Fixed Point Theory and Applications
Volume 2006 (2006), Article ID 57950, 13 pages
Fixed point theorems in locally convex spaces—the Schauder mapping method
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
Copyright © 2006 S. Cobzaş. 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.
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.