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# Fully history-dependent evolution hemivariational inequalities with constraints

Dedicated to Professor Meir Shillor on the occasion of his 70th birthday

• In this paper we study a new class of abstract evolution first order hemivariational inequalities which involves constraints and history-dependent operators. First, we prove the existence and uniqueness of solution by using a mixed equilibrium formulation with suitable selected bifunctions combined with a fixed-point principle for history-dependent operators. Next, we deduce existence, uniqueness and regularity results for some special subclasses of problems which include a constrained history-dependent variational–hemivariational inequality, an evolution quasi-variational inequality with constraints, and an evolution second order hemivariational inequality with constraints. Then, we provide an application of the results to a dynamic unilateral viscoelastic frictional contact problem and show its unique weak solvability.

Mathematics Subject Classification: Primary: 47J20, 47J22, 49J40, 49J45; Secondary: 74G25, 74G30, 74M15.

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