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Suppose $F:X \to Y$ is a map from Banach spaces $X=\widetilde{C}^{k+2, \alpha}(S)$ to $Y = \widetilde{C}^{k, \alpha}(S)$ where $S = I \times [0,T].$

Suppose the derivative $DF(u):X \to Y$ exists and is continuous.

(1) Am I right that the statement

$DF[u]^{-1}$ are uniformly bounded for bounded $u$

means $\lVert DF[u]^{-1}\rVert \leq M$ holds for all bounded (in what?) $u$ and $M$ is a constant not depending on $u$?

(2) If I write $DF[u]h = f$, and get a bound $\lVert h \rVert \leq C\lVert h_0\rVert + C\lVert f \rVert$ where $h_0 = h(\cdot, 0)$, then how does this show that $DF[u]^{-1}$ is uniformly bounded? (If the constant doesn't depend on $u$).

(3) Willie Wong said that if the $DF[u]^{-1}$ are uniformly bounded then applying the inverse function theorem to F, the size of the neighbourhood of $F(u^0)$ that is invertible doesn't depend on $F(u^0)$. Can someone give me a reference for this fact?

Thanks.

(These questions stem from Inverse function theorem in Banach space to prove short time existence of PDE (explanation of statements), and I didn't want to keep asking questions there!)

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    All the proofs that I know (could be just my own ignorance) of this fact requires using something which states that \exists \delta > 0 such that for all $u,v,w\in X$ with \|v - u\| + \|w - u\| < \delta, $ \| (v-w) - DF^{-1}[u] (F(v) - F(w)) \| \leq \frac12 \|v-w\| $ This suggests that we can probably weaken the $C^{1,1}$ condition on $F$ to something something similar to uniform continuity of $DF$ and still have the proof go through. Which also suggests that the additional regularity properties may be unnecessary if you restrict to $K\subset X$ compact.2012-07-26

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