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

• Main Author: Jungck Gerald F
• Format: Article
• Language: English
• Published: SpringerOpen 2005-01-01
• Series: Fixed Point Theory and Applications
• Online Access: http://www.fixedpointtheoryandapplications.com/content/2005/645183

<p/> <p>The concept of <it>proper orbits</it> of a map <inline-formula><graphic file="1687-1812-2005-645183-i1.gif"/></inline-formula> is introduced and results of the following type are obtained. If a continuous self-map <inline-formula><graphic file="1687-1812-2005-645183-i2.gif"/></inline-formula> of a Hausdorff topological space <inline-formula><graphic file="1687-1812-2005-645183-i3.gif"/></inline-formula> has relatively compact proper orbits, then <inline-formula><graphic file="1687-1812-2005-645183-i4.gif"/></inline-formula> has a fixed point. In fact, <inline-formula><graphic file="1687-1812-2005-645183-i5.gif"/></inline-formula> has a common fixed point with every continuous self-map <inline-formula><graphic file="1687-1812-2005-645183-i6.gif"/></inline-formula> of <inline-formula><graphic file="1687-1812-2005-645183-i7.gif"/></inline-formula> which is nontrivially compatible with <inline-formula><graphic file="1687-1812-2005-645183-i8.gif"/></inline-formula>. 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.</p>