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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 540381, 24 pages
http://dx.doi.org/10.1155/2014/540381
Research Article

Iterative Algorithms for Systems of Generalized Equilibrium Problems with the Constraints of Variational Inclusion and Fixed Point Problems

1Department of Mathematics, Shanghai Normal University, and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China
2Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 6 December 2013; Accepted 11 January 2014; Published 12 March 2014

Academic Editor: Jen-Chih Yao

Copyright © 2014 Lu-Chuan Ceng 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. J. L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites non Linéaire, Dunod, Paris, France, 1969.
  2. J.-W. Peng and J.-C. Yao, “A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems,” Taiwanese Journal of Mathematics, vol. 12, no. 6, pp. 1401–1432, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. Q. H. Ansari and J.-C. Yao, “Systems of generalized variational inequalities and their applications,” Applicable Analysis, vol. 76, no. 3-4, pp. 203–217, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. Q. H. Ansari, S. Schaible, and J.-C. Yao, “The system of generalized vector equilibrium problems with applications,” Journal of Global Optimization, vol. 22, no. 1–4, pp. 3–16, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. L.-J. Lin and Q. H. Ansari, “Collective fixed points and maximal elements with applications to abstract economies,” Journal of Mathematical Analysis and Applications, vol. 296, no. 2, pp. 455–472, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. S. Al-Homidan, Q. H. Ansari, and S. Schaible, “Existence of solutions of systems of generalized implicit vector variational inequalities,” Journal of Optimization Theory and Applications, vol. 134, no. 3, pp. 515–531, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. S. Al-Homidan, Q. H. Ansari, and J.-C. Yao, “Collectively fixed point and maximal element theorems in topological semilattice spaces,” Applicable Analysis, vol. 90, no. 6, pp. 865–888, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. S. Al-Homidan and Q. H. Ansari, “Fixed point theorems on product topological semilattice spaces, generalized abstract economies and systems of generalized vector quasi-equilibrium problems,” Taiwanese Journal of Mathematics, vol. 15, no. 1, pp. 307–330, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. G. Cai and S. Q. Bu, “Strong and weak convergence theorems for general mixed equilibrium problems and variational inequality problems and fixed point problems in Hilbert spaces,” Journal of Computational and Applied Mathematics, vol. 247, pp. 34–52, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. L. C. Ceng, H.-Y. Hu, and M. M. Wong, “Strong and weak convergence theorems for generalized mixed equilibrium problem with perturbation and fixed pointed problem of infinitely many nonexpansive mappings,” Taiwanese Journal of Mathematics, vol. 15, no. 3, pp. 1341–1367, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. L.-C. Ceng, Q. H. Ansari, and S. Schaible, “Hybrid extragradient-like methods for generalized mixed equilibrium problems, systems of generalized equilibrium problems and optimization problems,” Journal of Global Optimization, vol. 53, no. 1, pp. 69–96, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. L.-C. Ceng, S.-M. Guu, and J.-C. Yao, “Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems,” Fixed Point Theory and Applications, vol. 2012, article 92, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. L.-C. Ceng and A. Petruşel, “Relaxed extragradient-like method for general system of generalized mixed equilibria and fixed point problem,” Taiwanese Journal of Mathematics, vol. 16, no. 2, pp. 445–478, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. Y. Yao, Y.-C. Liou, and J.-C. Yao, “New relaxed hybrid-extragradient method for fixed point problems, a general system of variational inequality problems and generalized mixed equilibrium problems,” Optimization, vol. 60, no. 3, pp. 395–412, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. 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 · View at Zentralblatt MATH · View at MathSciNet
  16. L.-C. Ceng, Q. H. Ansari, S. Schaible, and J.-C. Yao, “Iterative methods for generalized equilibrium problems, systems of general generalized equilibrium problems and fixed point problems for nonexpansive mappings in Hilbert spaces,” Fixed Point Theory, vol. 12, no. 2, pp. 293–308, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. L.-C. Ceng and J.-C. Yao, “A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1922–1937, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. L.-C. Ceng, C.-Y. Wang, and J.-C. Yao, “Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities,” Mathematical Methods of Operations Research, vol. 67, no. 3, pp. 375–390, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. R. T. Rockafellar, “Monotone operators and the proximal point algorithm,” SIAM Journal on Control and Optimization, vol. 14, no. 5, pp. 877–898, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. 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 · View at Zentralblatt MATH · View at MathSciNet
  21. J. G. O'Hara, P. Pillay, and H.-K. Xu, “Iterative approaches to convex feasibility problems in Banach spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 64, no. 9, pp. 2022–2042, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. Y. Yao, Y.-C. Liou, and J.-C. Yao, “Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2007, Article ID 64363, 12 pages, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28, Cambridge University Press, Cambridge, UK, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  24. T. Suzuki, “Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227–239, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. N.-J. Huang, “A new completely general class of variational inclusions with noncompact valued mappings,” Computers & Mathematics with Applications, vol. 35, no. 10, pp. 9–14, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. L.-C. Zeng, S.-M. Guu, and J.-C. Yao, “Characterization of H-monotone operators with applications to variational inclusions,” Computers & Mathematics with Applications, vol. 50, no. 3-4, pp. 329–337, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  28. L.-C. Ceng, Q. H. Ansari, M. M. Wong, and J.-C. Yao, “Mann type hybrid extragradient method for variational inequalities, variational inclusions and fixed point problems,” Fixed Point Theory, vol. 13, no. 2, pp. 403–422, 2012. View at Google Scholar · View at MathSciNet
  29. Y. Yao, M. A. Noor, S. Zainab, and Y. C. Liou, “Mixed equilibrium problems and optimization problems,” Journal of Mathematical Analysis and Applications, vol. 354, pp. 319–329, 2009. View at Google Scholar