Approximation for the Hierarchical Constrained Variational Inequalities over the Fixed Points of Nonexpansive Semigroups

Abstract

The purpose of the present paper is to study the hierarchical constrained variational inequalities of finding a point $x^{*}$ such that $x^{*}\in \Omega ,\langle (A-\gamma f)x^{*}-(I-B)S{x}^{*},x-x^{*}\rangle \ge 0,\forall x\in \Omega$, where $\Omega$ is the set of the solutions of the following variational inequality: $x^{*}\in Ϝ,\langle (A-S)x^{*},x-x^{*}\rangle \ge 0,\forall x\in Ϝ$, where $A,B$ are two strongly positive bounded linear operators, $f$ is a $\rho$-contraction, $S$ is a nonexpansive mapping, and $Ϝ$ is the fixed points set of a nonexpansive semigroup $\{T(s){\}}_{s\ge 0}$. We present a double-net convergence hierarchical to some elements in $Ϝ$ which solves the above hierarchical constrained variational inequalities.