A point on fixpoints in posets

Given a non-empty strictly inductive poset X, that is, a non-empty partially ordered set such that every non-empty chain has a least upper bound (a chain being a totally ordered subset), we are interested in sufficient conditions such that, given an element a_0 and a function f:X->X, there is some ordinal k such that a_{k+1}=a_k, where (a_k) is the transfinite sequence of iterates of f starting from a_0. This note summarizes known results about this problem and provides a slight generalization of some of them