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

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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.

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, China

    L. C. Ceng

  2. Scientific Computing Key Laboratory of Shanghai Universities, Shanghai, 200234, China

    L. C. Ceng

  3. A.G. Anderson Graduate School of Management, University of California, Riverside, CA, USA

    S. Schaible

  4. Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan

    J. C. Yao

Authors
  1. L. C. Ceng
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  2. S. Schaible
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  3. J. C. Yao
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Corresponding author

Correspondence to J. C. Yao.

Additional information

In this research, the first author was partially supported by the National Science Foundation China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commision of Shanghai Municipality Grant (075105118).

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Ceng, L.C., Schaible, S. & Yao, J.C. Implicit Iteration Scheme with Perturbed Mapping for Equilibrium Problems and Fixed Point Problems of Finitely Many Nonexpansive Mappings. J Optim Theory Appl 139, 403–418 (2008). https://doi.org/10.1007/s10957-008-9361-y

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  • DOI: https://doi.org/10.1007/s10957-008-9361-y

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