Abstract and Applied Analysis
Volume 2003 (2003), Issue 5, Pages 311-324
Transfinite methods in metric fixed-point theory
Department of Mathematics, The University of Iowa, Iowa City 52242-1419, IA, USA
Received 28 September 2001
Copyright © 2003 W. A. Kirk. 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.
This is a brief survey of the use of transfinite induction in metric fixed-point theory. Among the results discussed in some detail is the author's 1989 result on directionally nonexpansive mappings (which is somewhat sharpened), a result of Kulesza and Lim giving conditions when countable compactness implies compactness, a recent inwardness result for contractions due to Lim, and a recent extension of Caristi's theorem due to Saliga and the author. In each instance, transfinite methods seem necessary.