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Journal of Applied Mathematics
Volume 2014 (2014), Article ID 193749, 10 pages
http://dx.doi.org/10.1155/2014/193749
Research Article

New Mixed Equilibrium Problems and Iterative Algorithms for Fixed Point Problems in Banach Spaces

1School of Mathematics and Sciences, Shijiazhuang University of Economics, Shijiazhuang 050031, China
2Department of Mathematics and Physics, North China Electric Power University, Baoding 071003, China

Received 30 October 2013; Accepted 16 December 2013; Published 9 January 2014

Academic Editor: Li Wei

Copyright © 2014 Minjiang Chen et al. 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. Takahashi and K. Zembayashi, “Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces,” Nonlinear Analysis, Theory, Methods and Applications, vol. 70, no. 1, pp. 45–57, 2009. View at Publisher · View at Google Scholar · View at Scopus
  2. X. L. Qin, Y. J. Cho, and S. M. Kang, “Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 225, no. 1, pp. 20–30, 2009. View at Publisher · View at Google Scholar · View at Scopus
  3. Y. P. Fang and N. J. Huang, “Variational-like inequalities with generalized monotone mappings in Banach spaces,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 327–338, 2003. View at Publisher · View at Google Scholar · View at Scopus
  4. D. Goeleven and D. Motreanu, “Eigenvalue and dynamic problems for variational and hemivariational inequalities,” Communications on Applied Nonlinear Analysis, vol. 3, pp. 1–21, 1996. View at Google Scholar
  5. A. H. Saddqi, Q. H. Ansari, and K. R. Kazmi, “On nonlinear variational inequalities,” Indian Journal of Pure and Applied Mathematics, vol. 25, pp. 969–973, 1994. View at Google Scholar
  6. R. U. Verma, “Nonlinear variational inequalities on convex subsets of banach spaces,” Applied Mathematics Letters, vol. 10, no. 4, pp. 25–27, 1997. View at Google Scholar · View at Scopus
  7. S. H. Wang, G. Marino, and F. H. Wang, “Strong convergence theorems for a generalized equilibrium problem with a relaxed monotone mapping and a countable family of nonexpansive mappings in a Hilbert space,” Fixed Point Theory and Applications, vol. 2010, Article ID 230304, 22 pages, 2010. View at Publisher · View at Google Scholar · View at Scopus
  8. Y. L. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theorem and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G. Kartsatos, Ed., pp. 15–50, Marcel Dekker, New York, NY, USA, 1996. View at Google Scholar
  9. I. Cioranescu, Geometry of Banach Spaces, Duaity Mappings and Nonlinear Problems, Kluwer, Dordrecht, The Netherlands, 1990.
  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, 2003. View at Publisher · View at Google Scholar · View at Scopus
  11. W. Takahashi, Nonlinear Functional Analysis, Yokohama, Kanagawa, Japan, 2000.
  12. Y. L. Alber and S. Guerre-Delabriere, “On the projection methods for fixed point problems,” Analysis, vol. 21, pp. 17–39, 2001. View at Google Scholar
  13. S. Reich, “A weak convergence theorem for the alternating method with Bregman distance,” in Theory and Applications of NonlInear Operators of Accretive and Montone Type, A. G. Kartsatos, Ed., pp. 313–318, Marcel Dekker, New York, NY, USA, 1996. View at Google Scholar
  14. D. Butnariu, S. Reich, and A. J. Zaslavski, “Asymptotic behavior of relatively nonexpansiv operators in Banach spaces,” Journal of Applied Analysis, vol. 7, pp. 151–174, 2001. View at Google Scholar
  15. 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 Google Scholar · View at Scopus
  16. 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 Google Scholar · View at Scopus
  17. H. Y. Zhou and X. H. Gao, “A strong convergence theorem for a family of quasi-ϕ-nonexpansive mappings in a banach space,” Fixed Point Theory and Applications, vol. 2009, Article ID 351265, 12 pages, 2009. View at Publisher · View at Google Scholar · View at Scopus
  18. Y. J. Cho, H. Zhou, and G. Guo, “Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings,” Computers and Mathematics with Applications, vol. 47, no. 4-5, pp. 707–717, 2004. View at Google Scholar · View at Scopus
  19. K. Fan, “Some properties of convex sets related to fixed point theorems,” Mathematische Annalen, vol. 266, no. 4, pp. 519–537, 1984. View at Publisher · View at Google Scholar · View at Scopus
  20. R. Wangkeeree and R. Wangkeeree, “A general iterative method for variational inequality problems, mixed equilibrium problems, and fixed point problems of strictly pseudocontractive mappings in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 519065, 32 pages, 2009. View at Publisher · View at Google Scholar · View at Scopus
  21. 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 Scopus