  In this paper we investigate a system of coupled inequalities consisting of a
variational-hemivariational inequality and a quasi-hemivariational inequality
on Banach spaces. The approach is topological, and a wide variety of existence
results is established for both bounded and unbounded constraint sets in real
reflexive Banach spaces. The main point of interest is that no linearity
condition is imposed on the coupling functional, therefore making the system
fully nonlinear. Applications to Contact Mechanics are provided in the last
section of the paper. More precisely, we consider a contact model with
(possibly) multivalued constitutive law whose variational formulation leads to
a coupled system of inequalities. The weak solvability of the problem is proved
via employing the theoretical results obtained in the previous section. The
novelty of our approach comes from the fact that we consider two potential
contact zones and the variational formulation allows us to determine
simultaneously the displacement field and the Cauchy stress tensor.
