Fixed point theorems for topological contractions and the Hutchinson operator

TL;DR

This paper extends fixed point theorems to topological spaces using topological contractions, introducing weak contractions and applying results to iterated function systems, broadening the scope beyond metric spaces.

Contribution

It introduces weak topological contractions and proves fixed point theorems in non-metric, non-compact spaces, including new classes like peripherally Hausdorff spaces.

Findings

Unique fixed points exist for topological contractions in compact T1 spaces.

Weak topological contractions have unique fixed points without requiring completeness.

The Hutchinson operator of a contractive IFS has a unique fixed point despite not always being closed.

Abstract

For a topological space $X$ a topological contraction on $X$ is a closed mapping $f:X\to X$ such that for every open cover of $X$ there is a positive integer $n$ such that the image of the space $X$ via the $n$th iteration of $f$ is a subset of some element of the cover. Every topological contraction in a compact $T_1$ space has a unique fixed point. As in the case of metric spaces and the classical Banach fixed point theorem, this analogue of Banach's theorem is true not only in compact but also in complete (here in the sense of \v{C}ech) $T_1$ spaces. We introduce a notion of weak topological contraction and in Hausdorff spaces we prove the existence of a unique fixed point for such continuous and closed mappings without assuming completeness or compactness of the space considered. These theorems are applied to prove existence of fixed points for mappings on compact subsets of linear spaces with weak topologies and for compact monoids. We also prove some fixed point results for $T_1$ locally Hausdorff spaces and, introduced here, peripherally Hausdorff spaces. An iterated function system on a topological space, IFS, is a finite family of closed mappings from the space into itself. It is contractive if for every open cover of $X$ for some positive integer $n$ the image of $X$ via a composition of $n$ mappings from the IFS is contained in an element of the cover. We show that in $T_1$ compact topological spaces the Hutchinson operator of a contractive IFS may not be closed as the mapping in the hyperspace of closed subsets of the space. Nevertheless, the Hutchinson operator of a contractive IFS has always a unique fixed point.