A new class of hyperbolic variational-hemivariational inequalities driven by nonlinear evolution equations

The aim of the paper is to introduce and investigate a dynamical system which consists of avariational–hemivariational inequality of hyperbolic type combined with a non-linear evolutionequation. Such a dynamical system arises in studies of complicated contact problems in mechan-ics. Existence, uniqueness and regularity of a global solution to the system are established. Theapproach is based on a new semi-discrete approximation with an application of a surjectivity resultfor a pseudomonotone perturbation of a maximal monotone operator. A new dynamic viscoelasticfrictional contact model with adhesion is studied as an application, in which the contact boundarycondition is described by a generalised normal damped response condition with unilateral constraintand a multivalued frictional contact law.