  We consider a class of time-dependent inclusions in Hilbert spaces for which
we state and prove an existence and uniqueness result. The proof is based on
arguments of variational inequalities, convex analysis and fixed point theory.
Then we use this result to prove the unique weak solvability of a new class of
Moreau's sweeping processes with constraints in velocity. Our results are
useful in the study of mathematical models which describe the quasistatic
evolution of deformable bodies in contact with an obstacle. To provide some
examples we consider three viscoelastic contact problems which lead to
time-dependent inclusions and sweeping processes in which the unknowns are the
displacement and the velocity fields, respectively. Then we apply our abstract
results in order to prove the unique weak solvability of the corresponding
contact problems.
