Mildly Inertial Subgradient Extragradient Method for Variational Inequalities Involving an Asymptotically Nonexpansive and Finitely Many Nonexpansive Mappings

Lu-Chuan Ceng, Xiaolong Qin, Yekini Shehu and Jen-Chih Yao
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Lu-Chuan Ceng: Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
Xiaolong Qin: General Education Center, National Yunlin University of Science and Technology, Douliou 64002, Taiwan
Yekini Shehu: Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
Jen-Chih Yao: Research Center for Interneural Computing, China Medical University Hospital, Taichung 40447, Taiwan

Mathematics, 2019, vol. 7, issue 10, 1-19

Abstract: In a real Hilbert space, let the notation VIP indicate a variational inequality problem for a Lipschitzian, pseudomonotone operator, and let CFPP denote a common fixed-point problem of an asymptotically nonexpansive mapping and finitely many nonexpansive mappings. This paper introduces mildly inertial algorithms with linesearch process for finding a common solution of the VIP and the CFPP by using a subgradient approach. These fully absorb hybrid steepest-descent ideas, viscosity iteration ideas, and composite Mann-type iterative ideas. With suitable conditions on real parameters, it is shown that the sequences generated our algorithms converge to a common solution in norm, which is a unique solution of a hierarchical variational inequality (HVI).

Keywords: inertial subgradient extragradient method; pseudomonotone variational inequality; asymptotically nonexpansive mapping; sequentially weak continuity (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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