Common fixed point theorems for compatible self-maps of Hausdorff topological spaces
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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. 1. Introduction Let g be a mapping of a topological space X into itself. Let N denote the set of positive integers and ω = N ∪ {0}. For x ∈ X, ᏻ(x) is called the orbit of g at x and defined by ᏻ(x) = {g k (x) : k ∈ ω}, where g o (x) = x. Thus, if n ∈ ω, the orbit of g at g n (x) is the set ᏻ(g n (x)) = {g k (x) : k ∈ ω and k ≥ n}. (Clearly, ᏻ(g n (x)) ⊂ ᏻ(x) for n ∈ N.) And if X has a metric or semimetric d, we will designate the diameter of a set M ⊂ X by δ(M) which of course is defined δ(M) = sup{d(x, y) : x, y ∈ M }. 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. The orbit ᏻ(x) of g at x is proper if and only if ᏻ(x) = {x} or there exists n = nx ∈ N such that cl(ᏻ(g n (x))) is a proper subset of cl(ᏻ(x)). If ᏻ(x) is proper for each x ∈ M ⊂ X, we will say that g has proper orbits on M. If M = X, we say g has proper orbits. The concept of proper orbits generalizes the concept of diminishing orbital diameters, which was introduced by Belluce and Kirk [1] in 1969. They introduced the concept of mappings with diminishing orbital diameters to obtain fixed point theorems for nonexpansive self-maps of metric spaces. A self-map g of a metric space X has diminishing orbital diameters if for each x ∈ X, δ(ᏻ(x)) < ∞, and whenever δ(ᏻ(x)) > 0, there exists n = nx ∈ N such that δ(ᏻ(x)) > δ(ᏻ(g n (x))). (If the given property holds for a specific x, Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:3 (2005) 355–363 DOI: 10.1155/FPTA.2005.355
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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. In particular, he proved the following interesting result for a metric space M. Theorem 1.2 (Kirk [13]). Suppose M is compact and g : M → M is continuous with diminishing orbital diameters. Then for each x ∈ M, some subsequence {g nk (x)} of the sequence {g n (x)} of iterates of x has a limit which is a fixed point of g. One purpose of this paper is to extend Theorem 1.2, and in the following manner. The underlying space will be a Hausdorff topological space. Instead of requiring that the space M be compact, we will require that the orbits be relatively compact. And we will replace the requirement
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