Approximation for the hierarchical constrained variational inequalities over the fixed points of nonexpansive semigroups (Q370064)

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