Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2012, Article ID 136134, 12 pages
http://dx.doi.org/10.1155/2012/136134
Research Article

Strong Convergence Theorems for a Countable Family of Total Quasi- -Asymptotically Nonexpansive Nonself Mappings

1College of Mathematics, Yibin University, Yibin 644000, China
2College of Statistics and Mathematics, Yunnan University of Finance and Economics, Yunnan, Kunming 650221, China

Received 3 June 2012; Revised 22 July 2012; Accepted 23 July 2012

Academic Editor: Naseer Shahzad

Copyright © 2012 Liang-cai Zhao and Shih-sen Chang. 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.

Linked References

  1. Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G. Kartosator, Ed., pp. 15–50, Marcel Dekker, New York, NY, USA, 1996. View at Google Scholar
  2. W. Nilsrakoo and S. Saejung, “Strong convergence theorems by Halpern-Mann iterations for relatively nonexpansive mappings in Banach spaces,” Applied Mathematics and Computation, vol. 217, no. 14, pp. 6577–6586, 2011. View at Publisher · View at Google Scholar
  3. C. E. Chidume, E. U. Ofoedu, and H. Zegeye, “Strong and weak convergence theorems for asymptotically nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 280, no. 2, pp. 364–374, 2003. View at Publisher · View at Google Scholar
  4. S. S. Chang, C. K. Chan, and H. W. Joseph Lee, “Modified block iterative algorithm for quasi-ϕ-asymptotically nonexpansive mappings and equilibrium problem in Banach spaces,” Applied Mathematics and Computation, vol. 217, no. 18, pp. 7520–7530, 2011. View at Publisher · View at Google Scholar
  5. S. S. Chang, H. W. Joseph Lee, and C. K. Chan, “A new hybrid method for solving a generalized equilibrium problem, solving a variational inequality problem and obtaining common fixed points in Banach spaces, with applications,” Nonlinear Analysis, vol. 73, no. 7, pp. 2260–2270, 2010. View at Publisher · View at Google Scholar
  6. S. S. Chang, H. W. Joseph Lee, C. K. Chan, and L. Yang, “Approximation theorems for total quasi-ϕ-asymptotically nonexpansive mappings with applications,” Applied Mathematics and Computation, vol. 218, no. 6, pp. 2921–2931, 2011. View at Publisher · View at Google Scholar
  7. S. S. Chang, L. Wang, Y. K. Tang, B. Wang, and L. J. Qin, “Strong convergence theorems for a countable family of quasi-ϕ-asymptotically nonexpansive nonself mappings,” Applied Mathematics and Computation, vol. 218, no. 15, pp. 7864–7870, 2012. View at Publisher · View at Google Scholar
  8. W. P. Guo and W. Guo, “Weak convergence theorems for asymptotically nonexpansive nonself-mappings,” Applied Mathematics Letters, vol. 24, no. 12, pp. 2181–2185, 2011. View at Publisher · View at Google Scholar
  9. Y. Hao, S. Y. Cho, and X. Qin, “Some weak convergence theorems for a family of asymptotically nonexpansive nonself mappings,” Fixed Point Theory and Applications, Article ID 218573, 11 pages, 2010. View at Google Scholar
  10. S. Kamimura and W. Takahashi, “Strong convergence of a proximal-type algorithm in a Banach space,” SIAM Journal on Optimization, vol. 13, no. 3, pp. 938–945, 2002. View at Publisher · View at Google Scholar
  11. H. Kiziltunc and S. Temir, “Convergence theorems by a new iteration process for a finite family of nonself asymptotically nonexpansive mappings with errors in Banach spaces,” Computers & Mathematics with Applications, vol. 61, no. 9, pp. 2480–2489, 2011. View at Publisher · View at Google Scholar
  12. H. K. Pathak, Y. J. Cho, and S. M. Kang, “Strong and weak convergence theorems for nonself-asymptotically perturbed nonexpansive mappings,” Nonlinear Analysis, vol. 70, no. 5, pp. 1929–1938, 2009. View at Publisher · View at Google Scholar
