Fixed points, intersection theorems, variational inequalities, and equilibrium theorems

Abstract

Abstract. From a fixed point theorem for compact acyclic maps defined on admissible convex sets in the sense of Klee, we first deduce collectively fixed point theorems, intersection theorems for sets with convex sections, and quasi-equilibrium theorems. These quasi-equilibrium theorems are applied to give simple and unified proofs of the known variational inequalities of the Hartman-Stampacchia-Browder type. Moreover, from our new fixed point theorem, we deduce new variational inequalities which can be used to obtain fixed point results for convex-valued maps. Finally, various general economic equilibrium theorems are deduced in the forms of the Nash type, the Tarafdar type, and the Yannelis-Prabhakar type. Our results are stated for not-necessarily locally convex topological vector spaces and for abstract economies with arbitrary number of commodities and agents. Our new results extend a lot of known works with much simpler proofs.

Generalizations of the Nash Equilibrium Theorem in the KKM Theory

“…This procedure can be called "from the KKM principle to the Nash equilibria" simply, "K to N" ; see 27 . In 1999 17 , we obtained a "K to N" for G-convex spaces. These results extended and unified a number of known results for particular types of G-convex spaces; see also [18][19][20][21]42 . Therefore, the procedure also holds for Lassonde type-convex spaces, Horvath's cspace, hyperconvex metric spaces, and others.…”

An essential map theory for ${\mathcal U}_c^{\kappa}$ and PK maps

“…In this paper we extend this notion (the proofs here are different) to enable us to discuss a more general class of maps, namely the P K maps. The theory and results in this paper complement and extend previously known results in the literature (see [1], [2], [6], [8], [10], [11] and the references therein).…”

Compact Browder maps and equilibria of abstract economies

“…Note also that Theorems 5-8 improve corresponding ones in our previous works [18] and [22]. Finally, we notice that Lemma 3 of Kim and Ding [14] is an incorrectly stated version of our Corollary 4.1 and they had to assume local convexity of the space E.…”