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Implicit Iteration Scheme with Perturbed Mapping for Equilibrium Problems and Fixed Point Problems of Finitely Many Nonexpansive Mappings

Author

Listed:
  • L. C. Ceng

    (Shanghai Normal University
    Scientific Computing Key Laboratory of Shanghai Universities)

  • S. Schaible

    (University of California)

  • J. C. Yao

    (National Sun Yat-Sen University)

Abstract

We introduce an implicit iteration scheme with a perturbed mapping for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of finitely many nonexpansive mappings in a Hilbert space. Then, we establish some convergence theorems for this implicit iteration scheme which are connected with results by Xu and Ori (Numer. Funct. Analysis Optim. 22:767–772, 2001), Zeng and Yao (Nonlinear Analysis, Theory, Methods Appl. 64:2507–2515, 2006) and Takahashi and Takahashi (J. Math. Analysis Appl. 331:506–515, 2007). In particular, necessary and sufficient conditions for strong convergence of this implicit iteration scheme are obtained.

Suggested Citation

  • L. C. Ceng & S. Schaible & J. C. Yao, 2008. "Implicit Iteration Scheme with Perturbed Mapping for Equilibrium Problems and Fixed Point Problems of Finitely Many Nonexpansive Mappings," Journal of Optimization Theory and Applications, Springer, vol. 139(2), pages 403-418, November.
    • Handle: RePEc:spr:joptap:v:139:y:2008:i:2:d:10.1007_s10957-008-9361-y
    DOI: 10.1007/s10957-008-9361-y
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    References listed on IDEAS

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    1. H. K. Xu & T. H. Kim, 2003. "Convergence of Hybrid Steepest-Descent Methods for Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 119(1), pages 185-201, October.
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    Cited by:

    1. P. N. Anh, 2012. "Strong Convergence Theorems for Nonexpansive Mappings and Ky Fan Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 303-320, July.
    2. Yonghong Yao & Yeong-Cheng Liou & Shin Kang, 2010. "Minimization of equilibrium problems, variational inequality problems and fixed point problems," Journal of Global Optimization, Springer, vol. 48(4), pages 643-656, December.

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