  We prove existence and comparison results for multi-valued variational
inequalities in a bounded domain $\Omega$ of the form \begin{equation*} u\in
K\,:\, 0 \in Au+\partial
I_K(u)+\mathcal{F}(u)+\mathcal{F}_\Gamma(u)\quad\text{in
}W^{1,\mathcal{H}}(\Omega)^*, \end{equation*} where $A\colon W^{1,
\mathcal{H}}(\Omega) \to W^{1, \mathcal{H}}(\Omega)^*$ given by
\begin{equation*} Au:=-\text{div}\left(|\nabla u|^{p(x)-2} \nabla u+ \mu(x)
|\nabla u|^{q(x)-2} \nabla u\right) \end{equation*} for $u \in W^{1,
\mathcal{H}}(\Omega)$, is the double phase operator with variable exponents and
$W^{1, \mathcal{H}}(\Omega)$ is the associated Musielak-Orlicz Sobolev space.
First, an existence result is proved under some weak coercivity condition. Our
main focus aims at the treatment of the problem under consideration when
coercivity fails. To this end we establish the method of sub-supersolution for
the multi-valued variational inequality in the space $W^{1,
\mathcal{H}}(\Omega)$ based on appropriately defined sub- and supersolutions,
which yields the existence of solutions within an ordered interval of
sub-supersolution. Moreover, the existence of extremal solutions will be shown
provided the closed, convex subset $K$ of $W^{1, \mathcal{H}}(\Omega)$
satisfies a lattice condition. As an application of the sub-supersolution
method we are able to show that a class of generalized
variational-hemivariational inequalities with a leading double phase operator
are included as a special case of the multi-valued variational inequality
considered here. Based on a fixed point argument, we also study the case when
the corresponding Nemytskij operators $\mathcal{F}, \mathcal{F}_\Gamma$ need
not be continuous. At the end, we give a nontrivial example of the construction
of sub- and supersolutions related to the problem above.
