Analysis of a frictional contact problem for viscoelastic materials with long memory

We consider a mathematical model whichdescribes the frictional contact between a deformable body and afoundation. The process is time-dependent, the material behavioris described with a viscoelastic constitutive law with long memoryand the contact is modeled with subdifferential boundaryconditions. We derive the variational formulation of the problemwhich is of the form of a hemivariational inequality with Volterraintegral term for the displacement field. Then we prove existenceand uniqueness results in the study of abstract inclusions as wellas in the study of abstract hemivariational inequalities withVolterra integral term. The proofs are based on arguments onpseudomonotone operators, compactness and fixed point. We use theabstract results to prove the unique solvability of the frictionalcontact problem. Finally, we present examples of contact andfrictional boundary conditions for which our results work.