The Study of Minimax Inequalities, Abstract Economics and Applications to Variational Inequalities and Nash Equilibria

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In this survey, a new minimax inequality and one equivalent geometric form are proved. Next, a theorem concerning the existence of maximal elements for an LC-majorized correspondence is obtained. By the maximal element theorem, existence theorems of equilibrium point for a noncompact one-person game and for a noncompact qualitative game with LC-majorized correspondences are given. Using the last result and employing ‘approximation approach’, we prove the existence of equilibria for abstract economies in which the constraint correspondence is lower (upper) semicontinuous instead of having lower (upper) open sections or open graphs in infinite-dimensional topological spaces. Then, as the applications, the existence theorems of solutions for the quasi-variational inequalities and generalized quasi-variational inequalities for noncompact cases are also proven. Finally, with the applications of quasi-variational inequalities, the existence theorems of Nash equilibrium of constrained games with noncompact are given. Our results include many results in the literature as special cases.

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