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This work concerns the unilateral problem for the Klein-Gordon operator $$ \mathbb{L}=\frac{\partial^2 u}{\partial t^2}-M(|\nabla u|^2)\Delta u+M_1(|u|^2)u-f. $$ Using an appropriate penalization, we obtain a variational inequality for a perturbed equation, and then show the existence and uniqueness of solutions.