A mixed hemivariational-variational problem and applications

We consider an abstract system with Lagrange multipliers which consists of a hemivariational inequality and a variational inequality. The hemivariational inequality is governed by a hemicontinuous, generalized monotone, possibly nonlinear operator as well as by a bilinear form. This bilinear form also governs the variational inequality. We are looking for a pair solution in a subset of a product of two real reflexive Banach spaces. In order to illustrate the applicability of the abstract results, two examples in terms of PDEs are delivered. Each example is related to the weak solvability of a boundary value problem arising in contact mechanics. (C) 2018 Elsevier Ltd. All rights reserved.