I have been told the following.
Suppose $\Omega\subseteq\mathbb{R}^n$ is a bounded borel set, $f$ is Carathéodory function on $\Omega\times\mathbb{R}=\{(x,s):x\in\Omega,s\in\mathbb{R}\}$, $f_s$, the partial derivative of $f$ w.r.t. the last variable $s$, is also Carathéodory, and $f_s$ is bounded. Then if the Nemyckij operator associated to $f$, which maps $L^2(\Omega)$ into itself, is Fréchet differentiable, there exist measurable functions $a,b:\Omega\to\mathbb{R}$ such that:
$$f(x,s)=a(x)s+b(x).$$
I have no idea how to prove this, and the proof is (in the teacher's own words) "difficult to find in books". The only reference I have is to a book by Ambrosetti-Prodi. I have found one, but it seems to be the wrong one, though "A primer of nonlinear analysis" combined with a sort of abstract I found online, seem to say otherwise. In any case, it is not completely available online. So can anyone either prove this or point me to a reference which is available online? (Possibly not Google books, I have learnt not to trust on those since the needed page can always be outside my preview, but maybe if there is a Google book with this then the pointer can post a screenshot of just the proof…)
Update
Thanks Tomás for your comment. The reason I asked for an online reference is that I had rather avoid buying a book for a single proof, and if I bought Krasnoel'skii I would probably end up only ever reading that single proof, since I believe my notes already contain enough info on Nemyckij operators. Krasnoel'skii doesn't seem available online.