Abstract
AbstractThe concept of proper orbits of a map "Equation missing" is introduced and results of the following type are obtained. If a continuous self-map "Equation missing" of a Hausdorff topological space "Equation missing" has relatively compact proper orbits, then "Equation missing" has a fixed point. In fact, "Equation missing" has a common fixed point with every continuous self-map "Equation missing" of "Equation missing" which is nontrivially compatible with "Equation missing". 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.
Highlights
Let g be a mapping of a topological space X into itself
The purpose of this paper is to introduce the concept of proper orbits and to demonstrate its role in obtaining fixed points. (We use cl(A) to denote the closure of the set A.) Definition 1.1
Let g be a self-map of a topological space X and let x ∈ X
Summary
Let g be a mapping of a topological space X into itself. Let N denote the set of positive integers and ω = N ∪ {0}. 356 Common fixed points for compatible maps we will say that ᏻ(x) has diminishing diameters.) Subsequently Kirk [13] (1969) extended the concept to more general mappings. He proved the following interesting result for a metric space M. It will be shown that g has a common fixed point with every f ∈ Kg , a set we define. If the function g of Theorem 1.4 has proper orbits, g has a common fixed point with each f ∈ Kg. We show that this implication extends to very general settings
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