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Journal of Global Optimization

, Volume 43, Issue 4, pp 487–502 | Cite as

Viscosity approximation methods for generalized equilibrium problems and fixed point problems

  • Lu-Chuan Ceng
  • Qamrul Hasan Ansari
  • Jen-Chih Yao
Article

Abstract

The purpose of this paper is to investigate the problem of finding a common element of the set of solutions of a generalized equilibrium problem (for short, GEP) and the set of fixed points of a nonexpansive mapping in the setting of Hilbert spaces. By using well-known Fan-KKM lemma, we derive the existence and uniqueness of a solution of the auxiliary problem for GEP. On account of this result and Nadler’s theorem, we propose an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of GEP and the set of fixed points of a nonexpansive mapping. Furthermore, it is proven that the sequences generated by this iterative scheme converge strongly to a common element of the set of solutions of GEP and the set of fixed points of a nonexpansive mapping.

Keywords

Viscosity approximation method Generalized equilibrium problem Fixed points Nonexpansive mappings Strong convergence 

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Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  • Lu-Chuan Ceng
    • 1
  • Qamrul Hasan Ansari
    • 2
    • 3
  • Jen-Chih Yao
    • 4
  1. 1.Department of MathematicsShanghai Normal UniversityShanghaiChina
  2. 2.Department of Mathematics and StatisticsKing Fahd University of Petroleum & MineralsDhahranSaudi Arabia
  3. 3.Department of MathematicsAligarh Muslim UniversityAligarhIndia
  4. 4.Department of Applied MathematicsNational Sun Yat-sen UniversityKaohsiungTaiwan, ROC

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