Regularity results for time-dependent variational and quasi-variational inequalities and application to the calculation of dynamic traffic network

The aim of this paper is to consider time-dependent variational and quasi-variational inequalities and to study under which assumptions the continuity of solutions with respect to time can be ensured. Making an appropriate use of the set convergence in Mosco's sense, we are able to prove continuity results for strongly monotone variational and quasi-variational inequalities. The continuity results allow us to provide a discretization procedure for the calculation of solutions to the variational inequalities and, as a consequence, we can solve the time-dependent traffic network equilibrium problem.