Convergence of a generalized penalty method for variational–hemivariational inequalities

The aim the paper is to study a large class of variational-hemivariational inequalities involving constraints in a Banach space. First, we establish a general existence theorem for this class. Second, we introduce a sequence of penalized problems without constraints. Under the suitable assumptions, we prove that the Kuratowski upper limit with respect to the weak topology of the sets of solutions to penalized problems, is nonempty and is contained in the set of solutions to original inequality problem. Also, we prove the identity, when operator A satisfies -property. Finally, we illustrate the applicability of the theoretical results and we explore two complicated partial differential systems of elliptic type, which are an elliptic mixed boundary value problem involving a nonlinear nonhomogeneous differential operator with an obstacle effect, and a nonlinear elastic contact problem in mechanics with unilateral constraints, respectively.