We investigate a generalized Lagrange multiplier system in a Banach space, a mixed variational-hemivariational inequality (MVHVI, for short), which a hemivariational inequality and a variational inequality. First, we the Minty technique and a monotonicity argument to establish an theorem, which provides three different equivalent formulations of inequality problem. Without compactness for one of operators in the, a general existence theorem for (MVHVI) is proved by using the-Knaster-Kuratowski-Mazurkiewicz principle combined with methods of analysis. Furthermore, we demonstrate several crucial properties of solution set to (MVHVI) which include boundedness, convexity, weak, and continuity. Finally, a uniqueness result with respect to the component of the solution for the inequality problem is proved by using Ladyzhenskaya-Babuska-Brezzi (LBB) condition. All results are obtained in a functional framework in reflexive Banach spaces.
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Bai et al. (2019) studied this question.