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Modified Mann-Type Subgradient Extragradient Rules for Variational Inequalities and Common Fixed Points Implicating Countably Many Nonexpansive Operators

Author

Listed:
  • Yun-Ling Cui

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

  • Lu-Chuan Ceng

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

  • Fang-Fei Zhang

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

  • Cong-Shan Wang

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

  • Jian-Ye Li

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

  • Hui-Ying Hu

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

  • Long He

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

Abstract

In a real Hilbert space, let the CFPP, VIP, and HFPP denote the common fixed-point problem of countable nonexpansive operators and asymptotically nonexpansive operator, variational inequality problem, and hierarchical fixed point problem, respectively. With the help of the Mann iteration method, a subgradient extragradient approach with a linear-search process, and a hybrid deepest-descent technique, we construct two modified Mann-type subgradient extragradient rules with a linear-search process for finding a common solution of the CFPP and VIP. Under suitable assumptions, we demonstrate the strong convergence of the suggested rules to a common solution of the CFPP and VIP, which is only a solution of a certain HFPP.

Suggested Citation

  • Yun-Ling Cui & Lu-Chuan Ceng & Fang-Fei Zhang & Cong-Shan Wang & Jian-Ye Li & Hui-Ying Hu & Long He, 2022. "Modified Mann-Type Subgradient Extragradient Rules for Variational Inequalities and Common Fixed Points Implicating Countably Many Nonexpansive Operators," Mathematics, MDPI, vol. 10(11), pages 1-26, June.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:11:p:1949-:d:832736
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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.
    2. 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.
    3. Gang Cai & Qiao-Li Dong & Yu Peng, 2021. "Strong Convergence Theorems for Solving Variational Inequality Problems with Pseudo-monotone and Non-Lipschitz Operators," Journal of Optimization Theory and Applications, Springer, vol. 188(2), pages 447-472, February.
    4. 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.
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