Common fixed point theorems for compatible self-maps of Hausdorff topological spaces

Abstract

The concept of proper orbits of a map is introduced and results of the following type are obtained. If a continuous self-map of a Hausdorff topological space has relatively compact proper orbits, then has a fixed point. In fact, has a common fixed point with every continuous self-map of which is nontrivially compatible with . 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.