  We investigate a generalized Lagrange multiplier system in a Banach space,
called a mixed variational-hemivariational inequality (MVHVI, for short), which
contains a hemivariational inequality and a variational inequality. First, we
employ the Minty technique and a monotonicity argument to establish an
equivalence theorem, which provides three different equivalent formulations of
the inequality problem. Without compactness for one of operators in the
problem, a general existence theorem for (MVHVI) is proved by using the
Fan-Knaster-Kuratowski-Mazurkiewicz principle combined with methods of
nonsmooth analysis. Furthermore, we demonstrate several crucial properties of
the solution set to (MVHVI) which include boundedness, convexity, weak
closedness, and continuity. Finally, a uniqueness result with respect to the
first component of the solution for the inequality problem is proved by using
the Ladyzhenskaya-Babuska-Brezzi (LBB) condition. All results are obtained in a
general functional framework in reflexive Banach spaces.
