Convergence and some control conditions of hybrid steepest-descent methods for systems of variational inequalities and hierarchical variational inequalities

The purpose of this paper is to find a solution of a general system of variational inequalities (for short, GSVI), which is also a unique solution of a hierarchical variational inequality (for short, HVI) for an infinite family of nonexpansive mappings in Banach spaces. We introduce general implicit and explicit iterative algorithms, which are based on the hybrid steepest-descent method and the Mann iteration method. Under some appropriate conditions, we prove the strong convergence of the sequences generated by the proposed iterative algorithms to a solution of the GSVI, which is also a unique solution of the HVI.