Abstract
We consider a new class of mixed variational problems arising in Contact Mechanics. The problems are formulated on the unbounded interval of time $$[0,+\infty )$$ [ 0 , + ? ) and involve history-dependent operators. For such problems we prove existence, uniqueness and continuous dependence results. The proofs are based on results on generalized saddle point problems and various estimates, combined with a fixed point argument. Then, we apply the abstract results in the study of a mathematical model which describes the frictionless contact between a viscoplastic body and an obstacle, the so-called foundation. The process is quasistatic and the contact is modelled with normal compliance and unilateral constraint, in such a way that the stiffness coefficient depends on the history of the penetration. We prove the unique weak solvability of the contact problem, as well as the continuous dependence of its weak solution with respect to the viscoplastic constitutive function, the applied forces, the contact conditions and the initial data.
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