Relaxed hybrid steepest-descent methods with variable parameters for triple-hierarchical variational inequalities

In this article, we consider a monotone variational inequality with variational inequality constraint over the set of fixed points of a nonexpansive mapping, which is called a triple-hierarchical variational inequality (THVI). A relaxed hybrid steepest-descent method with variable parameters is introduced for solving the THVI. Strong convergence of the method to a unique solution of the THVI is studied under certain assumptions. We also investigate another monotone variational inequality with the variational inequality constraint over the set of common fixed points of a finite family of nonexpansive mappings, and present an iterative algorithm for solving such kind of problems. It is proven that under mild conditions the sequence generated by the proposed algorithm converges strongly to a unique solution of later problem.