  A system of a first order history-dependent evolutionary
variational-hemivariational inequality with unilateral constraints coupled with
a nonlinear ordinary differential equation in a Banach space is studied. Based
on a fixed point theorem for history dependent operators, results on the
well-posedness of the system are proved. Existence, uniqueness, continuous
dependence of the solution on the data, and the solution regularity are
established. Two applications of dynamic problems from contact mechanics
illustrate the abstract results. First application is a unilateral viscoplastic
frictionless contact problem which leads to a hemivariational inequality for
the velocity field, and the second one deals with a viscoelastic frictional
contact problem which is described by a variational inequality.

Analysis of PDEs