I am working on a perturbation h of the identity I on a Banach space X, i.e. h(x) = x – f(x), with f(0)=0.
f is regular enough on X, but I do not know if the regularity holds on a subspace Y of X.
More precisely:
f is C^1_{Lip} and the operator (I – f_x(0))^{-1} exists and it is bounded in L(X), so that the classic conditions for the existence of a local inverse in a neighborhood of x=0 are satisfied. Let this inverse function be h^{-1}: B(0,R_1) \to B(0,R).
Now, the question.
Let Y be a Banach subspace of X, embedded continuously with norm \| \cdot \|_Y in X.
f is Lipschitz in the Y-norm, moreover we have h(Y) \subseteq Y
and if y \in B(0,R_1) \cap Y then h^{-1}(y) \in Y.
We have also that (I – f_x(0))^{-1}|_Y \in L(Y).
Are these conditions sufficient for the continuity of h^{-1}(y)|_Y , (or at least for the boundedness) without assuming that h|_Y in C^1 ?
I would like to suggest to you to read the book by A. Dontchev and R.T. Rockafellar entitled "Implicit function Theorem" second edition appeard in Springer. If you need help you can contact me.
Thank you very much.
I have found an answer in the book "Applied Nonlinear Analysis" by Ekeland and Aubin, and also in the Notes by Ralph Howard: "The Inverse Function Theorem for Lipschitz Maps" (inverse.pdf)
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