Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings

ABSTRACT

In this paper, we introduce hybrid inertial subgradient extragradient algorithms with line-search process for solving a variational inequality problem (VIP) with pseudomonotone and Lipschitz continuous mapping and a common fixed-point problem (CFPP) of finitely many nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. The proposed algorithms are based on inertial subgradient extragradient method with line-search process, hybrid steepest-descent method, and viscosity approximation method. Under mild conditions, we prove strong convergence of the proposed algorithms to an element in the common solution set of the VIP and CFPP, which solves a certain hierarchical VIP defined on this common solution set.

Keywords:

• Hybrid inertial subgradient extragradient method
• pseudomonotone variational inequality
• common fixed point
• asymptotically nonexpansive mapping
• sequentially weak continuity

Mathematics Subject Classifications (2010):

• 47H09
• 47H10
• 47J20
• 47J25

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This second author's research was supported by the National Natural Science Foundation of China under Grant 11601348.