Approximating fixed points of non-self nonexpansive mappings in Banach spaces. Nonlinear Analysis 2005,61(6):1031–1039. 10.1016/j.na.2005.01.092

    Article  MathSciNet  MATH  Google Scholar 

  13. Takahashi W, Kim G-E: Strong convergence of approximants to fixed points of nonexpansive nonself-mappings in Banach spaces. Nonlinear Analysis 1998,32(3):447–454. 10.1016/S0362-546X(97)00482-3

    Article  MathSciNet  MATH  Google Scholar 

  14. Takahashi W, Tamura T: Convergence theorems for a pair of nonexpansive mappings. Journal of Convex Analysis 1998,5(1):45–56.

    MathSciNet  MATH  Google Scholar 

  15. 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.1993.1309

    Article  MathSciNet  MATH  Google Scholar 

  16. 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/S0002-9939-1994-1203993-5

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Xu Y: Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations. Journal of Mathematical Analysis and Applications 1998,224(1):91–101. 10.1006/jmaa.1998.5987

    Article  MathSciNet  MATH  Google Scholar 

  19. Xu HK, Yin XM: Strong convergence theorems for nonexpansive non-self-mappings. Nonlinear Analysis 1995,24(2):223–228. 10.1016/0362-546X(94)E0059-P

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand

    S Plubtieng & K Ungchittrakool

Authors
  1. S Plubtieng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K Ungchittrakool
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S Plubtieng.

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

Plubtieng, S., Ungchittrakool, K. Weak and strong convergence of finite family with errors of nonexpansive nonself-mappings. Fixed Point Theory Appl 2006, 81493 (2006). https://doi.org/10.1155/FPTA/2006/81493

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/FPTA/2006/81493

Keywords