  We consider an abstract mixed variational problem governed by a nonlinear
operator $A$ and a bifunctional $J$, in a real reflexive Banach space $X$. The
operator $A$ is assumed to be continuous, Lipschitz continuous on each bounded
subset of $X,$ and generalized monotone. First, we pay attention to the unique
solvability of the problem. Next, we prove a continuous dependence result of
the solution with respect to the data. Based on this result we prove the
existence of at least one solution for an associated optimization problem.
Finally, we apply our abstract results to the well-posedness and the
optimization of an antiplane frictional contact model for nonlinearly elastic
materials of Hencky-type.
