  In this paper we establish existence, uniqueness, and boundedness results for
an elliptic variational inequality coupled with a nonlinear ordinary
differential equation. Under the general framework, we present a new
application modelling the antiplane shear deformation of a static frictional
adhesive contact problem. The adhesion process has been extensively studied,
but it is usual to assume a priori that the intensity of adhesion is bounded by
introducing truncation operators. The aim of this article is to remove this
restriction.
  The proof is based on an iterative approximation scheme showing that the
problem has a unique solution. A key ingredient is finding uniform a priori
bounds for each iterate. These are obtained by adapting versions of the Moser
iteration to our system of equations.
