Double phase obstacle problems with multivalued convection and mixed boundary value conditions

<p style='text-indent:20px;'>In this paper, we consider a mixed boundary value problem with double phase partial differential operator, an obstacle effect and multivalued reaction convection term. Under very general assumptions, existence theorem for the under consideration is proved by using surjectivity pseudomonotone operators together approximation method of Moreau-Yosida. Then, introduce family approximating problems without constraints corresponding to problem. Denoting <inline-formula><tex-math id="M1">\begin{document}$ \mathcal S $\end{document}</tex-math></inline-formula> solution set id="M2">\begin{document}$ S_n sets problems, establish following convergence relation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} \emptyset\neq w-\limsup\limits_{n\to\infty}{\mathcal S}_n = s-\limsup\limits_{n\to\infty}{\mathcal S}_n\subset S, \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where id="M3">\begin{document}$ w $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M4">\begin{document}$ \limsup_{n\to\infty}\mathcal id="M5">\begin{document}$ s id="M6">\begin{document}$ stand weak strong Kuratowski upper limit id="M7">\begin{document}$ $\end{document}</tex-math></inline-formula>, respectively.</p>