On Mann Implicit Composite Subgradient Extragradient Methods for General Systems of Variational Inequalities with Hierarchical Variational Inequality Constraints

In a real Hilbert space, let the VIP, GSVI, HVI and CFPP denote a variational inequality problem, a general system of variational inequalities, a hierarchical variational inequality and a common fixed-point problem of a countable family of uniformly Lipschitzian pseudocontractive mappings and an asymptotically nonexpansive mapping, respectively. We design two Mann implicit composite subgradient extragradient algorithms with line-search process for finding a common solution of the CFPP, GSVI and VIP. The suggested algorithms are based on the Mann implicit iteration method, subgradient extragradient method with line-search process, and viscosity approximation method. Under mild assumptions, we prove the strong convergence of the suggested algorithms to a common solution of the CFPP, GSVI and VIP, which solves a certain HVI defined on their common solutions set.