Convergence theorems for composite viscosity approaches to systems variational inequalities in Banach spaces

Abstract

The purpose of this paper is to find a solution of a general system of variational inequalities (GSVI) with a hierarchical variational inequality (HVI) constraint for an infinite family of nonexpansive mappings. We
introduce general implicit and explicit iterative algorithms. Under some mild control conditions, we prove the strong convergence of the sequences generated by the proposed iterative algorithms to a solution of
the GSVI with a HVI constraint. Furthermore, we also present a weak convergence
theorem for the proposed explicit iterative algorithm involving an infinite family of nonexpansive mappings in a real Hilbert space.