Parallel Subgradient-like Extragradient Approaches for Variational Inequality and Fixed-Point Problems with Bregman Relatively Asymptotical Nonexpansivity

• Published in: Symmetry (Basel) 2023-09, Vol.15 (9), p.1749
• Main Authors: Ceng, Lu-Chuan, Cui, Yun-Ling, Cao, Sheng-Long, Li, Bing, Wang, Cong-Shan, Hu, Hui-Ying
• Format: Article
• Language: English
• Subjects: Algorithms; Asymptotic properties; Banach spaces; Bregman distance; Bregman projection; Bregman relatively asymptotically nonexpansive mapping; inertial effect; parallel subgradient-like extragradient approach; variational inequality problem
• ISSN: 2073-8994; 2073-8994
• DOI: 10.3390/sym15091749

In a uniformly smooth and p-uniformly convex Banach space, let the pair of variational inequality and fixed-point problems (VIFPPs) consist of two variational inequality problems (VIPs) involving two uniformly continuous and pseudomonotone mappings and two fixed-point problems implicating two uniformly continuous and Bregman relatively asymptotically nonexpansive mappings. This article designs two parallel subgradient-like extragradient algorithms with an inertial effect for solving this pair of VIFPPs, where each algorithm consists of two parts which are of a mutually symmetric structure. With the help of suitable registrations, it is proven that the sequences generated by the suggested algorithms converge weakly and strongly to a solution of this pair of VIFPPs, respectively. Lastly, an illustrative instance is presented to verify the implementability and applicability of the suggested approaches.