Skip to content


  • Research Article
  • Open Access

A General Iterative Method for Equilibrium Problems and Fixed Point Problems in Hilbert Spaces

Fixed Point Theory and Applications20082007:095412

  • Received: 14 May 2007
  • Accepted: 18 September 2007
  • Published:


We introduce a general iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. Our results improve and extend the corresponding ones announced by S. Takahashi and W. Takahashi in 2007, Marino and Xu in 2006, Combettes and Hirstoaga in 2005, and many others.


  • Hilbert Space
  • Approximation Method
  • Differential Geometry
  • Equilibrium Problem
  • Nonexpansive Mapping


Authors’ Affiliations

Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300160, China
Department of Mathematics, Shijiazhuang University, Shijiazhuang, 050035, China
Department of Mathematics, Gyeongsang National University, Chinju, 660-701, Korea


  1. Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. Journal of Nonlinear and Convex Analysis 2005,6(1):117–136.MATHMathSciNetGoogle Scholar
  2. Flåm SD, Antipin AS: Equilibrium programming using proximal-like algorithms. Mathematical Programming 1997,78(1):29–41.MATHMathSciNetView ArticleGoogle Scholar
  3. Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007,331(1):506–515. 10.1016/j.jmaa.2006.08.036MATHMathSciNetView ArticleGoogle Scholar
  4. Deutsch F, Yamada I: Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numerical Functional Analysis and Optimization 1998,19(1–2):33–56.MATHMathSciNetView ArticleGoogle Scholar
  5. Xu H-K: Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society 2002,66(1):240–256. 10.1112/S0024610702003332MATHMathSciNetView ArticleGoogle Scholar
  6. Xu HK: An iterative approach to quadratic optimization. Journal of Optimization Theory and Applications 2003,116(3):659–678. 10.1023/A:1023073621589MATHMathSciNetView ArticleGoogle Scholar
  7. Yamada I: 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 (Haifa, 2000), Studies in Computational Mathematics. Volume 8. Edited by: Butnariu D, Censor Y, Reich S. North-Holland, Amsterdam, The Netherlands; 2001:473–504.View ArticleGoogle Scholar
  8. Marino G, Xu H-K: A general iterative method for nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications 2006,318(1):43–52. 10.1016/j.jmaa.2005.05.028MATHMathSciNetView ArticleGoogle Scholar
  9. Moudafi A: Viscosity approximation methods for fixed-points problems. Journal of Mathematical Analysis and Applications 2000,241(1):46–55. 10.1006/jmaa.1999.6615MATHMathSciNetView ArticleGoogle Scholar
  10. Wittmann R: Approximation of fixed points of nonexpansive mappings. Archiv der Mathematik 1992,58(5):486–491. 10.1007/BF01190119MATHMathSciNetView ArticleGoogle Scholar
  11. Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bulletin of the American Mathematical Society 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0MATHMathSciNetView ArticleGoogle Scholar
  12. Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994,63(1–4):123–145.MATHMathSciNetGoogle Scholar
  13. Solodov MV, Svaiter BF: A truly globally convergent Newton-type method for the monotone nonlinear complementarity problem. SIAM Journal on Optimization 2000,10(2):605–625. 10.1137/S1052623498337546MATHMathSciNetView ArticleGoogle Scholar


© Meijuan Shang 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.