  In this paper we consider a mixed boundary value problem with a
nonhomogeneous, nonlinear differential operator (called double phase operator),
a nonlinear convection term (a reaction term depending on the gradient), three
multivalued terms and an implicit obstacle constraint. Under very general
assumptions on the data, we prove that the solution set of such implicit
obstacle problem is nonempty (so there is at least one solution) and weakly
compact. The proof of our main result uses the Kakutani-Ky Fan fixed point
theorem for multivalued operators along with the theory of nonsmooth analysis
and variational methods for pseudomonotone operators.
