This question reminds me of Peter Freyd's work on fixed points.  I forget the reference for that but Google will show you some of the follow-up work that other people have done since.

Fixed points of this nature live in domains, which live in categories that do not have pullbacks and equalisers, so the notion of kernel is not going to help you.

Be warned, however: there are results in this area that are extremely clever tricks, but such tricks are possibly only discovered once per decade.  In the meantime, you will cover thousands of pages with calculations that lead to absolutely nothing.

The two examples I had in mind were **Pataraia's** and [**Bekić**'s](https://mathoverflow.net/questions/179790/is-fixed-point-property-for-posets-preserved-by-products/179890#179890) fixed point theorems. (The late Georgian Dito) Pataraia's theorem is that in a DCPO $P$ with $\bot$, every monotone $f:P\to P$ has a least fixed point. As the comments indicate, the proof is very clever, very short and available in various places on the Web, though Pataraia never published it in a journal.