On inertial subgradient extragradient algorithms for equilibria systems and hierarchical variational inequalities for countable nonexpansive mappings

Abstract

In a real Hilbert space, let the SGEP, VIP, HVI and CFPP represent a system of generalized equilibrium problems, a pseudomonotone variational inequality problem, a hierarchical variational inequality and a common fixed-point problem of countable nonexpansive mappings and an asymptotically nonexpansive mapping, respectively. We propose two relaxed inertial subgradient extragradient implicit rules with line-search process for finding a solution of the HVI with the SGEP, VIP and CFPP constraints. The designed algorithms are on the basis of the subgradient extragradient rule with line-search process, inertial iteration approach, hybrid deepest-descent method and Mann implicit iteration technique. Under suitable conditions, we prove the strong convergence of the designed algorithms to a solution of the HVI with the SGEP, VIP and CFPP constraints.