  The overarching goal of this paper is to introduce and investigate a new
nonlinear system driven by a nonlinear differential equation, a
history-dependent quasivariational inequality, and a parabolic variational
inequality in Banach spaces. Such a system can be used to model quasistatic
frictional contact problems for viscoelastic materials with long memory, damage
and wear. By using the Banach fixed point theorem, we prove an existence and
uniqueness theorem of solution for such a system under some mild conditions. As
a novel application, we obtain a unique solvability of a quasistatic
viscoelastic frictional contact problem with long memory, damage and wear.
