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

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