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Journal of Applied Mathematics
Volume 2014, Article ID 478172, 9 pages
http://dx.doi.org/10.1155/2014/478172
Research Article

Iterative Computation for Solving the Variational Inequality and the Generalized Equilibrium Problem

1School of Science, Tianjin Polytechnic University, Tianjin 300387, China
2College of Management and Economics, Tianjin University, Tianjin 300072, China
3Department of Mathematics and the RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea
4Department of Mathematics, Dong-A University, Pusan 614-714, Republic of Korea

Received 22 March 2014; Revised 7 May 2014; Accepted 13 May 2014; Published 27 May 2014

Academic Editor: Giuseppe Marino

Copyright © 2014 Xiujuan Pan 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. 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 · View at MathSciNet
  2. P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 117–136, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. 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
  4. G. Stampacchia, “Formes bilinéaires coercitives sur les ensembles convexes,” Comptes Rendus de l'Académie des Sciences, vol. 258, pp. 4413–4416, 1964. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. 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
  6. R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer, New York, NY, USA, 1984. View at MathSciNet
  7. A. N. Iusem, “An iterative algorithm for the variational inequality problem,” Computational and Applied Mathematics, vol. 13, no. 2, pp. 103–114, 1994. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. 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 · View at Zentralblatt MATH · View at MathSciNet
  9. F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume 1, Springer Series in Operations Research, Springer, New York, NY, USA, 2003. View at MathSciNet
  10. F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume 2, Springer Series in Operations Research, Springer, New York, NY, USA, 2003. View at MathSciNet
  11. H. K. Xu and T. H. Kim, “Convergence of hybrid steepest-descent methods for variational inequalities,” Journal of Optimization Theory and Applications, vol. 119, no. 1, pp. 185–201, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. C. Yao, “Variational inequalities with generalized monotone operators,” Mathematics of Operations Research, vol. 19, no. 3, pp. 691–705, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. 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
  14. L.-C. Ceng, S. Al-Homidan, Q. H. Ansari, and J.-C. Yao, “An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings,” Journal of Computational and Applied Mathematics, vol. 223, no. 2, pp. 967–974, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. M. A. Noor, “New approximation schemes for general variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 217–229, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, vol. 9 of SIAM Studies in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1989. View at MathSciNet
  17. S. Haubruge, V. H. Nguyen, and J. J. Strodiot, “Convergence analysis and applications of the Glowinski-Le Tallec splitting method for finding a zero of the sum of two maximal monotone operators,” Journal of Optimization Theory and Applications, vol. 97, no. 3, pp. 645–673, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. Y. Censor and T. Elfving, “A multiprojection algorithm using Bregman projections in a product space,” Numerical Algorithms, vol. 8, no. 2–4, pp. 221–239, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. C. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem,” Inverse Problems, vol. 18, no. 2, pp. 441–453, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. H.-K. Xu, “Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces,” Inverse Problems, vol. 26, no. 10, Article ID 105018, 17 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. T. Suzuki, “Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces,” Fixed Point Theory and Applications, no. 1, pp. 103–123, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. 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