# The Knaster-Tarski theorem versus monotone nonexpansive mappings

## TL;DR

This paper extends fixed point theorems in partially ordered spaces with compact order intervals, showing the existence of common fixed points for monotone families, with applications to integral equations.

## Contribution

It introduces a new fixed point theorem for monotone maps in topological partially ordered spaces with compact intervals, generalizing recent results.

## Key findings

- Existence of supremum for directed subsets in certain ordered spaces
- Nonempty set of common fixed points with a maximal element for monotone families
- Application to Urysohn-type integral equations

## Abstract

Let $X$ be a partially ordered set with the property that each family of order intervals of the form $[a,b],[a,\rightarrow )$ with the finite intersection property has a nonempty intersection. We show that every directed subset of $X$ has a supremum. Then we apply the above result to prove that if $X$ is a topological space with a partial order $\preceq $ for which the order intervals are compact, $\mathcal{F}$ a nonempty commutative family of monotone maps from $X$ into $X$ and there exists $c\in X$ such that $c\preceq Tc$ for every $T\in \mathcal{F}$, then the set of common fixed points of $\mathcal{F}$ is nonempty and has a maximal element. The result, specialized to the case of Banach spaces gives a general fixed point theorem that drops almost all assumptions from the recent results in this area. An application to the theory of integral equations of Urysohn's type is also given.