Open Access

Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces

Fixed Point Theory and Applications20062006:18909

https://doi.org/10.1155/FPTA/2006/18909

Received: 21 April 2005

Accepted: 18 July 2005

Published: 23 February 2006

Abstract

Suppose is a nonempty closed convex subset of a real Banach space . Let be two asymptotically quasi-nonexpansive maps with sequences such that and , and . Suppose is generated iteratively by where and are real sequences in . It is proved that (a) converges strongly to some if and only if ; (b) if is uniformly convex and if either or is compact, then converges strongly to some . Furthermore, if is uniformly convex, either or is compact and is generated by , where , are bounded, are real sequences in such that and , are summable; it is established that the sequence (with error member terms) converges strongly to some .

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Authors’ Affiliations

(1)
Department of Mathematics, King Abdul Aziz University
(2)
Department of Mathematics/Statistics/Computer Science, University of Port Harcourt, PMB

References

  1. Chidume CE: Iterative algorithms for nonexpansive mappings and some of their generalizations. In Nonlinear Analysis and Applications: to V. Lakshmikantham on His 80th Birthday. Vol. 1, 2. Edited by: Agarwal RPet al.. Kluwer Academic, Dordrecht; 2003:383–429.Google Scholar
  2. 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.1997.5268MathSciNetView ArticleMATHGoogle Scholar
  3. Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proceedings of the American Mathematical Society 1972, 35: 171–174. 10.1090/S0002-9939-1972-0298500-3MathSciNetView ArticleMATHGoogle Scholar
  4. Goebel K, Kirk WA: A fixed point theorem for transformations whose iterates have uniform Lipschitz constant. Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica 1973, 47: 135–140.MathSciNetMATHGoogle Scholar
  5. Ishikawa S: Fixed points by a new iteration method. Proceedings of the American Mathematical Society 1974, 44: 147–150. 10.1090/S0002-9939-1974-0336469-5MathSciNetView ArticleMATHGoogle Scholar
  6. Khan SH, Takahashi W: Approximating common fixed points of two asymptotically nonexpansive mappings. Scientiae Mathematicae Japonicae 2001,53(1):143–148.MathSciNetMATHGoogle Scholar
  7. Mann WR: Mean value methods in iteration. Proceedings of the American Mathematical Society 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3MathSciNetView ArticleMATHGoogle Scholar
  8. 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: 459–497. 10.1016/0022-247X(73)90087-5MathSciNetView ArticleMATHGoogle Scholar
  9. Qihou L: Iterative sequences for asymptotically quasi-nonexpansive mappings. Journal of Mathematical Analysis and Applications 2001,259(1):1–7. 10.1006/jmaa.2000.6980MathSciNetView ArticleMATHGoogle Scholar
  10. 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.2000.7353MathSciNetView ArticleMATHGoogle Scholar
  11. 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.2001.7629MathSciNetView ArticleMATHGoogle Scholar
  12. Rhoades BE: Fixed point iterations for certain nonlinear mappings. Journal of Mathematical Analysis and Applications 1994,183(1):118–120. 10.1006/jmaa.1994.1135MathSciNetView ArticleMATHGoogle Scholar
  13. 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
  14. 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.1993.1309MathSciNetView ArticleMATHGoogle Scholar
  15. 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/S0002-9939-1994-1203993-5MathSciNetView ArticleMATHGoogle Scholar
  16. Xu HK: Inequalities in Banach spaces with applications. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 1991,16(12):1127–1138. 10.1016/0362-546X(91)90200-KMathSciNetView ArticleMATHGoogle Scholar

Copyright

© Shahzad and Udomene 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.