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

Lu-Chuan Ceng, Ching-Feng Wen, Yeong-Cheng Liou and Jen-Chih Yao
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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
Yeong-Cheng Liou: Department of Medical Research, Kaohsiung Medical University Hospital, Kaohsiung 80708, Taiwan
Jen-Chih Yao: Research Center for Interneural Computing, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan

Mathematics, 2022, vol. 10, issue 6, 1-21

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).

Keywords: strengthened inertial-type subgradient extragradient rules; adaptive step sizes; variational inequality problem; asymptotically nonexpansive mapping; Lipschitz continuity (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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