Two abstract mixed variational problems and applications in contact mechanics

We consider two abstract mixed variational problems. Each of them consists of a system of two variational inequalities. The first problem involves two convex functionals while the second one involves a convex functional and a bifunctional which depends on a Lagrange multiplier in the first argument and is convex in the second argument. We obtain existence and uniqueness results for the first problem. Then, we combine the results we get with a fixed point technique in order to investigate the existence and the uniqueness of the solution of the second problem. The abstract results we obtain can be applied to the weak solvability of frictional contact models for nonlinearly elastic materials. To illustrate the applicability, three examples of frictional contact models are discussed.