Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014, Article ID 513678, 25 pages
http://dx.doi.org/10.1155/2014/513678
Research Article

Strong and Weak Convergence Criteria of Composite Iterative Algorithms for Systems of Generalized Equilibria

1Department of Mathematics, Shanghai Normal University and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China
2Department of Food and Beverage Management, Vanung University, Chung-Li 320061, Taiwan
3Department of Information Management, Yuan Ze University, Chung-Li 32003, Taiwan
4Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan
5Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

Received 20 January 2014; Accepted 12 February 2014; Published 25 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 Linéaires, Dunod, Paris, 1969.
  2. R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer, New York, NY, USA, 1984. View at MathSciNet
  3. W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000. View at MathSciNet
  4. J. T. Oden, Quantitative Methods on Nonlinear Mechanics, Prentice-Hall, Englewood Cliffs, NJ, USA, 1986.
  5. E. Zeidler, Nonlinear Functional Analysis and Its Applications, Springer, New York, NY, USA, 1985. View at MathSciNet
  6. L. C. Zeng, “Iterative algorithms for finding approximate solutions for general strongly nonlinear variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 187, no. 2, pp. 352–360, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. L. C. Zeng, “Iterative algorithm for finding approximate solutions to completely generalized strongly nonlinear quasivariational inequalities,” Journal of Mathematical Analysis and Applications, vol. 201, no. 1, pp. 180–194, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Y. Yao and J. C. Yao, “On modified iterative method for nonexpansive mappings and monotone mappings,” Applied Mathematics and Computation, vol. 186, no. 2, pp. 1551–1558, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. L. C. Ceng, Q. H. Ansari, and J. C. Yao, “Iterative methods for triple hierarchical variational inequalities in Hilbert spaces,” Journal of Optimization Theory and Applications, vol. 151, no. 3, pp. 489–512, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. L. C. Ceng, M. Teboulle, and J. C. Yao, “Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed-point problems,” Journal of Optimization Theory and Applications, vol. 146, no. 1, pp. 19–31, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. 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 Zentralblatt MATH · View at MathSciNet
  12. G. M. Korpelevič, “An extragradient method for finding saddle points and for other problems,” Èkonomika i Matematicheskie Metody, vol. 12, no. 4, pp. 747–756, 1976. View at Google Scholar · View at MathSciNet
  13. 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
  14. L. C. Zeng and J. C. Yao, “Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems,” Taiwanese Journal of Mathematics, vol. 10, no. 5, pp. 1293–1303, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. L. C. Ceng and J. C. Yao, “An extragradient-like approximation method for variational inequality problems and fixed point problems,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 205–215, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. N. Nadezhkina and W. Takahashi, “Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol. 128, no. 1, pp. 191–201, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. N. Nadezhkina and W. Takahashi, “Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings,” SIAM Journal on Optimization, vol. 16, no. 4, pp. 1230–1241, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. 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
  19. 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
  20. 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
  21. L. C. Ceng, N. Hadjisavvas, and N. C. Wong, “Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems,” Journal of Global Optimization, vol. 46, no. 4, pp. 635–646, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. S. Huang, “Hybrid extragradient methods for asymptotically strict pseudo-contractions in the intermediate sense and variational inequality problems,” Optimization, vol. 60, no. 6, pp. 739–754, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. 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
  24. 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
  25. 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
  26. G. Cai and S. 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
  27. 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
  28. 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, 19 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. S. Takahashi and W. Takahashi, “Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 3, pp. 1025–1033, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. Y. Yao, H. Zhou, and Y. C. Liou, “Weak and strong convergence theorems for an asymptotically k-strict pseudo-contraction and a mixed equilibrium problem,” Journal of the Korean Mathematical Society, vol. 46, no. 3, pp. 561–576, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  31. 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, no. 1, pp. 319–329, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. 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
  33. L. C. Ceng and J. C. Yao, “Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings,” Applied Mathematics and Computation, vol. 198, no. 2, pp. 729–741, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. L. C. Ceng, A. Petruşel, and J. C. Yao, “Iterative approaches to solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings,” Journal of Optimization Theory and Applications, vol. 143, no. 1, pp. 37–58, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. T. H. Kim and H. K. Xu, “Convergence of the modified Mann's iteration method for asymptotically strict pseudo-contractions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 9, pp. 2828–2836, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. D. R. Sahu, H. K. Xu, and J. C. Yao, “Asymptotically strict pseudocontractive mappings in the intermediate sense,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 10, pp. 3502–3511, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. 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
  38. K. Goebel and W. A. Kirk, Topics on Metric Fixed-Point Theory, Cambridge University Press, Cambridge, UK, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  39. 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
  40. J. Górnicki, “Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces,” Commentationes Mathematicae Universitatis Carolinae, vol. 30, no. 2, pp. 249–252, 1989. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. H. K. Xu, “Existence and convergence for fixed points of mappings of asymptotically nonexpansive type,” Nonlinear Analysis: Theory, Methods & Applications, vol. 16, no. 12, pp. 1139–1146, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  42. 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 · View at MathSciNet
  43. M. O. Osilike, S. C. Aniagbosor, and B. G. Akuchu, “Fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces,” Panamerican Mathematical Journal, vol. 12, no. 2, pp. 77–88, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  44. C. Martinez-Yanes and H. K. Xu, “Strong convergence of the CQ method for fixed point iteration processes,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 11, pp. 2400–2411, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  45. 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 Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet