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  • Research Article
  • Open Access

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

Fixed Point Theory and Applications20072007:048174

  • Received: 21 December 2006
  • Accepted: 19 March 2007
  • Published:


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.


  • Hilbert Space
  • Differential Geometry
  • Convergence Theorem
  • Hybrid Method
  • Nonexpansive Mapping


Authors’ Affiliations

Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300160, China


  1. Byrne C: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems 2004,20(1):103–120. 10.1088/0266-5611/20/1/006MathSciNetView ArticleMATHGoogle 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.7.000517View ArticleGoogle 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-LView ArticleGoogle 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)90024-6MathSciNetView ArticleMATHGoogle Scholar
  7. Genel A, Lindenstrauss J: An example concerning fixed points. Israel Journal of Mathematics 1975,22(1):81–86. 10.1007/BF02757276MathSciNetView ArticleMATHGoogle 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)00458-4MathSciNetView ArticleMATHGoogle 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-100105317MathSciNetView ArticleMATHGoogle 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.2004.01.038MathSciNetView ArticleMATHGoogle 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/ ArticleMATHGoogle 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.View ArticleMATHGoogle Scholar


© F. Zhang and Y. Su. 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.