Existence and Uniqueness of Solutions of Generalized Mixed Variational Inequalities

In this paper, we study the generalized mixed variational inequality, which encompasses both the generalized variational inequality and the mixed variational inequality. The core contribution of this paper is twofold. Firstly, by utilizing the principles of degree theory, we establish certain sufficient conditions for the existence of solutions to the generalized mixed variational inequality. Additionally, we formulate a sufficient condition that ensures the uniqueness of these solutions. Secondly, we recognize that the conditions outlined in our theorem are inapplicable to the generalized mixed polynomial variational inequality, a subclass within the broader family of generalized mixed variational inequalities. To address this, we employ an exceptional family of elements and establish an existence and uniqueness theorem specifically tailored for the generalized mixed polynomial variational inequality.