Open Access

On the Existence and Convergence of Approximate Solutions for Equilibrium Problems in Banach Spaces

Journal of Inequalities and Applications20072007:017294

https://doi.org/10.1155/2007/17294

Received: 15 June 2006

Accepted: 20 February 2007

Published: 21 March 2007

Abstract

We introduce and study a new class of auxiliary problems for solving the equilibrium problem in Banach spaces. Not only the existence of approximate solutions of the equilibrium problem is proven, but also the strong convergence of approximate solutions to an exact solution of the equilibrium problem is shown. Furthermore, we give some iterative schemes for solving some generalized mixed variational-like inequalities to illuminate our results.

[12345678910111213141516171819202122232425]

Authors’ Affiliations

(1)
Department of Mathematics, Sichuan University
(2)
Department of Mathematics, Sichuan University of Sciences and Engineering
(3)
Department of Mathematics and Statistics, Curtin University of Technology

References

  1. Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994,63(1–4):123–145.MathSciNetMATHGoogle Scholar
  2. Huang N-J, Deng C-X: Auxiliary principle and iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities. Journal of Mathematical Analysis and Applications 2001,256(2):345–359. 10.1006/jmaa.2000.6988MathSciNetView ArticleMATHGoogle Scholar
  3. Ansari QH, Yao JC: Iterative schemes for solving mixed variational-like inequalities. Journal of Optimization Theory and Applications 2001,108(3):527–541. 10.1023/A:1017531323904MathSciNetView ArticleMATHGoogle Scholar
  4. Ding XP: Algorithm of solutions for mixed-nonlinear variational-like inequalities in reflexive Banach space. Applied Mathematics and Mechanics 1998,19(6):489–496.MathSciNetGoogle Scholar
  5. Ding XP: Algorithms of solutions for completely generalized mixed implicit quasi-variational inclusions. Applied Mathematics and Computation 2004,148(1):47–66. 10.1016/S0096-3003(02)00825-1MathSciNetView ArticleMATHGoogle Scholar
  6. Guo JS, Yao JC: Variational inequalities with nonmonotone operators. Journal of Optimization Theory and Applications 1994,80(1):63–74. 10.1007/BF02196593MathSciNetView ArticleMATHGoogle Scholar
  7. Harker PT, Pang J-S: Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Mathematical Programming 1990,48(2):161–220.MathSciNetView ArticleMATHGoogle Scholar
  8. Iusem AN, Sosa W: Iterative algorithms for equilibrium problems. Optimization 2003,52(3):301–316. 10.1080/0233193031000120039MathSciNetView ArticleMATHGoogle Scholar
  9. Cavazzuti E, Pappalardo M, Passacantando M: Nash equilibria, variational inequalities, and dynamical systems. Journal of Optimization Theory and Applications 2002,114(3):491–506. 10.1023/A:1016056327692MathSciNetView ArticleMATHGoogle Scholar
  10. Göpfert A, Riahi H, Tammer C, Zălinescu C: Variational Methods in Partially Ordered Spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Volume 17. Springer, New York, NY, USA; 2003:xiv+350.Google Scholar
  11. Oettli W, Schläger D: Existence of equilibria for monotone multivalued mappings. Mathematical Methods of Operations Research 1998,48(2):219–228. 10.1007/s001860050024MathSciNetView ArticleMATHGoogle Scholar
  12. Yuan GX-Z: KKM Theory and Applications in Nonlinear Analysis, Monographs and Textbooks in Pure and Applied Mathematics. Volume 218. Marcel Dekker, New York, NY, USA; 1999:xiv+621.Google Scholar
  13. Chen GY, Teboulle M: Convergence analysis of a proximal-like minimization algorithm using Bregman functions. SIAM Journal on Optimization 1993,3(3):538–543. 10.1137/0803026MathSciNetView ArticleMATHGoogle Scholar
  14. Censor Y, Zenios SA: Proximal minimization algorithm with-functions. Journal of Optimization Theory and Applications 1992,73(3):451–464. 10.1007/BF00940051MathSciNetView ArticleMATHGoogle Scholar
  15. Flåm SD, Antipin AS: Equilibrium programming using proximal-like algorithms. Mathematical Programming 1997,78(1):29–41.MathSciNetView ArticleMATHGoogle Scholar
  16. Iusem AN: On some properties of generalized proximal point methods for variational inequalities. Journal of Optimization Theory and Applications 1998,96(2):337–362. 10.1023/A:1022670114963MathSciNetView ArticleMATHGoogle Scholar
  17. Iusem AN, Svaiter BF, Teboulle M: Entropy-like proximal methods in convex programming. Mathematics of Operations Research 1994,19(4):790–814. 10.1287/moor.19.4.790MathSciNetView ArticleMATHGoogle Scholar
  18. Moudafi A, Théra M: Proximal and dynamical approaches to equilibrium problems. In Ill-Posed Variational Problems and Regularization Techniques (Trier, 1998), Lecture Notes in Economics and Mathematical Systems. Volume 477. Edited by: Théra M, Tichatschke R. Springer, Berlin, Germany; 1999:187–201.View ArticleGoogle Scholar
  19. Chen GY, Wu YN: An iterative approach for solving equilibrium problems. Advances in Nonlinear Variational Inequalities 2003,6(1):41–48.MathSciNetMATHGoogle Scholar
  20. Glowinski R, Lions J-L, Trémolières R: Numerical Analysis of Variational Inequalities, Studies in Mathematics and Its Applications. Volume 8. North-Holland, Amsterdam, The Netherlands; 1981:xxix+776.Google Scholar
  21. Hanson MA: On sufficiency of the Kuhn-Tucker conditions. Journal of Mathematical Analysis and Applications 1981,80(2):545–550. 10.1016/0022-247X(81)90123-2MathSciNetView ArticleMATHGoogle Scholar
  22. Kothe G: Topological Vector Spaces. I. Springer, New York, NY, USA; 1983.View ArticleGoogle Scholar
  23. Fan K: A generalization of Tychonoff's fixed point theorem. Mathematische Annalen 1961,142(3):305–310. 10.1007/BF01353421MathSciNetView ArticleMATHGoogle Scholar
  24. Ding XP, Tan K-K: A minimax inequality with applications to existence of equilibrium point and fixed point theorems. Colloquium Mathematicum 1992,63(2):233–247.MathSciNetMATHGoogle Scholar
  25. Zhu DL, Marcotte P: Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities. SIAM Journal on Optimization 1996,6(3):714–726. 10.1137/S1052623494250415MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Nan-Jing Huang et al. 2007

This article is published under license to BioMed Central Ltd. 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.