Modified Mann-Type Subgradient Extragradient Rules for Variational Inequalities and Common Fixed Points Implicating Countably Many Nonexpansive Operators
Yun-Ling Cui, Lu-Chuan Ceng, Fang-Fei Zhang, Cong-Shan Wang, Jianye Li, Huiying Hu, Long He · Mathematics · 2022
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.