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On Mann implicit composite subgradient extragradient methods for general systems of variational inequalities with hierarchical variational inequality constraints. (English) Zbl 1511.47070

Summary: 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.

MSC:

47J25 Iterative procedures involving nonlinear operators
49J40 Variational inequalities
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
65K15 Numerical methods for variational inequalities and related problems
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