  We consider a mixed variational problem in real Hilbert spaces, defined on on
the unbounded interval of time and governed by a history-dependent operator. We
state the unique solvability of the problem, which follows from a general
existence and uniqueness result obtained in our previous paper. Then, we state
and prove a general convergence result. The proof is based on arguments of
monotonicity, compactness, lower semicontinuity and Mosco convergence. Finally,
we consider a general optimization problem for which we prove the existence of
minimizers. The mathematical tools developed in this paper are useful in the
analysis of a large class of nonlinear boundary value problems which, in a weak
formulation, lead to history-dependent mixed variational problems. To provide
an example, we illustrate our abstract results in the study of a frictional
contact problem for viscoelastic materials with long memory.
