Well-posedness by perturbations of variational-hemivariational inequalities with perturbations

In this paper, we consider an extension of the notion of well-posedness by perturbations, introduced by Zolezzi for a minimization problem, to a class of variational-hemivariational inequalities with perturbations in Banach spaces, which includes as a special case the class of mixed variational inequalities. Under very mild conditions, we establish some metric characterizations for the well-posed variational-hemivariational inequality, and show that the well-posedness by perturbations of a variational-hemivariational inequality is closely related to the well-posedness by perturbations of the corresponding inclusion problem. Furthermore, in the setting of finite-dimensional spaces we also derive some conditions under which the variational-hemivariational inequality is strongly generalized well-posed-like by perturbations.