In this paper we prove the existence and uniqueness of the weak solution for a dynamic thermoviscoelastic problem which describes frictional contact between a body and a foundation. We employ the nonlinear constitutive viscoelastic law with a long-term memory, which include the thermal effects and consider the general nonmonotone and multivalued subdifferential boundary conditions for the contact, friction and heat flux. The model consists of the system of the hemivariational inequality of hyperbolic type for the displacement and the parabolic hemivariational inequality for the temperature. The existence of solutions is proved by using recent results from the theory of hemivariational inequalities and a fixed point argument.

We study an abstract second order inclusion involving two nonlinear single-valued operators and a nonlinear multivalued term. Our goal is to establish the existence of solutions to the problem by applying numerical scheme based on time discretization. We show that the sequence of approximate solution converges weakly to a solution of the exact problem. We apply our abstract result to a dynamic, second order in time differential inclusion involving Clarke subdifferential of a locally Lipschitz, possibly nonconvex and nonsmooth potential. In two presented examples the Clarke subdifferential appears either in a source term or in a boundary term.

In this paper we prove the existence and uniqueness of the weak solution for a dynamic thermoviscoelastic problem describing contact problem between the body and foundation. The process is dynamic, the material behaviour is described by nonlinear viscoelastic law, strongly coupled with the thermal effects. The contact is modelled by nonmonotone subdifferential boundary conditions. The mechanical damage of the material is described by a parabolic equation. We use recent results from the theory of hemivariational inequailities and fixed point theorems.

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