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Journal of Function Spaces and Applications
Volume 2013, Article ID 374268, 5 pages
http://dx.doi.org/10.1155/2013/374268
Research Article

Mixed Equilibrium Problems with Weakly Relaxed α-Monotone Bifunction in Banach Spaces

Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani 12121, Thailand

Received 28 May 2013; Accepted 1 August 2013

Academic Editor: Sompong Dhompongsa

Copyright © 2013 Wutiphol Sintunavarat. 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. M. R. Bai, S. Z. Zhou, and G. Y. Ni, “Variational-like inequalities with relaxed η-α pseudomonotone mappings in Banach spaces,” Applied Mathematics Letters, vol. 19, no. 6, pp. 547–554, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  2. M. Bianchi and S. Schaible, “Generalized monotone bifunctions and equilibrium problems,” Journal of Optimization Theory and Applications, vol. 90, no. 1, pp. 31–43, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M. Bianchi and S. Schaible, “Equilibrium problems under generalized convexity and generalized monotonicity,” Journal of Global Optimization, vol. 30, no. 2-3, pp. 121–134, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. Bianchi and R. Pini, “A note on equilibrium problems with properly quasimonotone bifunctions,” Journal of Global Optimization, vol. 20, no. 1, pp. 67–76, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. L. C. Ceng and J. C. Yao, “On generalized variational-like inequalities with generalized monotone multivalued mappings,” Applied Mathematics Letters, vol. 22, no. 3, pp. 428–434, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. Y. Q. Chen, “On the semi-monotone operator theory and applications,” Journal of Mathematical Analysis and Applications, vol. 231, no. 1, pp. 177–192, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. R. W. Cottle and J. C. Yao, “Pseudomonotone complementarity problems in Hilbert spaces,” Journal of Optimization Theory and Applications, vol. 75, no. 2, pp. 281–295, 1992. View at Publisher · View at Google Scholar
  8. X. P. Ding, “Existence and algorithm of solutions for generalized mixed implicit quasi-variational inequalities,” Applied Mathematics and Computation, vol. 113, no. 1, pp. 67–80, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. K. Fan, “Some properties of convex sets related to fixed point theorems,” Mathematische Annalen, vol. 266, no. 4, pp. 519–537, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. Y. P. Fang and N. J. Huang, “Variational-like inequalities with generalized monotone mappings in Banach spaces,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 327–338, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. F. Giannessi, Vector Variational Inequalities and Vector Equilibria, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  12. R. Glowinski, J. L. Lions, and R. Trémolières, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, The Netherlands, 1981. View at MathSciNet
  13. D. Goeleven and D. Motreanu, “Eigenvalue and dynamic problems for variational and hemivariational inequalities,” Communications on Applied Nonlinear Analysis, vol. 3, no. 4, pp. 1–21, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. S. Karamardian and S. Schaible, “Seven kinds of monotone maps,” Journal of Optimization Theory and Applications, vol. 66, no. 1, pp. 37–46, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. R. U. Verma, “On generalized variational inequalities involving relaxed Lipschitz and relaxed monotone operators,” Journal of Mathematical Analysis and Applications, vol. 213, no. 1, pp. 387–392, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. R. U. Verma, “On monotone nonlinear variational inequality problems,” Commentationes Mathematicae Universitatis Carolinae, vol. 39, no. 1, pp. 91–98, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. X. Q. Yang and G. Y. Chen, “A class of nonconvex functions and pre-variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 169, no. 2, pp. 359–373, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. N. K. Mahato and C. Nahak, “Weakly relaxed α-pseudomonotonicity and equilibrium problem in Banach spaces,” Journal of Applied Mathematics and Computing, vol. 40, no. 1-2, pp. 499–509, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  19. N. K. Mahato and C. Nahak, “Mixed equilibrium problems with relaxed α-monotone mapping in Banach spaces,” Rendiconti del Circolo Matematico di Palermo, 2013. View at Publisher · View at Google Scholar
  20. K. Fan, “A generalization of Tychonoff's fixed point theorem,” Mathematische Annalen, vol. 142, pp. 305–310, 1961. View at Google Scholar · View at MathSciNet