Maximal Element Theorem Research Articles

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 geometricform are proved. Next, a theorem concerning the existence of maximalelements for an LC-majorized correspondence is obtained.By the maximal element theorem, existence theorems of equilibrium point fora noncompact one-person game and for a noncompact qualitative game withLC-majorized correspondences are given. Using the lastresult and employing 'approximation approach', we prove theexistence of equilibria for abstract economies in which the constraintcorrespondence is lower (upper) semicontinuous instead of having lower(upper) open sections or open graphs in infinite-dimensional topologicalspaces. Then, as the applications, the existence theorems of solutions forthe quasi-variational inequalities and generalized quasi-variationalinequalities for noncompact cases are also proven. Finally, with theapplications of quasi-variational inequalities, the existence theorems ofNash equilibrium of constrained games with noncompact are given. Our resultsinclude many results in the literature as special cases.

A random section theorem and its applications

One random section theorem is obtained. As applications, random fixed point and random maximal element theorems are derived which are then used to give applications to the equilibria of random generalized games.

A minimax inequality with applications to existence of equilibrium points

A new minimax inequality is first proved. As a consequence, five equivalent fixed point theorems are formulated. Next a theorem concerning the existence of maximal elements for an Lc-majorised correspondence is obtained. By the maximal element theorem, existence theorems of equilibrium points for a non-compact one-person game and for a non-compact qualitative game with Lc-majorised correspondences are given. Using the latter result and employing an “approximation” technique used by Tulcea, we deduce equilibrium existence theorems for a non-compact generalised game with LC correspondences in topological vector spaces and in locally convex topological vector spaces. Our results generalise the corresponding results due to Border, Borglin-Keiding, Chang, Ding-Kim-Tan, Ding-Tan, Shafer-Sonnenschein, Shih-Tan, Toussaint, Tulcea and Yannelis-Prabhakar.

Differential inequalities via maximal element techniques

A VERY useful and effective tool of solving a wide class of problems concerning differential equations is that offered by the so-called differential inequalities (see, e.g., as classical references, Bellman [ 11, Coddington and Levinson [2], Corduneanu [3], Lakshmikantham and Leela [4], Walter [5], among many others). Taking into account the great importance of this topic, a considerable number of extensions of these classical results were obtained in the last years. In this direction, an important way ofextension seems to be that employed by Bebernes and Meisters [6], in which, an enumerable exceptional set is allowed in the formulation of the problem. However, the “supremum” technique used by the above quoted authors depends essentially on the continuity properties of the considered functions and therefore, it is a natural question to ask whether this continuity assumption cannot be removed. It is the main aim of this note to give a positive answer to this question, extending the above Bebernes-Meisters’ results to a class of order-semi-continuous functions and, in this context, it is not without importance to emphasize that the basic instrument in proving our main result is represented by a maximal element theorem on ordered metric spaces that may be considered as a metric variant of a general ordering principle due to Brbis and Browder [7]. Finally, it should be noted that our main results have a number of important applications to mean value theorems; this question will be treated in a forthcoming paper, under preparation.