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

Abstract

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.

System of Variational Inclusions and Fixed Points of Pseudocontractive Mappings in Banach Spaces

“…The general system of variational inclusions (GSVI) is to find (x * , y * ) ∈ C × C such that 0 ∈ x * − y * + ρ 1 (A 1 y * + M 1 x * ), 0 ∈ y * − x * + ρ 2 (A 2 x * + M 2 y * ), (4) where ρ 1 and ρ 2 are two positive constants. In 2010, Qin et al [4] introduced a relaxed extragradient-type method for solving GSVI (4), and proved a strong convergence theorem for the proposed method (for its related results in the literature, see, e.g., [1,[5][6][7][8][9][10][11][12][13][14][15][16][17][18]). Furthermore, Aoyama et al [19] considered the following variational inequality: Find…”

“…where η > 0 is a constant and Π C is a sunny nonexpansive retraction from E onto C. In particular, if E = H a Hilbert space, then Π C coincides with the metric projection P C from H onto C. Recently, many authors have studied the problem of finding a common element of the set of fixed points of nonlinear mappings and the set of solutions to variational inequalities by iterative methods (see, e.g., [1][2][3]5,6,[8][9][10]12,[14][15][16][18][19][20][21][22][23][24]).…”