Generalized penalty method for semilinear differential variational inequalities

We consider a semilinear differential variational inequality in reflexive Banach spaces, governed by a set of constraints K. We associate to a sequence of problems where, for each , is a differential variational inequality governed by a set of constraints and a penalty parameter . We use a result in [Liu ZH, Zeng SD. Penalty method for a class of differential variational inequalities. Appl Anal. 2019;1-16. doi:] to prove the unique solvability of problems and . Then, we prove that, under appropriate assumptions, the sequence of solutions to Problem converges to the solution of the original problem . The proof is based on arguments of compactness, pseudomonotonicity and Mosco convergence. We also present two relevant particular case of our convergence result, including a recent result obtained in [Liu ZH, Zeng SD. Penalty method for a class of differential variational inequalities. Appl Anal. 2019;1-16. doi:], in the case . Finally, we provide an example of initial and boundary value problem for which our abstract results can be applied.