International Journal of Mathematics and Mathematical Sciences
Volume 24 (2000), Issue 2, Pages 73-93

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

Sehie Park

Department of Mathematics, Seoul National University, Seoul 151-742, Korea

Received 20 October 1998

Copyright © 2000 Sehie Park. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.