A point on fixpoints in posets

Let (X,≤) be a non-empty strictly inductive poset, that is, a non-empty partially ordered set such that every non-empty chain Y has a least upper bound lub(Y ) ∈ X , a chain being a subset of X totally ordered by ≤. We are interested in sufficient conditions such that, given an element a0 ∈ X and a function f : X → X , there is some ordinal k such that ak+1 = ak, where ak is the transfinite sequence of iterates of f starting from a0 (implying that ak is a fixpoint of f):