Editing Knaster–Tarski theorem (section)

==Weaker versions of the theorem==
Weaker versions of the Knaster–Tarski theorem can be formulated for ordered sets, but involve more complicated assumptions. For example:{{citation needed|date=February 2024}}

:''Let L be a [[partially ordered set]] with a [[least element]] (bottom) and let f'' : ''L'' → ''L be an [[Monotonic function#In order theory|monotonic function]]. Further, suppose there exists u in L such that f''(''u'') ≤ ''u and that any [[Total order#Chains|chain]] in the subset <math>\{x \in L \mid x \leq f(x), x \leq u\}</math> has a supremum. Then f admits a [[least fixed point]].''

This can be applied to obtain various theorems on [[invariant set]]s, e.g. the Ok's theorem:

:''For the monotone map F'' : ''P''(''X''{{hairsp}}) → ''P''(''X''{{hairsp}}) ''on the [[powerset|family]] of (closed) nonempty subsets of X, the following are equivalent: (o) F admits A in P''(''X''{{hairsp}}) ''s.t. <math>A \subseteq F(A)</math>, (i) F admits invariant set A in P''(''X''{{hairsp}}) ''i.e. <math>A = F(A)</math>, (ii) F admits maximal invariant set A, (iii) F admits the greatest invariant set A.''

In particular, using the Knaster-Tarski principle one can develop the theory of global attractors for noncontractive discontinuous (multivalued) [[iterated function system]]s. For weakly contractive iterated function systems the [[Kantorovich theorem]] (known also as Tarski-Kantorovich fixpoint principle) suffices.

Other applications of fixed-point principles for ordered sets come from the theory of [[differential equation|differential]], [[integral equation|integral]] and [[operator equation|operator]] equations.