Skip to main content
  • Research Article
  • Open access
  • Published:

Strong Convergence of Modified Implicit Iteration Processes for Common Fixed Points of Nonexpansive Mappings

Abstract

Strong convergence theorems are obtained by hybrid method for modified composite implicit iteration process of nonexpansive mappings in Hilbert spaces. The results presented in this paper generalize and improve the corresponding results of Nakajo and Takahashi (2003) and others.

[123456789101112]

References

  1. Byrne C: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems 2004,20(1):103–120. 10.1088/0266-5611/20/1/006

    Article  MathSciNet  MATH  Google Scholar 

  2. Podilchuk CI, Mammone RJ: Image recovery by convex projections using a least-squares constraint. Journal of the Optical Society of America A 1990,7(3):517–521. 10.1364/JOSAA.7.000517

    Article  Google Scholar 

  3. Sezan MI, Stark H: Applications of convex projection theory to image recovery in tomography and related areas. In Image Recovery: Theory and Application. Edited by: Stark H. Academic Press, Orlando, Fla, USA; 1987:415–462.

    Google Scholar 

  4. Youla D: On deterministic convergence of iteration of relaxed projection operators. Journal of Visual Communication and Image Representation 1990,1(1):12–20. 10.1016/1047-3203(90)90013-L

    Article  Google Scholar 

  5. Youla D: Mathematical theory of image restoration by the method of convex projections. In Image Recovery: Theory and Application. Edited by: Stark H. Academic Press, Orlando, Fla, USA; 1987:29–77.

    Google Scholar 

  6. 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

    Article  MathSciNet  MATH  Google Scholar 

  7. Genel A, Lindenstrauss J: An example concerning fixed points. Israel Journal of Mathematics 1975,22(1):81–86. 10.1007/BF02757276

    Article  MathSciNet  MATH  Google Scholar 

  8. Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. Journal of Mathematical Analysis and Applications 2003,279(2):372–379. 10.1016/S0022-247X(02)00458-4

    Article  MathSciNet  MATH  Google Scholar 

  9. Xu H-K, Ori RG: An implicit iteration process for nonexpansive mappings. Numerical Functional Analysis and Optimization 2001,22(5–6):767–773. 10.1081/NFA-100105317

    Article  MathSciNet  MATH  Google Scholar 

  10. Osilike MO: Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps. Journal of Mathematical Analysis and Applications 2004,294(1):73–81. 10.1016/j.jmaa.2004.01.038

    Article  MathSciNet  MATH  Google Scholar 

  11. Martinez-Yanes C, Xu H-K: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Analysis 2006,64(11):2400–2411. 10.1016/j.na.2005.08.018

    Article  MathSciNet  MATH  Google Scholar 

  12. Goebel K, Kirk WA: Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics. Volume 28. Cambridge University Press, Cambridge, UK; 1990:viii+244.

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Fang Zhang.

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

Zhang, F., Su, Y. Strong Convergence of Modified Implicit Iteration Processes for Common Fixed Points of Nonexpansive Mappings. Fixed Point Theory Appl 2007, 048174 (2007). https://doi.org/10.1155/2007/48174

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/2007/48174

Keywords