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Hybrid Methods for Solving Simultaneously an Equilibrium Problem and Countably Many Fixed Point Problems in a Hilbert Space

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Abstract

This paper presents a framework of iterative methods for finding a common solution to an equilibrium problem and a countable number of fixed point problems defined in a Hilbert space. A general strong convergence theorem is established under mild conditions. Two hybrid methods are derived from the proposed framework in coupling the fixed point iterations with the iterations of the proximal point method or the extragradient method, which are well-known methods for solving equilibrium problems. The strategy is to obtain the strong convergence from the weak convergence of the iterates without additional assumptions on the problem data. To achieve this aim, the solution set of the problem is outer approximated by a sequence of polyhedral subsets.

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Acknowledgements

The authors would like to thank the Editor, the Associate Editor, and the two anonymous referees for their comments and suggestions on improving significantly the presentation of an earlier version of the paper. This research is funded by the Department of Science and Technology at Ho Chi Minh City, Vietnam. Computing resources and support provided by the Institute for Computational Science and Technology—Ho Chi Minh City (ICST) is gratefully acknowledged. The work of Thi Thu Van Nguyen is also supported by the Vietnam National University at HCMC.

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Authors and Affiliations

  1. Institute for Computational Science and Technology at Ho Chi Minh City (ICST), Ho Chi Minh City, Vietnam

    Thi Thu Van Nguyen, Jean Jacques Strodiot & Van Hien Nguyen

  2. Faculty of Mathematics and Computer Science, University of Science, VNU-HCM, Ho Chi Minh City, Vietnam

    Thi Thu Van Nguyen

  3. University of Namur, Namur, Belgium

    Jean Jacques Strodiot & Van Hien Nguyen

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  1. Thi Thu Van Nguyen
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  2. Jean Jacques Strodiot
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  3. Van Hien Nguyen
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Correspondence to Jean Jacques Strodiot.

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Communicated by Jen-Chih Yao.

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Nguyen, T.T.V., Strodiot, J.J. & Nguyen, V.H. Hybrid Methods for Solving Simultaneously an Equilibrium Problem and Countably Many Fixed Point Problems in a Hilbert Space. J Optim Theory Appl 160, 809–831 (2014). https://doi.org/10.1007/s10957-013-0400-y

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  • DOI: https://doi.org/10.1007/s10957-013-0400-y

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