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On Strengthened Inertial-Type Subgradient Extragradient Rule with Adaptive Step Sizes for Variational Inequalities and Fixed Points of Asymptotically Nonexpansive Mappings

Author

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  • Lu-Chuan Ceng

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

  • Ching-Feng Wen

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

  • Yeong-Cheng Liou

    (Department of Medical Research, Kaohsiung Medical University Hospital, Kaohsiung 80708, Taiwan
    Department of Healthcare Administration and Medical Informatics and Research Center of Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung 807, Taiwan)

  • Jen-Chih Yao

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

Abstract

In a real Hilbert space, let the VIP denote a pseudomonotone variational inequality problem with Lipschitz continuity operator, and let the CFPP indicate a common fixed-point problem of finitely many nonexpansive mappings and an asymptotically nonexpansive mapping. On the basis of the Mann iteration method, the viscosity approximation method and the hybrid steepest-descent method, we propose and analyze two strengthened inertial-type subgradient extragradient rules with adaptive step sizes for solving the VIP and CFPP. With the help of suitable restrictions, we show the strong convergence of the suggested rules to a common solution of the VIP and CFPP, which is the unique solution of a hierarchical variational inequality (HVI).

Suggested Citation

  • Lu-Chuan Ceng & Ching-Feng Wen & Yeong-Cheng Liou & Jen-Chih Yao, 2022. "On Strengthened Inertial-Type Subgradient Extragradient Rule with Adaptive Step Sizes for Variational Inequalities and Fixed Points of Asymptotically Nonexpansive Mappings," Mathematics, MDPI, vol. 10(6), pages 1-21, March.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:6:p:958-:d:773246
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    References listed on IDEAS

    as
    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. Jolaoso, Lateef O. & Shehu, Yekini & Yao, Jen-Chih, 2022. "Inertial extragradient type method for mixed variational inequalities without monotonicity," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 192(C), pages 353-369.
    3. Yekini Shehu & Aviv Gibali & Simone Sagratella, 2020. "Inertial Projection-Type Methods for Solving Quasi-Variational Inequalities in Real Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 877-894, March.
    4. 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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    6. 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.
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