The study of minimax inequalities, abstract economics and applications to variational inequalities and Nash equilibria

In this survey, a new minimax inequality and one equivalent geometric form are proved. Next, a theorem concerning the existence of maximal elements fo r an L-C-majorized correspondence is obtained. By the maximal element theor em, existence theorems of equilibrium point for a noncompact one-person gam e and for a noncompact qualitative game with L-C-majorized correspondences are given. Using the last result and employing 'approximation approach', we prove the existence of equilibria for abstract economies in which the cons traint correspondence is lower (upper) semicontinuous instead of having low er (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 inequ alities for noncompact cases are also proven. Finally, with the application s of quasi-variational inequalities, the existence theorems of Nash equilib rium of constrained games with noncompact are given. Our results include ma ny results in the literature as special cases.