Title: The Knaster-Tarski theorem versus monotone nonexpansive mappings,
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.