  In this paper, we consider a mixed boundary value problem with a double phase
partial differential operator, an obstacle effect and a multivalued reaction
convection term. Under very general assumptions, an existence theorem for the
mixed boundary value problem under consideration is proved by using a
surjectivity theorem for multivalued pseudomonotone operators together with the
approximation method of Moreau-Yosida. Then, we introduce a family of the
approximating problems without constraints corresponding to the mixed boundary
value problem. Denoting by $\mathcal S$ the solution set of the mixed boundary
value problem and by $\mathcal S_n$ the solution sets of the approximating
problems, we establish the following convergence relation \begin{align*}
  \emptyset\neq w\text{-}\limsup\limits_{n\to\infty}{\mathcal
S}_n=s\text{-}\limsup\limits_{n\to\infty}{\mathcal S}_n\subset \mathcal S,
  \end{align*}
  where $w$-$\limsup_{n\to\infty}\mathcal S_n$ and
$s$-$\limsup_{n\to\infty}\mathcal S_n$ stand for the weak and the strong
Kuratowski upper limit of $\mathcal S_n$, respectively.
