Remarks on Simplicial Spaces and L*-spaces of Kulpa and Szymanski

摘要

In a recent paper, Kulpa and Szymanski [15] introduced a series of theorems called In-fimum Principles in simplicial spaces. As applications, they derive fixed point theorems due to Schauder, Tychonoff, Kakutani, and Fan-Browder; minimax theorems; the Nash equilibrium theorem; the Gale-Nikaido-Debreu theorem; and the Ky Fan minimax inequality. Their study is based on and utilizes the techniques of simplicial structure and the Fan-Browder map. In this paper, we recall that for any abstract convex spaces satisfying abstract KKM principle, we can deduce such classical theorems without using any Infimum Principles. Moreover, we note that the newly defined L*-spaces in [15] are particular types of abstract convex spaces satisfying the abstract KKM principle, and add some remarks on them.