SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research Article

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

Yonghong Yao1*, Yeong-Cheng Liou2 and Jen-Chih Yao3

Author Affiliations

1 Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China

2 Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan

3 Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan

For all author emails, please log on.

Fixed Point Theory and Applications 2007, 2007:064363  doi:10.1155/2007/64363

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2007/1/064363

Received:17 March 2007
Accepted:20 August 2007
Published:4 December 2007

© 2007 Yao 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.

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.


  1. Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student. 63(1–4), 123–145 (1994)

  2. Combettes, PL, Hirstoaga, SA: Equilibrium programming in Hilbert spaces. Journal of Nonlinear and Convex Analysis. 6(1), 117–136 (2005)

  3. Flåm, SD, Antipin, AS: Equilibrium programming using proximal-like algorithms. Mathematical Programming. 78(1), 29–41 (1997)

  4. Takahashi, S, Takahashi, W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. Journal of Mathematical Analysis and Applications. 331(1), 506–515 (2007). Publisher Full Text OpenURL

  5. Moudafi, A: Viscosity approximation methods for fixed-points problems. Journal of Mathematical Analysis and Applications. 241(1), 46–55 (2000). Publisher Full Text OpenURL

  6. Wittmann, R: Approximation of fixed points of nonexpansive mappings. Archiv der Mathematik. 58(5), 486–491 (1992). Publisher Full Text OpenURL

  7. Tada, A, Takahashi, W: Strong convergence theorem for an equilibrium problem and a nonexpansive mapping. In: Takahashi W, Tanaka T (eds.) Nonlinear Analysis and Convex Analysis, pp. 609–617. Yokohama, Yokohama, Japan (2007)

  8. Takahashi, W: Nonlinear Functional Analysis,p. iv+276. Yokohama, Yokohama, Japan (2000)

  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. 305(1), 227–239 (2005). Publisher Full Text OpenURL

  10. Xu, H-K: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications. 298(1), 279–291 (2004). Publisher Full Text OpenURL

  11. Takahashi, W: Weak and strong convergence theorems for families of nonexpansive mappings and their applications. Annales Universitatis Mariae Curie-Skłodowska. Sectio A. 51(2), 277–292 (1997)

  12. Takahashi, W, Shimoji, K: Convergence theorems for nonexpansive mappings and feasibility problems. Mathematical and Computer Modelling. 32(11–13), 1463–1471 (2000)

  13. Shimoji, K, Takahashi, W: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese Journal of Mathematics. 5(2), 387–404 (2001)