Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2012, Article ID 536283, 20 pages
http://dx.doi.org/10.1155/2012/536283
Research Article

An Iterative Shrinking Projection Method for Solving Fixed Point Problems of Closed and 𝜙-Quasi-Strict Pseudocontractions along with Generalized Mixed Equilibrium Problems in Banach Spaces

1Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 22 April 2012; Revised 4 July 2012; Accepted 29 July 2012

Academic Editor: Simeon Reich

Copyright © 2012 Kasamsuk Ungchittrakool. 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. W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. H. H. Bauschke, E. Matoušková, and S. Reich, “Projection and proximal point methods: convergence results and counterexamples,” Nonlinear Analysis, vol. 56, no. 5, pp. 715–738, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. S. Reich, “Weak convergence theorems for nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 274–276, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372–379, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. T.-H. Kim and H.-K. Xu, “Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups,” Nonlinear Analysis, vol. 64, no. 5, pp. 1140–1152, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. G. Marino and H.-K. Xu, “Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 336–346, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. C. Martinez-Yanes and H.-K. Xu, “Strong convergence of the CQ method for fixed point iteration processes,” Nonlinear Analysis, vol. 64, no. 11, pp. 2400–2411, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. S. Plubtieng and K. Ungchittrakool, “Strong convergence of modified Ishikawa iteration for two asymptotically nonexpansive mappings and semigroups,” Nonlinear Analysis, vol. 67, no. 7, pp. 2306–2315, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. W. Takahashi, Y. Takeuchi, and R. Kubota, “Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 276–286, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. 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. Kartsatos, Ed., pp. 15–50, Marcel Dekker, New York, NY, USA, 1996. View at Google Scholar · View at Zentralblatt MATH
  11. 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
  12. S. Reich, “A weak convergence theorem for the alternating method with Bregman distances,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G. Kartsatos, Ed., pp. 313–318, Marcel Dekker, New York, NY, USA, 1996. View at Google Scholar · View at Zentralblatt MATH
  13. D. Butnariu, S. Reich, and A. J. Zaslavski, “Asymptotic behavior of relatively nonexpansive operators in Banach spaces,” Journal of Applied Analysis, vol. 7, no. 2, pp. 151–174, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. D. Butnariu, S. Reich, and A. J. Zaslavski, “Weak convergence of orbits of nonlinear operators in reflexive Banach spaces,” Numerical Functional Analysis and Optimization, vol. 24, no. 5-6, pp. 489–508, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. Y. Censor and S. Reich, “Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization,” Optimization, vol. 37, no. 4, pp. 323–339, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. S. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257–266, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. S. Plubtieng and K. Ungchittrakool, “Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 149, no. 2, pp. 103–115, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. S. Plubtieng and K. Ungchittrakool, “Strong convergence theorems of block iterative methods for a finite family of relatively nonexpansive mappings in Banach spaces,” Journal of Nonlinear and Convex Analysis, vol. 8, no. 3, pp. 431–450, 2007. View at Google Scholar · View at Zentralblatt MATH
  19. E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994. View at Google Scholar · View at Zentralblatt MATH
  20. S. Takahashi and W. Takahashi, “Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 506–515, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. A. Tada and W. Takahashi, “Strong convergence theorem for an equilibrium problem and a nonexpansive mapping,” in Nonlinear Analysis and Convex Analysis, W. Takahashi and T. Tanaka, Eds., pp. 609–617, Yokohama Publishers, Yokohama, Japan, 2007. View at Google Scholar · View at Zentralblatt MATH
  22. A. Tada and W. Takahashi, “Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem,” Journal of Optimization Theory and Applications, vol. 133, no. 3, pp. 359–370, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. W. Takahashi and K. Zembayashi, “Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2008, Article ID 528476, 11 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. W. Takahashi and K. Zembayashi, “Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces,” Nonlinear Analysis, vol. 70, no. 1, pp. 45–57, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. H. Zhou and X. Gao, “An iterative method of fixed points for closed and quasi-strict pseudo-contractions in Banach spaces,” Journal of Applied Mathematics and Computing, vol. 33, no. 1-2, pp. 227–237, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. H. Hudzik, W. Kowalewski, and G. Lewicki, “Approximate compactness and full rotundity in Musielak-Orlicz spaces and Lorentz-Orlicz spaces,” Zeitschrift für Analysis und ihre Anwendungen, vol. 25, no. 2, pp. 163–192, 2006. View at Publisher · View at Google Scholar
  27. I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers, Dodrecht, The Netherlands, 1990. View at Publisher · View at Google Scholar
  28. S. Reich, “Book review: geometry of Banach spaces, duality mappings and nonlinear problems by loana Cioranescu, Kluwer Academic Publishers, Dordrecht, 1990,” Bulletin of the American Mathematical Society, vol. 26, no. 2, pp. 367–370, 1992. View at Publisher · View at Google Scholar
  29. W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, 2000.
  30. Y. I. Alber and S. Reich, “An iterative method for solving a class of nonlinear operator equations in Banach spaces,” Panamerican Mathematical Journal, vol. 4, no. 2, pp. 39–54, 1994. View at Google Scholar
  31. S.-S. Zhang, “Generalized mixed equilibrium problem in Banach spaces,” Applied Mathematics and Mechanics, vol. 30, no. 9, pp. 1105–1112, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH