Open Access

Implicit predictor-corrector iteration process for finitely many asymptotically (quasi-)nonexpansive mappings

Journal of Inequalities and Applications20062006:65983

https://doi.org/10.1155/JIA/2006/65983

Received: 13 February 2006

Accepted: 5 June 2006

Published: 3 October 2006

Abstract

We study an implicit predictor-corrector iteration process for finitely many asymptotically quasi-nonexpansive self-mappings on a nonempty closed convex subset of a Banach space . We derive a necessary and sufficient condition for the strong convergence of this iteration process to a common fixed point of these mappings. In the case is a uniformly convex Banach space and the mappings are asymptotically nonexpansive, we verify the weak (resp., strong) convergence of this iteration process to a common fixed point of these mappings if Opial's condition is satisfied (resp., one of these mappings is semicompact). Our results improve and extend earlier and recent ones in the literature.

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Authors’ Affiliations

(1)
Department of Mathematics, Shanghai Normal University
(2)
Department of Applied Mathematics, National Sun Yat-sen University

References

  1. 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/S0002-9939-1978-0493543-4MathSciNetView ArticleMATHGoogle Scholar
  2. Chang S-S, Cho YJ, Zhou H: Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings. Journal of the Korean Mathematical Society 2001,38(6):1245–1260.MathSciNetMATHGoogle Scholar
  3. 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/S0002-9939-1972-0298500-3MathSciNetView ArticleMATHGoogle Scholar
  4. Köthe G: Topological Vector Spaces. I, Die Grundlehren der mathematischen Wissenschaften. Volume 159. Springer, New York; 1969:xv+456.Google Scholar
  5. 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/S0002-9904-1967-11761-0MathSciNetView ArticleMATHGoogle Scholar
  6. 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/S0004972700028884MathSciNetView ArticleMATHGoogle Scholar
  7. Sun Z-H: 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)00537-7MathSciNetView ArticleMATHGoogle Scholar
  8. 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
  9. Zhang SS: On the iterative approximation problem of fixed points for asymptotically nonexpansive type mappings in Banach spaces. Applied Mathematics and Mechanics (English Edition) 2001,22(1):25–34.MathSciNetView ArticleMATHGoogle Scholar
  10. Zhou Y, Chang S-S: Convergence of implicit iteration process for a finite family of asymptotically nonexpansive mappings in Banach spaces. Numerical Functional Analysis and Optimization 2002,23(7–8):911–921. 10.1081/NFA-120016276MathSciNetView ArticleMATHGoogle Scholar

Copyright

© L. C. Ceng et al. 2006

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.