We study a class of abstract hemivariational inequalities in a reflexive Banach space. For this class, using the theory of multivalued pseudomonotone mappings and a fixed-point argument, we provide a result on the existence and uniqueness of the solution. Next, we investigate a static frictional contact problem with unilateral constraints between a piezoelastic body and a conductive foundation. The contact, friction and electrical conductivity condition on the contact surface are described with the Clarke generalized subgradient multivalued boundary relations. We derive the variational formulation of the contact problem which is a coupled system of two hemivariational inequalities. Finally, for such system we apply our abstract result and prove its unique weak solvability.
In this paper we study boundary optimal control of an evolutionary system governed by a history-dependent variational-hemivariational inequality. The inequality is a weak formulation of a dynamic frictional contact problem for a viscoelastic body with a multivalued normal damped response condition and a simplified version of the Coulomb law of dry friction. A continuous dependence result for the solution map is proved and the existence of optimal solutions to the control problem is established.
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