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I'm looking for properties of the Newton-derivative, defined as follows: A function $F \colon X \to Y$ is Newton differentiable at $x\in X$ if there exists $\varepsilon>0$ and a function $G\colon B_\varepsilon(x) \to L(X,Y)$ such that \begin{align*} \lim_{y\to0} \frac{\|F(x + y) -F(x) -G(x+y)[y]\|_Y}{\|y\|_X} = 0. \end{align*}

$G$ is called Newton-derivative. Can you provide me with a proof (or any reference) for a chain rule for this differential?

Actually, I need to know about the Newton-derivative of the function \begin{align*} L^2(\Omega) \ni u \mapsto \max(-1,\min(1,u)), \end{align*} which hopefully exists... Thanks a lot, Malte.

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    $\begingroup$ I assume that $v+y$ should be something else, because $v$ is undefined, but it looks like this is another name for the Fréchet dferivative. $\endgroup$
    – Federico Poloni
    Commented Nov 15, 2017 at 14:51
  • $\begingroup$ It was a typo. I edited the question. And regarding your comment with the Fréchet derivative, I think it's a little weaker and especially only a local property $\endgroup$
    – Malte Winckler
    Commented Nov 15, 2017 at 15:30
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    $\begingroup$ To anybody who has doubts: This is not the Frechet derivative and the $x+y$ in the argument of G is correct. More to the question: First, the Newton derivative is not unique. It could be that a Newton derivative in this case may be u where it is between -1 and +1 and zero elsewhere but I remember vaguely that there may be a problem with the norms here... (Buzzword "norm gap", probably in a paper by Kunisch or Ulbrich). $\endgroup$
    – Dirk
    Commented Nov 15, 2017 at 17:42
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    $\begingroup$ That's a standard result that can be found in, e.g., the books by Ulbrich (Theorem 3.69), Ito and Kunisch (Lemma 8.4), or (shameless plug) my lecture notes (Theorem 9.2). Note that there is no chain rule for two Gâteaux differentiable functions. And a(!) Newton derivative is given by the pointwise multiplication with $v(x) = 1$ if $u(x)\in [-1,1]$ and $0$ else -- iff, as @Dirk writes, your function is considered as mapping to $L^p$ for $p<2$, see again the mentioned literature. (For both results, the proof is virtually the same as for Fréchet derivatives.) $\endgroup$
    – Christian Clason
    Commented Nov 15, 2017 at 19:54
  • $\begingroup$ Thank you both for your remarks and this extensive literature! I consider this question solved. $\endgroup$
    – Malte Winckler
    Commented Nov 16, 2017 at 9:02

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