Fixed Point Theory and Applications 
Volume 2006 (2006), Article ID 57950, 13 pages
doi:10.1155/FPTA/2006/57950

Fixed point theorems in locally convex spaces—the Schauder mapping method

S. Cobzaş

 Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca 400084, Romania
Received 22 March 2005; Revised 22 July 2005; Accepted 6 September 2005

Abstract

In the appendix to the book by F. F. Bonsal, Lectures on Some Fixed Point Theorems of Functional Analysis (Tata Institute, Bombay, 1962) a proof by Singbal of the Schauder-Tychonoff fixed point theorem, based on a locally convex variant of Schauder mapping method, is included. The aim of this note is to show that this method can be adapted to yield a proof of Kakutani fixed point theorem in the locally convex case. For the sake of completeness we include also the proof of Schauder-Tychonoff theorem based on this method. As applications, one proves a theorem of von Neumann and a minimax result in game theory.