Skip to main content


  • Research Article
  • Open Access

Convergence Theorem for Equilibrium Problems and Fixed Point Problems of Infinite Family of Nonexpansive Mappings

Fixed Point Theory and Applications20072007:064363

  • Received: 17 March 2007
  • Accepted: 20 August 2007
  • Published:


We introduce an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of infinite nonexpansive mappings in a Hilbert space. We prove a strong-convergence theorem under mild assumptions on parameters.


  • Hilbert Space
  • Differential Geometry
  • Convergence Theorem
  • Equilibrium Problem
  • Nonexpansive Mapping


Authors’ Affiliations

Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300160, China
Department of Information Management, Cheng Shiu University, Kaohsiung, 833, Taiwan
Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, 804, Taiwan


  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. Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. Journal of Nonlinear and Convex Analysis 2005,6(1):117–136.MathSciNetMATHGoogle Scholar
  3. Flåm SD, Antipin AS: Equilibrium programming using proximal-like algorithms. Mathematical Programming 1997,78(1):29–41.MathSciNetView ArticleMATHGoogle Scholar
  4. 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.2006.08.036MathSciNetView ArticleMATHGoogle Scholar
  5. Moudafi A: Viscosity approximation methods for fixed-points problems. Journal of Mathematical Analysis and Applications 2000,241(1):46–55. 10.1006/jmaa.1999.6615MathSciNetView ArticleMATHGoogle Scholar
  6. Wittmann R: Approximation of fixed points of nonexpansive mappings. Archiv der Mathematik 1992,58(5):486–491. 10.1007/BF01190119MathSciNetView ArticleMATHGoogle Scholar
  7. Tada A, Takahashi W: Strong convergence theorem for an equilibrium problem and a nonexpansive mapping. In Nonlinear Analysis and Convex Analysis. Edited by: Takahashi W, Tanaka T. Yokohama, Yokohama, Japan; 2007:609–617.Google Scholar
  8. Takahashi W: Nonlinear Functional Analysis. Yokohama, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar
  9. Suzuki T: Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals. Journal of Mathematical Analysis and Applications 2005,305(1):227–239. 10.1016/j.jmaa.2004.11.017MathSciNetView ArticleMATHGoogle Scholar
  10. Xu H-K: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2004,298(1):279–291. 10.1016/j.jmaa.2004.04.059MathSciNetView ArticleMATHGoogle Scholar
  11. Takahashi W: Weak and strong convergence theorems for families of nonexpansive mappings and their applications. Annales Universitatis Mariae Curie-Skłodowska. Sectio A 1997,51(2):277–292.MathSciNetMATHGoogle Scholar
  12. Takahashi W, Shimoji K: Convergence theorems for nonexpansive mappings and feasibility problems. Mathematical and Computer Modelling 2000,32(11–13):1463–1471.MathSciNetView ArticleMATHGoogle Scholar
  13. Shimoji K, Takahashi W: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese Journal of Mathematics 2001,5(2):387–404.MathSciNetMATHGoogle Scholar


© Yonghong Yao 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.