About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 153456, 16 pages
http://dx.doi.org/10.1155/2012/153456
Research Article

Some Algorithms for Finding Fixed Points and Solutions of Variational Inequalities

Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea

Received 1 December 2011; Accepted 9 January 2012

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 Jong Soo Jung. 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. G. L. Acedo and H.-K. Xu, “Iterative methods for strict pseudo-contractions in Hilbert spaces,” Nonlinear Analysis, vol. 67, no. 7, pp. 2258–2271, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. Y. J. Cho, S. M. Kang, and X. Qin, “Some results on k-pseudo-contractive mappings in Hilbert spaces,” Nonlinear Analysis, vol. 70, no. 5, pp. 1956–1964, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  3. J. S. Jung, “Strong convergence of iterative methods for k-strictly pseudo-contractive mappings in Hilbert spaces,” Applied Mathematics and Computation, vol. 215, no. 10, pp. 3746–3753, 2010. View at Publisher · View at Google Scholar
  4. J. S. Jung, “Some results on a general iterative method for k-strictly pseudo-contractive mappings,” Fixed Point Theory and Applications, vol. 2011, article 24, 2011. View at Publisher · View at Google Scholar
  5. H. Zhou, “Convergence theorems of fixed points for κ-strict pseudo-contractions in Hilbert spaces,” Nonlinear Analysis, vol. 69, no. 2, pp. 456–462, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free Boundary Problems, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1984.
  7. F. Giannessi and A. Maugeri, Variational Inequalities and Network Equilibrium Problems, Plenum Press, New York, NY, USA, 1995.
  8. D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, vol. 88 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1980.
  9. M. A. Noor, “Some aspects of extended general variational inequalities,” Abstract and Applied Analysis. In press.
  10. M. Aslam 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
  11. 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 · View at Zentralblatt MATH
  12. 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, 2010.
  13. M. A. Noor, K. I. Noor, Z. Y. Huang, and E. Al-Said, “Implicit schemes for solving extended general nonconvex vatiational inequalities,” Journal of Applied Mathematics, vol. 2012, Article ID 646259, 10 pages, 2012. View at Publisher · View at Google Scholar
  14. I. Yamada, “The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, D. Butnariu, Y. Censor, and S. Reich, Eds., vol. 8 of Studies in Computational Mathematics, pp. 473–504, North-Holland, Amsterdam, The Netherlands, 2001. View at Zentralblatt MATH
  15. L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Some iterative methods for finding fixed points and for solving constrained convex minimization problems,” Nonlinear Analysis, vol. 74, no. 16, pp. 5286–5302, 2011. View at Publisher · View at Google Scholar
  16. X. Lu, H.-K. Xu, and X. Yin, “Hybrid methods for a class of monotone variational inequalities,” Nonlinear Analysis, vol. 71, no. 3-4, pp. 1032–1041, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. P.-E. Maingé and A. Moudafi, “Strong convergence of an iterative method for hierarchical fixed-point problems,” Pacific Journal of Optimization, vol. 3, no. 3, pp. 529–538, 2007. View at Zentralblatt MATH
  18. G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. G. Marino and H.-K. Xu, “Explicit hierarchical fixed point approach to variational inequalities,” Journal of Optimization Theory and Applications, vol. 149, no. 1, pp. 61–78, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. A. Moudafi and P.-E. Maingé, “Towards viscosity approximations of hierarchical fixed-point problems,” Fixed Point Theory and Applications, vol. 2006, Article ID 646259, 10 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. M. Tian, “A general iterative algorithm for nonexpansive mappings in Hilbert spaces,” Nonlinear Analysis, vol. 73, no. 3, pp. 689–694, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. M. Tian, “A general iterative method based on the hybrid steepest descent scheme for nonexpansive mappings in Hilbert spaces,” in Proceedings of the International Conference on Computational Intelligence and Software Engineering (CiSE '10), pp. 1–4, December 2010. View at Publisher · View at Google Scholar
  23. 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
  24. Y. Yao, R. Chen, and H.-K. Xu, “Schemes for finding minimum-norm solutions of variational inequalities,” Nonlinear Analysis, vol. 72, no. 7-8, pp. 3447–3456, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. Y. Yao, Y. J. Cho, and Y.-C. Liou, “Hierarchical convergence of an implicit double-net algorithms for nonexpansive semiroups and variational inequalities problems,” Fixed Point Theory and Applications, vol. 2011, article 101, 2011.
  26. Y. Yao, R. Chen, and Y.-C. Liou, “A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1506–1515, 2012. View at Publisher · View at Google Scholar
  27. Y. Yao, W. Jiang, and Y.-C. Liou, “Regularized methods for the split feasibility problem,” Abstract and Applied Analysis. In press. View at Publisher · View at Google Scholar
  28. Y. Yao, M. A. Noor, and Y.-C. Liou, “Strong convergence of a modified extra-gradient method to the minimum-norm solution of variational inequalites,” Abstract and Applied Analysis, vol. 2012, Article ID 817436, 9 pages, 2012. View at Publisher · View at Google Scholar
  29. Y. Yao, M. A. Noor, Y.-C. Liou, and S. M. Kang, “Iterative algorithms for general multivalued variational inequalities,” Abstract and Applied Analysis, vol. 2012, Article ID 768277, 10 pages, 2012. View at Publisher · View at Google Scholar
  30. Y. Yao, Y.-C. Liou, and S. M. Kang, “Two-step projection methods for a system of variational inequality problems in Banach spaces,” Journal of Global Optimization. In press. View at Publisher · View at Google Scholar
  31. Y. Yao and N. Shahzad, “Strong convergence of a proximal point algorithm with general errors,” Optimization Letters. In press. View at Publisher · View at Google Scholar
  32. H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society. Second Series, vol. 66, no. 1, pp. 240–256, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. 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
  34. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990. View at Publisher · View at Google Scholar · View at MathSciNet