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Strong convergence theorems for equilibrium problems and fixed point problems in Banach spaces

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Abstract

The paper proposes two new iterative methods for solving pseudomonotone equilibrium problems involving fixed point problems for quasi-\(\phi \)-nonexpansive mappings in Banach spaces. The proposed algorithms combine the extended extragradient method or the linesearch method with the Halpern iteration. The strong convergence theorems are established under standard assumptions imposed on equilibrium bifunctions and mappings. The results in this paper have generalized and enriched existing algorithms for equilibrium problems in Banach spaces.

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Notes

  1. We randomly choose \(\lambda _{1k}\in [-m,0],~\lambda _{2k}\in [0,m],~ k=1,\ldots ,m\). We set \(\widehat{Q}_1\), \(\widehat{Q}_2\) as two diagonal matrices with eigenvalues \(\left\{ \lambda _{1k}\right\} _{k=1}^m\) and \(\left\{ \lambda _{2k}\right\} _{k=1}^m\), respectively. Then, we construct a positive semidefinite matrix Q and a negative semidefinite matrix T using random orthogonal matrices with \(\widehat{Q}_2\) and \(\widehat{Q}_1\), respectively. Finally, we set \(P=Q-T\)

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Acknowledgements

The authors would like to thank the Associate Editor and two anonymous referees for their valuable comments and suggestions which helped us very much in improving the original version of this paper. The first author is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the project: 101.01-2017.315

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Authors and Affiliations

  1. Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Dang Van Hieu

  2. Department of Mathematics, University of Namur, Namur, Belgium

    Jean Jacques Strodiot

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  1. Dang Van Hieu
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  2. Jean Jacques Strodiot
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Correspondence to Dang Van Hieu.

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Van Hieu, D., Strodiot, J.J. Strong convergence theorems for equilibrium problems and fixed point problems in Banach spaces. J. Fixed Point Theory Appl. 20, 131 (2018). https://doi.org/10.1007/s11784-018-0608-4

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