  13. X. Qin, S. Y. Cho, T. Wang, and S. M. Kang, “Convergence of an implicit iterative process for asymptotically pseudocontractive nonself-mappings,” Nonlinear Analysis, vol. 74, no. 17, pp. 5851–5862, 2011. View at Publisher · View at Google Scholar
  14. Y. F. Su, H. K. Xu, and X. Zhang, “Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications,” Nonlinear Analysis, vol. 73, no. 12, pp. 3890–3906, 2010. View at Publisher · View at Google Scholar
  15. S. Thianwan, “Common fixed points of new iterations for two asymptotically nonexpansive nonself-mappings in a Banach space,” Journal of Computational and Applied Mathematics, vol. 224, no. 2, pp. 688–695, 2009. View at Publisher · View at Google Scholar
  16. Z. M. Wang, Y. F. Su, D. X. Wang, and Y. C. Dong, “A modified Halpern-type iteration algorithm for a family of hemi-relatively nonexpansive mappings and systems of equilibrium problems in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 235, no. 8, pp. 2364–2371, 2011. View at Publisher · View at Google Scholar
  17. I. Yıldırım and M. Özdemir, “A new iterative process for common fixed points of finite families of non-self-asymptotically non-expansive mappings,” Nonlinear Analysis, vol. 71, no. 3-4, pp. 991–999, 2009. View at Publisher · View at Google Scholar
  18. L. P. Yang and X. S. Xie, “Weak and strong convergence theorems of three step iteration process with errors for nonself-asymptotically nonexpansive mappings,” Mathematical and Computer Modelling, vol. 52, no. 5-6, pp. 772–780, 2010. View at Publisher · View at Google Scholar
  19. H. Zegeye, E. U. Ofoedu, and N. Shahzad, “Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings,” Applied Mathematics and Computation, vol. 216, no. 12, pp. 3439–3449, 2010. View at Publisher · View at Google Scholar
  20. P. Kanjanasamranwong, P. Kumam, and S. Saewan, “A modified Halpern type iterative method of a system of equilibrium problems and a fixed point for a totally quasi-phi-asymptotically non expansive mapping in a Banach space,” Journal of Applied Mathematics, vol. 2012, Article ID 750732, 19 pages, 2012. View at Google Scholar
  21. S. Saewan and P. Kumam, “A strong convergence theorem concerning a hybrid projection method for finding common fixed points of a countable family of relatively quasi-nonexpansive mappings,” Journal of Nonlinear and Convex Analysis, vol. 13, no. 2, pp. 313–330, 2012. View at Google Scholar
  22. S. Saewan and P. Kumam, “Convergence theorems for mixed equilibrium problems, variational inequality problem and uniformly quasi-ϕ-asymptotically nonexpansive mappings,” Applied Mathematics and Computation, vol. 218, no. 7, pp. 3522–3538, 2011. View at Publisher · View at Google Scholar
  23. S. Saewan and P. Kumam, “Convergence theorems for uniformly quasi-ϕ- asymptotically nonexpansive mappings, generalized equilibrium problems and variational inequalities,” Journal of Inequalities and Applications, vol. 2011, article 96, 2011. View at Google Scholar
  24. S. Saewan and P. Kumam, “A new modified block iterative algorithm for uniformly quasi-ϕ-asymptotically nonexpansive mappings and a system of generalized mixed equilibrium problems,” Fixed Point Theory and Applications, vol. 2011, article 35, 2011. View at Google Scholar
  25. K. Wattanawitoon and P. Kumam, “The modified block iterative algorithms for asymptotically relatively nonexpansive mappings and the system of generalized mixed equilibrium problems,” Journal of Applied Mathematics, vol. 2012, Article ID 395760, 24 pages, 2012. View at Publisher · View at Google Scholar