Fixed Point Theory and Applications
Volume 2005 (2005), Issue 3, Pages 355-363
doi:10.1155/FPTA.2005.355
Common fixed point theorems for compatible self-maps of Hausdorff topological spaces
Department of Mathematics, Bradley University, Peoria 61625, IL, USA
Received 13 July 2004; Revised 3 February 2005
Copyright © 2005 Gerald F. Jungck. 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.
Abstract
The concept of proper orbits of a map g is introduced and results of the following type are obtained. If a continuous self-map g of a Hausdorff topological space X has relatively compact proper orbits, then g has a fixed point. In fact, g has a common fixed point with every continuous self-map f of X which is nontrivially compatible with g. A collection of metric and semimetric space fixed point theorems follows as a consequence. Specifically, a theorem by Kirk regarding diminishing orbital diameters is generalized, and a fixed point theorem for maps with no recurrent points is proved.