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

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.