Let $(E,\mathscr E)$ be a measurable space and denote by $\mathrm b\mathscr E$ the space of all Borel measurable bounded functions $f:E\to\mathbb R$. On this space the partial order is given by $ f\leq g \quad \Leftrightarrow \quad f(x)\leq g(x)\text{ for all }x\in E $ and let us use the notation $\mathrm b\mathscr E_{\geq f} = \{g\in \mathrm b\mathscr E:g\geq f\}$. Let $A$ be the monotone operator on the space$\mathrm b\mathscr E$, which is not necessarily linear, and let $\mathrm{fix}(A)$ be the set of fixpoints of the operator $A$: $ \mathrm{fix}(A) = \{g\in \mathrm b\mathscr E:g = Ag\}. $ Let $f_0\in \mathrm b\mathscr E$ be such that $f_0\leq Af_0$ and construct the sequence $f_{n} = A^n f_0$. This sequence converges point-wise non-decreasingly to a measurable function $f:E\to\mathbb R$. One can prove that whenever $f\in \mathrm{fix}(A)$ it holds that $f = \min\left\{\mathrm{fix}(A)\cap \mathrm b\mathscr E_{\geq f_0}\right\}$.
I guess that it is already known, and maybe a consequence of a more general result. I am looking for the classical reference to this fact.