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Hybrid Mann Viscosity Implicit Iteration Methods for Triple Hierarchical Variational Inequalities, Systems of Variational Inequalities and Fixed Point Problems

Author

Listed:
  • Lu-Chuan Ceng

    (Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
    These authors contributed equally to this work.)

  • Qing Yuan

    (School of Mathematics and Statistics, Linyi University, Linyi 276000, China
    These authors contributed equally to this work.)

Abstract

In the present work, we introduce a hybrid Mann viscosity-like implicit iteration to find solutions of a monotone classical variational inequality with a variational inequality constraint over the common solution set of a general system of variational inequalities and a problem of common fixed points of an asymptotically nonexpansive mapping and a countable of uniformly Lipschitzian pseudocontractive mappings in Hilbert spaces, which is called the triple hierarchical constrained variational inequality. Strong convergence of the proposed method to the unique solution of the problem is guaranteed under some suitable assumptions. As a sub-result, we provide an algorithm to solve problem of common fixed points of pseudocontractive, nonexpansive mappings, variational inequality problems and generalized mixed bifunction equilibrium problems in Hilbert spaces.

Suggested Citation

  • Lu-Chuan Ceng & Qing Yuan, 2019. "Hybrid Mann Viscosity Implicit Iteration Methods for Triple Hierarchical Variational Inequalities, Systems of Variational Inequalities and Fixed Point Problems," Mathematics, MDPI, vol. 7(2), pages 1-24, February.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:2:p:142-:d:203176
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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.
    2. Hideaki Iiduka, 2011. "Iterative Algorithm for Solving Triple-Hierarchical Constrained Optimization Problem," Journal of Optimization Theory and Applications, Springer, vol. 148(3), pages 580-592, March.
    3. Lu-Chuan Ceng & Chang-yu Wang & Jen-Chih Yao, 2008. "Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 67(3), pages 375-390, June.
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    Cited by:

    1. Lu-Chuan Ceng & Adrian Petruşel & Jen-Chih Yao, 2019. "On Mann Viscosity Subgradient Extragradient Algorithms for Fixed Point Problems of Finitely Many Strict Pseudocontractions and Variational Inequalities," Mathematics, MDPI, vol. 7(10), pages 1-14, October.
    2. Lu-Chuan Ceng & Xiaoye Yang, 2019. "Some Mann-Type Implicit Iteration Methods for Triple Hierarchical Variational Inequalities, Systems Variational Inequalities and Fixed Point Problems," Mathematics, MDPI, vol. 7(3), pages 1-20, February.
    3. Lu-Chuan Ceng & Adrian Petruşel & Ching-Feng Wen & Jen-Chih Yao, 2019. "Inertial-Like Subgradient Extragradient Methods for Variational Inequalities and Fixed Points of Asymptotically Nonexpansive and Strictly Pseudocontractive Mappings," Mathematics, MDPI, vol. 7(9), pages 1-19, September.
    4. Lu-Chuan Ceng & Qing Yuan, 2019. "Triple Hierarchical Variational Inequalities, Systems of Variational Inequalities, and Fixed Point Problems," Mathematics, MDPI, vol. 7(2), pages 1-19, February.

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