Study of the existence and uniqueness of solutions for a class of Kirchhoff-type variational inequalities involving using Young measures

Abstract

This paper is devoted to discussing the existence of solutions for a class of Kirchhoff-type variational inequalities: \(-\mathcal {M}\biggl (\displaystyle \int _{\Omega }\mathcal {A}(z,\nabla u )\mathrm {~d}z\biggl )~\displaystyle \int _{\Omega }\mathcal {G}(z,\nabla u).(\nabla \vartheta -\nabla u)\mathrm {~d}z \ge \displaystyle \int _{\Omega }\Phi (z,u)(\vartheta -u)\mathrm {~d}z \), for \(\upsilon \) belonging to the following convex set \(\mathcal {S}_{\psi , \theta }\). By employing Young measure theory in conjunction with a theorem formulated by Kinderlehrer and Stampacchia, we attain the intended result.