Skip to main content

A New Iterative Algorithm for Approximating Common Fixed Points for Asymptotically Nonexpansive Mappings

Abstract

Suppose that is a nonempty closed convex subset of a real uniformly convex and smooth Banach space with as a sunny nonexpansive retraction. Let be two weakly inward and asymptotically nonexpansive mappings with respect to with sequences , , respectively. Suppose that is a sequence in generated iteratively by , , for all , where , , and are three real sequences in for some which satisfy condition . Then, we have the following. (1) If one of and is completely continuous or demicompact and , then the strong convergence of to some is established. (2) If is a real uniformly convex Banach space satisfying Opial's condition or whose norm is Fréchet differentiable, then the weak convergence of to some is proved.

[12345678910111213141516171819202122]

References

  1. Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proceedings of the American Mathematical Society 1972,35(1):171–174. 10.1090/S0002-9939-1972-0298500-3

    MATH  MathSciNet  Article  Google Scholar 

  2. 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/S0002-9939-1978-0493543-4

    MATH  MathSciNet  Article  Google Scholar 

  3. Passty GB: Construction of fixed points for asymptotically nonexpansive mappings. Proceedings of the American Mathematical Society 1982,84(2):212–216. 10.1090/S0002-9939-1982-0637171-7

    MATH  MathSciNet  Article  Google Scholar 

  4. 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/S0002-9939-1992-1068133-2

    MATH  MathSciNet  Article  Google Scholar 

  5. 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

    MATH  MathSciNet  Article  Google Scholar 

  6. 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

    MATH  MathSciNet  Article  Google Scholar 

  7. 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/S0004972700028884

    MATH  MathSciNet  Article  Google Scholar 

  8. 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

    MATH  MathSciNet  Article  Google Scholar 

  9. 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)00199-0

    MATH  MathSciNet  Article  Google Scholar 

  10. Chang S-S, Cho YJ, Zhou HY: Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings. Journal of the Korean Mathematical Society 2001,38(6):1245–1260.

    MATH  MathSciNet  Google Scholar 

  11. Górnicki J: Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces. Commentationes Mathematicae Universitatis Carolinae 1989,30(2):249–252.

    MATH  MathSciNet  Google Scholar 

  12. 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)90024-6

    MATH  MathSciNet  Article  Google Scholar 

  13. 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-X

    MATH  MathSciNet  Article  Google Scholar 

  14. 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

    MATH  MathSciNet  Article  Google Scholar 

  15. 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

    MATH  MathSciNet  Article  Google Scholar 

  16. 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

    MATH  MathSciNet  Article  Google Scholar 

  17. 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

    MATH  MathSciNet  Article  Google Scholar 

  18. Song Y, Chen R: Viscosity approximation methods for nonexpansive nonself-mappings. Journal of Mathematical Analysis and Applications 2006,321(1):316–326. 10.1016/j.jmaa.2005.07.025

    MATH  MathSciNet  Article  Google Scholar 

  19. 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)00061-1

    MATH  MathSciNet  Article  Google Scholar 

  20. Takahashi W: Nonlinear Functional Analysis. Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.

    MATH  Google Scholar 

  21. 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

    MATH  MathSciNet  Article  Google Scholar 

  22. Zhou HY, Guo GT, Hwang HJ, Cho YJ: On the iterative methods for nonlinear operator equations in Banach spaces. PanAmerican Mathematical Journal 2004,14(4):61–68.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to HY Zhou.

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

Zhou, H., Cho, Y. & Kang, S. A New Iterative Algorithm for Approximating Common Fixed Points for Asymptotically Nonexpansive Mappings. Fixed Point Theory Appl 2007, 064874 (2007). https://doi.org/10.1155/2007/64874

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/2007/64874

Keywords

  • Differential Geometry
  • Iterative Algorithm
  • Nonexpansive Mapping
  • Computational Biology
  • Asymptotically Nonexpansive