A class of time-dependent quasi-variational–hemivariational inequalities with applications

In this paper a class of time-dependent multivalued quasi-variational inequalities of elliptic type with a solution dependent constraint set, is studied. Its solvability and the closedness of the solution set are proved. Then, the results are applied to constrained quasi-variational– hemivariational inequalities for which the relaxed monotonicity condition is not required. Finally, the abstract results are illustrated by two applications. The first one is a time-dependent frictional contact problem with locking materials, and the second one is the stationary incompressible Navier–Stokes equation which models a generalized Newtonian fluid of Bingham type. Results on existence and the closedness of the solution sets are established for both applications.