Systems of variational inequalities with hierarchical variational inequality constraints for asymptotically nonexpansive and pseudocontractive mappings

Abstract

The purpose of this paper is to introduce and analyze a hybrid extragradient-like implicit rule for finding a common solution of a general system of variational inequalities (GSVI) and a common fixed point problem (CFPP) of a countable family of uniformly Lipschitzian pseudocontractive mappings and an asymptotically nonexpansive mapping in Hilbert spaces. Here the hybrid extragradient-like implicit rule is based on Korpelevich’s extragradient method, the viscosity approximation method and the Mann iteration method. We prove the strong convergence of this method to a common solution of the GSVI and the CFPP, which solves a certain variational inequality on their common solution set. As an application, we give an algorithm to solve common fixed point problems of nonexpansive and pseudocontractive mappings, variational inequality problems and generalized mixed equilibrium problems in Hilbert spaces.