We provide an existence result in the study of a new class of mixed variational problems. The problems are formulated on unbounded interval of time and involve history-dependent operators. The proof is based on generalized saddle point theory and various estimates, combined with fixed point arguments. Then, we consider a new mathematical model which describes the frictionless contact between a viscoelastic body and an obstacle. The process is quasistatic and the contact is modelled with a version of the normal compliance condition with unilateral constraint, which describes both the hardness and the softness of the foundation. We list the assumption on the data, derive a variational formulation of the problem, then we use our abstract result to prove its weak solvability.