Smooth dependence on parameters of solutions of variational inequalities

Variational inequalities on closed convex sets of a certain type (not cones in general) with a multidimensional parameter are considered, and local Ck-smooth dependence of their solutions on the parameter is proved. The basic idea is to show that under some assumptions, the variational inequality is in a neighbourhood of a given solution equivalent to an equation for which implicit function theorem can be used. The result is applied to a model of a beam unilaterally supported by a finite number of fixed obstacles at different heights, and to Nash equilibria with parameters.