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Inertial-Like Subgradient Extragradient Methods for Variational Inequalities and Fixed Points of Asymptotically Nonexpansive and Strictly Pseudocontractive Mappings

Author

Listed:
  • Lu-Chuan Ceng

    (Department of Mathematics, Shanghai Normal University, Shanghai 200234, China)

  • Adrian Petruşel

    (Department of Mathematics, Babes-Bolyai University, Cluj-Napoca 400084, Romania)

  • Ching-Feng Wen

    (Center for Fundamental Science and Research Center for Nonliear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung 80708, Taiwan
    Department of Medical Research, Kaohsiung Medical University Hospital, Kaohsiung 80708, Taiwan)

  • Jen-Chih Yao

    (Research Center for Interneural Computing, China Medical University Hospital, Taichung 40402, Taiwan)

Abstract

Let VIP indicate the variational inequality problem with Lipschitzian and pseudomonotone operator and let CFPP denote the common fixed-point problem of an asymptotically nonexpansive mapping and a strictly pseudocontractive mapping in a real Hilbert space. Our object in this article is to establish strong convergence results for solving the VIP and CFPP by utilizing an inertial-like gradient-like extragradient method with line-search process. Via suitable assumptions, it is shown that the sequences generated by such a method converge strongly to a common solution of the VIP and CFPP, which also solves a hierarchical variational inequality (HVI).

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:9:p:860-:d:268090
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    References listed on IDEAS

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    1. Rapeepan Kraikaew & Satit Saejung, 2014. "Strong Convergence of the Halpern Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 163(2), pages 399-412, November.
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    3. Y. Censor & A. Gibali & S. Reich, 2011. "The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space," Journal of Optimization Theory and Applications, Springer, vol. 148(2), pages 318-335, February.
    4. Alexander J. Zaslavski, 2016. "Numerical Optimization with Computational Errors," Springer Optimization and Its Applications, Springer, number 978-3-319-30921-7, June.
    5. Zhao-Rong Kong & Lu-Chuan Ceng & Ching-Feng Wen, 2012. "Some Modified Extragradient Methods for Solving Split Feasibility and Fixed Point Problems," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-32, December.
    6. 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.
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