Steinhaus Chessboard Theorem implies Equlibrium Theorem
by
Marian Turzanski
University of Silesia, Department of Mathematics, Bankowa 14, 40007 Katowice, Poland

W. Kulpa proved the existence of a stable-like point (Equilibrium Theorem)and applied this theorem to show the existence of rational divisions of bounded Lebesgue measurable sets in Euclidean spaces.

We present an algorithm for determining (on the plane) the place where equilibrium points are.For this purpose we use the Steinhaus Chessboard Theorem.

The existence of a market equilibrium is a classical problem in economics(Walras, von Neumann, Nash). The Brouwer fixed point theorem was a main mathematical tool in the Nash papers, for which he has won a Nobel prize in economics.The Brouwer Theorem is an easy consequence of Kulpa's Equilibrium Theoem. Consequently, an algorithm for determining a fixed point is also given.

Date received: June 15, 2000