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Journal of Applied Mathematics
Volume 2012, Article ID 437391, 9 pages
http://dx.doi.org/10.1155/2012/437391
Research Article

Implicit Methods for Equilibrium Problems on Hadamard Manifolds

1Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan
2Mathematics Department, HITEC University, Taxila Cantt, Pakistan
3Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China

Received 26 March 2012; Accepted 3 April 2012

Academic Editor: Rudong Chen

Copyright © 2012 Muhammad Aslam Noor 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. D. Azagra, J. Ferrera, and F. López-Mesas, “Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds,” Journal of Functional Analysis, vol. 220, no. 2, pp. 304–361, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. V. Colao, G. Lopez, G. Marino, and V. Martin-Marquez, “Equilibrium problems in Hadamard manifolds,” Journal of Mathematical Analysis and Applications, vol. 388, pp. 61–77, 2012. View at Publisher · View at Google Scholar
  3. M. P. DoCarmo, Riemannian Geometry, Mathematics: Theory & Applications, Birkhäuser, Boston, Mass, USA, 1992.
  4. O. P. Ferreira and P. R. Oliveira, “Proximal point algorithm on Riemannian manifolds,” Optimization, vol. 51, no. 2, pp. 257–270, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. M. A. Noor and K. I. Noor, “Proximal point methods for solving mixed variational inequalities on Hadamard manifolds,” Journal of Applied Mathematics. In press.
  6. T. Sakai, Riemannian Geometry, vol. 149 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1996.
  7. G. Tang, L. W. Zhou, and N. J. Huang, “The proximal point algorithm for pseudomonotone variational inequalities on Hadamard manifolds,” Optimization Letters. In press. View at Publisher · View at Google Scholar
  8. C. Udrişte, Convex Functions and Optimization Methods on Riemannian Manifolds, vol. 297 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994.
  9. S. Z. Németh, “Variational inequalities on Hadamard manifolds,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 52, no. 5, pp. 1491–1498, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. M. A. Noor and W. Oettli, “On general nonlinear complementarity problems and quasi-equilibria,” Le Matematiche, vol. 49, no. 2, pp. 313–331, 1994. View at Google Scholar · View at Zentralblatt MATH
  11. 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
  12. C. Li, G. López, V. Martín-Márquez, and J.-H. Wang, “Resolvents of set-valued monotone vector fields in Hadamard manifolds,” Set-Valued and Variational Analysis, vol. 19, no. 3, pp. 361–383, 2011. View at Publisher · View at Google Scholar
  13. M. A. Noor, “General variational inequalities,” Applied Mathematics Letters, vol. 1, no. 2, pp. 119–122, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. 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
  15. 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
  16. M. A. Noor, “Fundamentals of mixed quasi variational inequalities,” International Journal of Pure and Applied Mathematics, vol. 15, no. 2, pp. 137–258, 2004. View at Google Scholar · View at Zentralblatt MATH
  17. M. A. Noor, “Auxiliary principle technique for equilibrium problems,” Journal of Optimization Theory and Applications, vol. 122, no. 2, pp. 371–386, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. M. A. Noor, “Fundamentals of equilibrium problems,” Mathematical Inequalities & Applications, vol. 9, no. 3, pp. 529–566, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. 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
  20. M. A. Noor, “On an implicit method for nonconvex variational inequalities,” Journal of Optimization Theory and Applications, vol. 147, no. 2, pp. 411–417, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. M. A. Noor, “Auxiliary principle technique for solving general mixed variational inequalities,” Journal of Advanced Mathematical Studies, vol. 3, no. 2, pp. 89–96, 2010. View at Google Scholar · View at Zentralblatt MATH
  22. M. A. Noor, “Some aspects of extended general variational inequalities,” Abstract and Applied Analysis, vol. 2012, Article ID 303569, 16 pages, 2012. View at Publisher · View at Google Scholar
  23. R. Glowinski, J.-L. Lions, and R. Trémolières, Numerical Analysis of Variational Inequalities, vol. 8 of Studies in Mathematics and Its Applications, North-Holland Publishing, Amsterdam, The Netherlands, 1981.
  24. M. A. Noor and K. I. Noor, “On equilibrium problems,” Applied Mathematics E-Notes, vol. 4, pp. 125–132, 2004. View at Google Scholar · View at Zentralblatt MATH
  25. M. A. Noor, K. I. Noor, and T. M. Rassias, “Some aspects of variational inequalities,” Journal of Computational and Applied Mathematics, vol. 47, no. 3, pp. 285–312, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. G. Stampacchia, “Formes bilinéaires coercitives sur les ensembles convexes,” vol. 258, pp. 4413–4416, 1964. View at Google Scholar · View at Zentralblatt MATH
  27. Y. Yao, M. A. Noor, and Y. C. Liou, “Strong convergence of a modified extra-gradient method to the minimum-norm solution of variational inequalities,” Abstract and Applied Analysis, vol. 2012, Article ID 817436, 9. View at Publisher · View at Google Scholar
  28. Y. Yao, M. A. Noor, Y. C. Liou, and S. M. Kang, “Iterative algorithms for general multi-valued variational inequalities,” Abstract and Applied Analysis, vol. 2012, Article ID 768272, 10 pages, 2012. View at Publisher · View at Google Scholar