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Weak and Strong Convergence Theorems for a Nonexpansive Mapping and an Equilibrium Problem

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  • A. Tada

    (Tokyo Institute of Technology)

  • W. Takahashi

    (Tokyo Institute of Technology)

Abstract

In this paper, we introduce two iterative sequences for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in a Hilbert space. Then, we show that one of the sequences converges strongly and the other converges weakly.

Suggested Citation

  • A. Tada & W. Takahashi, 2007. "Weak and Strong Convergence Theorems for a Nonexpansive Mapping and an Equilibrium Problem," Journal of Optimization Theory and Applications, Springer, vol. 133(3), pages 359-370, June.
    • Handle: RePEc:spr:joptap:v:133:y:2007:i:3:d:10.1007_s10957-007-9187-z
    DOI: 10.1007/s10957-007-9187-z
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    References listed on IDEAS

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    1. W. Takahashi & M. Toyoda, 2003. "Weak Convergence Theorems for Nonexpansive Mappings and Monotone Mappings," Journal of Optimization Theory and Applications, Springer, vol. 118(2), pages 417-428, August.
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    Cited by:

    1. Prasit Cholamjiak & Suthep Suantai, 2013. "Iterative methods for solving equilibrium problems, variational inequalities and fixed points of nonexpansive semigroups," Journal of Global Optimization, Springer, vol. 57(4), pages 1277-1297, December.
    2. Jian-Wen Peng & Jen-Chih Yao, 2010. "Ishikawa iterative algorithms for a generalized equilibrium problem and fixed point problems of a pseudo-contraction mapping," Journal of Global Optimization, Springer, vol. 46(3), pages 331-345, March.
    3. Uthai Kamraksa & Rabian Wangkeeree, 2012. "Existence theorems and iterative approximation methods for generalized mixed equilibrium problems for a countable family of nonexpansive mappings," Journal of Global Optimization, Springer, vol. 54(1), pages 27-46, September.
    4. Ayed E. Hashoosh & Mohsen Alimohammady & M. K. Kalleji, 2016. "Existence Results for Some Equilibrium Problems Involving -Monotone Bifunction," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2016, pages 1-5, February.
    5. Mohammad Eslamian & Ali Abkar, 2014. "Viscosity iterative scheme for generalized mixed equilibrium problems and nonexpansive semigroups," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(2), pages 554-570, July.
    6. Phan Tu Vuong & Jean Jacques Strodiot & Van Hien Nguyen, 2012. "Extragradient Methods and Linesearch Algorithms for Solving Ky Fan Inequalities and Fixed Point Problems," Journal of Optimization Theory and Applications, Springer, vol. 155(2), pages 605-627, November.
    7. 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.
    8. Xiaolong Qin & Sun Cho & Shin Kang & Feng Gu, 2013. "On the hybrid projection method for fixed point and equilibrium problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(2), pages 341-354, July.
    9. Prasit Cholamjiak & Suthep Suantai, 2012. "A hybrid method for a family of relatively quasi-nonexpansive mappings and an equilibrium problem in Banach spaces," Journal of Global Optimization, Springer, vol. 54(1), pages 83-100, September.
    10. Xiaolong Qin & Sun Cho & Shin Kang, 2011. "An extragradient-type method for generalized equilibrium problems involving strictly pseudocontractive mappings," Journal of Global Optimization, Springer, vol. 49(4), pages 679-693, April.
    11. Zi-Ming Wang & Yongfu Su & Sun Cho & Wandong Lou, 2011. "A new iterative algorithm for equilibrium and fixed point problems of nonexpansive mapping," Journal of Global Optimization, Springer, vol. 50(3), pages 457-472, July.
    12. Yonghong Yao & Naseer Shahzad & Jen-Chih Yao, 2020. "Projected Subgradient Algorithms for Pseudomonotone Equilibrium Problems and Fixed Points of Pseudocontractive Operators," Mathematics, MDPI, vol. 8(4), pages 1-15, March.
    13. Hideaki Iiduka, 2021. "Inexact stochastic subgradient projection method for stochastic equilibrium problems with nonmonotone bifunctions: application to expected risk minimization in machine learning," Journal of Global Optimization, Springer, vol. 80(2), pages 479-505, June.
    14. Abkar, A. & Tavakkoli, M., 2015. "A new algorithm for two finite families of demicontractive mappings and equilibrium problems," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 491-500.
    15. Xiaolong Qin & Lai-Jiu Lin & Shin Min Kang, 2011. "On a Generalized Ky Fan Inequality and Asymptotically Strict Pseudocontractions in the Intermediate Sense," Journal of Optimization Theory and Applications, Springer, vol. 150(3), pages 553-579, September.
    16. J. W. Peng, 2010. "Iterative Algorithms for Mixed Equilibrium Problems, Strict Pseudocontractions and Monotone Mappings," Journal of Optimization Theory and Applications, Springer, vol. 144(1), pages 107-119, January.

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