Skip to main content

Convergece Theorems for Finite Families of Asymptotically Quasi-Nonexpansive Mappings

Abstract

Let be a real Banach space, a closed convex nonempty subset of, and asymptotically quasi-nonexpansive mappings with sequences (resp.) satisfying as, and. Let be a sequence in. Define a sequence by,,,,,. Let. Necessary and sufficient conditions for a strong convergence of the sequence to a common fixed point of the family are proved. Under some appropriate conditions, strong and weak convergence theorems are also proved.

[12345678910111213]

References

  1. 1.

    Chidume CE, Zegeye H, Shahzad N: Convergence theorems for a common fixed point of a finite family of nonself nonexpansive mappings. Fixed Point Theory and Applications 2005,2005(2):233–241.

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Chidume CE, Ali B: Approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications 2007,326(2):960–973. 10.1016/j.jmaa.2006.03.045

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Ghosh MK, Debnath L: Convergence of Ishikawa iterates of quasi-nonexpansive mappings. Journal of Mathematical Analysis and Applications 1997,207(1):96–103. 10.1006/jmaa.1997.5268

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Petryshyn WV, Williamson TE Jr.: Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings. Journal of Mathematical Analysis and Applications 1973,43(2):459–497. 10.1016/0022-247X(73)90087-5

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Qihou L: Iterative sequences for asymptotically quasi-nonexpansive mappings. Journal of Mathematical Analysis and Applications 2001,259(1):1–7. 10.1006/jmaa.2000.6980

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Qihou L: Iterative sequences for asymptotically quasi-nonexpansive mappings with error member. Journal of Mathematical Analysis and Applications 2001,259(1):18–24. 10.1006/jmaa.2000.7353

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Qihou L: Iteration sequences for asymptotically quasi-nonexpansive mapping with an error member of uniform convex Banach space. Journal of Mathematical Analysis and Applications 2002,266(2):468–471. 10.1006/jmaa.2001.7629

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Shahzad N, Udomene A: Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces. Fixed Point Theory and Applications 2006, 2006: 10 pages.

    MathSciNet  Article  Google Scholar 

  9. 9.

    Tan KK, 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

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    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

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    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

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Sun ZH: Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings. Journal of Mathematical Analysis and Applications 2003,286(1):351–358. 10.1016/S0022-247X(03)00537-7

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Xu Z-B, Roach GF: Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces. Journal of Mathematical Analysis and Applications 1991,157(1):189–210. 10.1016/0022-247X(91)90144-O

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to CE Chidume.

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

Chidume, C., Ali, B. Convergece Theorems for Finite Families of Asymptotically Quasi-Nonexpansive Mappings. J Inequal Appl 2007, 068616 (2007). https://doi.org/10.1155/2007/68616

Download citation

Keywords

  • Banach Space
  • Convergence Theorem
  • Weak Convergence
  • Strong Convergence
  • Nonempty Subset
\