Convergence of a generalized penalty and regularization method for quasi–variational–hemivariational inequalities

In the paper an elliptic quasi–variational–hemivariational inequality with constraints in a Banach space is studied. First, we apply the Minty technique, the KKM principle and the theory of nonsmooth analysis to establish the solvability of the inequality problem. Then, we employ a generalized penalty and regularization method for the inequality and introduce a family of penalized and regularized problems with no constraints and with Gâteaux differentiable potentials. Through a limit procedure, we prove that the Kuratowski upper limit with respect to the weak topology of the solution sets to penalized and regularized problems, is a nonempty subset of the solution set to the original inequality problem. Next, if a set-valued operator in the inequality has (S)+-property, then the Kuratowski upper limits with respect to the weak and strong topologies for the solution sets coincide. Finally, we illustrate our results by examining a nonlinear elliptic inclusion with the subgradient term of a locally Lipschitz function, mixed boundary conditions and an obstacle unilateral constraint which appears in a semipermeability problem.