Table of Contents
ISRN Applied Mathematics
Volume 2012, Article ID 482869, 22 pages
http://dx.doi.org/10.5402/2012/482869
Research Article

Hybrid Projection Algorithm for a New General System of Variational Inequalities in Hilbert Spaces

1Department of Mathematics and Statistics, Faculty of Science, Thaksin University, Phatthalung Campus, Phatthalung 93110, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 5 November 2012; Accepted 18 December 2012

Academic Editors: I. K. Argyros, H. Y. Chung, and Y.-G. Zhao

Copyright © 2012 S. Imnang. 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. H. Piri, “A general iterative method for finding common solutions of system of equilibrium problems, system of variational inequalities and fixed point problems,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1622–1638, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  2. X. Qin, M. Shang, and Y. Su, “Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems,” Mathematical and Computer Modelling, vol. 48, no. 7-8, pp. 1033–1046, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. Y. Shehu, “Iterative method for fixed point problem, variational inequality and generalized mixed equilibrium problems with applications,” Journal of Global Optimization, vol. 52, no. 1, pp. 57–77, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. R. Wangkeeree and P. Preechasilp, “A new iterative scheme for solving the equilibrium problems, variational inequality problems, and fixed point problems in Hilbert spaces,” Journal of Applied Mathematics, vol. 2012, Article ID 154968, 21 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. Y. Yao, Y. J. Cho, and R. Chen, “An iterative algorithm for solving fixed point problems, variational inequality problems and mixed equilibrium problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 3363–3373, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. Y. Yao, Y.-C. Liou, M.-M. Wong, and J.-C. Yao, “Strong convergence of a hybrid method for monotone variational inequalities and fixed point problems,” Fixed Point Theory and Applications, vol. 2011, article 53, 2011. View at Google Scholar · View at MathSciNet
  7. 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
  8. S.-S. Chang, H. W. Joseph Lee, and C. K. Chan, “A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3307–3319, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. K. R. Kazmi and S. H. Rizvi, “A hybrid extragradient method for approximating the common solutions of a variational inequality, a system of variational inequalities, a mixed equilibrium problem and a fixed point problem,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5439–5452, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. 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
  11. R. U. Verma, “On a new system of nonlinear variational inequalities and associated iterative algorithms,” Mathematical Sciences Research Hot-Line, vol. 3, no. 8, pp. 65–68, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. L. C. Ceng, M. M. Wong, and A. Latif, “Generalized extragradient iterative method for systems of variational inequalities,” Journal of Inequalities and Applications, vol. 2012, article 88, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  13. 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 · View at Zentralblatt MATH · View at MathSciNet
  14. P. Kumam and P. Katchang, “A system of mixed equilibrium problems, a general system of variational inequality problems for relaxed cocoercive, and fixed point problems for nonexpansive semigroup and strictly pseudocontractive mappings,” Journal of Applied Mathematics, vol. 2012, Article ID 414831, 35 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. R. Wangkeeree and U. Kamraksa, “An iterative approximation method for solving a general system of variational inequality problems and mixed equilibrium problems,” Nonlinear Analysis: Hybrid Systems, vol. 3, no. 4, pp. 615–630, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. Y. Yao, Y.-C. Liou, and S. M. Kang, “Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3472–3480, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. 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
  18. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, UK, 1990. View at MathSciNet
  19. J.-W. Peng and J.-C. Yao, “Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems,” Mathematical and Computer Modelling, vol. 49, no. 9-10, pp. 1816–1828, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. M. O. Osilike and D. I. Igbokwe, “Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations,” Computers & Mathematics with Applications, vol. 40, no. 4-5, pp. 559–567, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. 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
  23. A. Kangtunyakarn and S. Suantai, “Hybrid iterative scheme for generalized equilibrium problems and fixed point problems of finite family of nonexpansive mappings,” Nonlinear Analysis: Hybrid Systems, vol. 3, no. 3, pp. 296–309, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet