Hybrid Viscosity-like Approximation Methods for General Monotone Variational Inequalities

Abstract

In this paper, we introduce two implicit and explicit hybrid viscositylike approximation methods for solving a general monotone variational inequality, which covers their monotone variational inequality with $C = H$ as a special case. We use the contractions to regularize the general monotone variational inequality, where the monotone operators are the generalized complements of nonexpansive mappings and the solutions are sought in the set of fixed points of another nonexpansive mapping. Such general monotone variational inequality includes some monotone inclusions and some convex optimization problems to be solved over the fixed point sets of nonexpansive mappings. Both implicit and explicit hybrid viscosity-like approximation methods are shown to be strongly convergent. In the meantime, these results are applied to deriving the strong convergence theorems for a general monotone variational inequality with minimization constraint. An application in hierarchical minimization is also included.