I am aware of the inverse function theorem for Lipschitz maps, which uses the notion of generalised derivative $\delta_{x_0} f$ of a Lipschitz map $f$, due to F.H.Clarke in On the inverse function theorem:

Theorem. Let $f : U \subset \mathbb{R}^n \to \mathbb{R}^n$ be Lipschitz, and suppose that every matrix $A ∈ \delta_{x_0}f$ is invertible. Then there are neighbourhoods $U$ of $x_0$ and $V$ of $f(x_0)$ in $\mathbb{R}^n$ and a Lipschitz map $g : V → U$ such that $f \circ g = \mathbb{1}_V$ and $g \circ f = \mathbb{1}_U$.

I am looking for a reference to an inverse function theorem in the similar setting which provides a bound for the inner radius of the neighbourhoods $U$ and $V$.

The closest I have been able to find is due to Ralph Howard's note The inverse function theorem for Lipschitz maps. In the case of $B_r$ ball of radius $r$ centered in zero his result reads:

Theorem. Let $f : \overline{B_r} \subset \mathbb{R}^n \to \mathbb{R}^n$, $f(O)=O$, $L:\mathbb{R}^n \to \mathbb{R}^n$ invertible linear map and $\rho<1$ such that $$\forall x_1,x_2\in \overline{B_r},\quad \lVert L^{-1}(f(x_2)-f(x_1)) - (x_2-x_1) \rVert \leq \rho \lVert x_2 - x_1 \rVert\,.$$ Then $f$ is injective on $\overline{B_r}$ with $$\forall x_1,x_2\in \overline{B_r},\quad \frac{1-\rho}{\lVert L^{-1}\rVert} \lVert x_2-x_1\rVert \leq \lVert f(x_2)-f(x_1) \rVert \leq \lVert L\rVert (1+\rho) \lVert x_2-x_1 \rVert \,.$$ Furthermore, $f$ restricted to open ball $B_r$ is lipeomorphism onto $V=f[B_r]$, where $V$ contains the open ball $B_{r_1}$ of radius $r_1=\rho\frac{1-\rho}{\lVert L^{-1}\rVert}$.

In the case of differentiable $f$, results related to what I am looking for are usually based on Newton's iterates. For example, see Theorem 4.1 of Xinghua, Convergence of Newton's method and inverse function theorem in Banach space.


1 Answer 1


I have just found quite a good answer in Marco Papi, On the Domain of the Implicit Function and Applications, Journal of Inequalities and Applications, Vol. 3, 2005. ISSN: 1025-5834. It states and proves only the implicit function theorem, but the bounds in the case of inverse function theorem are easy to obtain via the standard relationship between the two related theorems.

It deals with Clarke's generalized derivatives, and gets the explicit bounds, though not of the same type as in mentioned Theorem 4.1 of Xinghua (which deals with the differentiable case).

It is interesting to see personally interesting and powerful result that was not referenced for almost 9 years after publication (first reference in 2014). It seems to me as if the paper is ahead of the time, or as if there are some other similar results which I have not come across.


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