Double phase obstacle problems with variable exponent

. This paper is devoted to the study of a quasilinear elliptic inclusion problem driven by a double phase differential operator with variable exponents, an obstacle effect and a multival-ued reaction term with gradient dependence. By using an existence result for mixed variational inequalities with multivalued pseudomonotone operators and the theory of nonsmooth analysis, we examine the nonemptiness, boundedness and closedness of the solution set to the problem under consideration. In the second part of the paper we present some convergence analysis for approximated problems. To be more precise, when the obstacle function is approximated by a suitable sequence, applying a generalized penalty technique, we introduce a family of perturbed problems without constraints associated to our problem and prove that the solution set of the original problem can be approached by the solution sets of the perturbed problems in the sense of Kuratowski.