Boundary Hemivariational Inequalities of Hyperbolic Type and Applications

In this paper we examine two classes of nonlinear hyperbolic initial boundary value problems with nonmonotone multivalued boundary conditions characterized by the Clarke subdifferential. We prove two existence results for multidimensional hemivariational inequalities: one for the inequalities with relation between reaction and velocity and the other for the expressions containing the reaction–displacement law. The existence of weak solutions is established by using a surjectivity result for pseudomonotone operators and a priori estimates. We present also an example of dynamic viscoelastic contact problem in mechanics which illustrate the applicability of our results.