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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 147915, 16 pages
http://dx.doi.org/10.1155/2012/147915
Research Article

Strong Convergence Theorems for a Generalized Mixed Equilibrium Problem and a Family of Total Quasi- 𝝓 -Asymptotically Nonexpansive Multivalued Mappings in Banach Spaces

1Department of Mathematics, Yibin University, Yibin 644007, China
2College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, China

Received 30 December 2011; Accepted 2 February 2012

Academic Editor: Khalida Inayat Noor

Copyright © 2012 J. F. Tan and S. S. 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. L.-C. Ceng and J.-C. Yao, “A hybrid iterative scheme for mixed equilibrium problems and fixed point problems,” Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 186–201, 2008. View at Publisher · View at Google Scholar
  2. F. E. Browder, “Existence and approximation of solutions of nonlinear variational inequalities,” Proceedings of the National Academy of Sciences of the United States of America, vol. 56, pp. 1080–1086, 1966.
  3. I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990.
  4. 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., vol. 178, pp. 15–50, Marcel Dekker, New York, NY, USA, 1996.
  5. S.-y. 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
  6. S. Plubtieng and K. Ungchittrakool, “Hybrid iterative methods for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2008, Article ID 583082, 19 pages, 2008. View at Publisher · View at Google Scholar
  7. S.-S. Chang, J. K. Kim, and X. R. Wang, “Modified block iterative algorithm for solving convex feasibility problems in Banach spaces,” Journal of Inequalities and Applications, vol. 2010, Article ID 869684, 14 pages, 2010. View at Publisher · View at Google Scholar
  8. 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
  9. S. Homaeipour and A. Razani, “Weak and strong convergence theorems for relatively nonexpansive multi-valued mappings in Banach spaces,” Fixed Point Theorem and Applications, vol. 73, 2011. View at Publisher · View at Google Scholar
  10. 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
  11. J. F. Tang, “Strong convergence theorem for a generalized mixed equilibrium problem and a family of quasi-φ-asymptotically nonexpansive mappings,” Acta Mathematicae Applicatae Sinica, vol. 33, no. 5, pp. 878–888, 2010.
  12. R. P. Agarwal, Y. J. Cho, and N. Petrot, “Systems of general nonlinear set-valued mixed variational inequalities problems in Hilbert spaces,” Fixed Point Theory and Applications, vol. 31, 2011. View at Publisher · View at Google Scholar
  13. Y. J. Cho and X. Qin, “Systems of generalized nonlinear variational inequalities and its projection methods,” Nonlinear Analysis, vol. 69, no. 12, pp. 4443–4451, 2008. View at Publisher · View at Google Scholar
  14. Y. J. Cho, X. Qin, and J. I. Kang, “Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems,” Nonlinear Analysis, vol. 71, no. 9, pp. 4203–4214, 2009. View at Publisher · View at Google Scholar
  15. Y. J. Cho, I. K. Argyros, and N. Petrot, “Approximation methods for common solutions of generalized equilibrium, systems of nonlinear variational inequalities and fixed point problems,” Computers & Mathematics with Applications, vol. 60, no. 8, pp. 2292–2301, 2010. View at Publisher · View at Google Scholar
  16. Y. J. Cho and N. Petrot, “On the system of nonlinear mixed implicit equilibrium problems in Hilbert spaces,” Journal of Inequalities and Applications, vol. 2010, Article ID 437976, 12 pages, 2010. View at Publisher · View at Google Scholar
  17. Y. J. Cho and N. Petrot, “Regularization and iterative method for general variational inequality problem in Hilbert spaces,” Journal of Inequalities and Applications, vol. 21, 2011. View at Publisher · View at Google Scholar
  18. H. He, S. Liu, and Y. J. Cho, “An explicit method for systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings,” Journal of Computational and Applied Mathematics, vol. 235, no. 14, pp. 4128–4139, 2011. View at Publisher · View at Google Scholar
  19. X. Qin, S.-s. Chang, and Y. J. Cho, “Iterative methods for generalized equilibrium problems and fixed point problems with applications,” Nonlinear Analysis, vol. 11, no. 4, pp. 2963–2972, 2010. View at Publisher · View at Google Scholar
  20. Y. Yao, Y. J. Cho, and Y.-C. Liou, “Iterative algorithms for variational inclusions, mixed equilibrium and fixed point problems with application to optimization problems,” Central European Journal of Mathematics, vol. 9, no. 3, pp. 640–656, 2011. View at Publisher · View at Google Scholar
  21. Y. Yao, Y. J. Cho, and Y.-C. Liou, “Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems,” European Journal of Operational Research, vol. 212, no. 2, pp. 242–250, 2011. View at Publisher · View at Google Scholar
  22. M. A. Noor, “Some developments in general variational inequalities,” Applied Mathematics and Computation, vol. 152, no. 1, pp. 199–277, 2004. View at Publisher · View at Google Scholar
  23. M. A. Noor, “General variational inequalities,” Applied Mathematics Letters, vol. 1, no. 2, pp. 119–122, 1988. View at Publisher · View at Google Scholar
  24. M. A. Noor, “Some aspects of extended general variational inequalities,” Abstract and Applied Analysis. In press.
  25. M. A. Noor, “Extended general variational inequalities,” Applied Mathematics Letters, vol. 22, no. 2, pp. 182–186, 2009. View at Publisher · View at Google Scholar
  26. M. A. Noor, K. I. Noor, and E. Al-Said, “Iterative methods for solving nonconvex equilibrium variational inequalities,” Applied Mathematics & Information Sciences, vol. 6, no. 1, pp. 65–69, 2012.
  27. 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.
  28. 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