Hybrid methods for a class of monotone variational inequalities

We use contractions to regularize a class of monotone variational inequalities, where the monotone operators are complements of nonexpansive mappings and the solutions are sought in the set of fixed points of another nonexpansive mapping. Such variational inequalities include monotone inclusions and convex optimization problems to be solved over the fixed point sets of nonexpansive mappings. Both implicit and explicit schemes are shown to be strongly convergent. An application in hierarchical minimization is included.