Some quantitative results on Lipschitz inverse and implicit functions theorems
This work offers refined quantitative bounds for known results in nonsmooth analysis, which may be useful for researchers working on Lipschitz mappings and implicit functions.
The paper provides quantitative estimates for Clarke's theorem on Lipschitz inverse functions and proves that the class of such mappings is open, along with a quantitative version of the Lipschitz implicit function theorem.
Let $ f: \mathbb{R} ^ n \rightarrow \mathbb{R}^n $ be a Lipschitz mapping with generalized Jacobian at $x_0$, denoted by $\partial f(x_0)$, is of maximal rank. F. H. Clarke (1976) proved that $f$ is locally invertible. In this paper, we give some quantitative assessments for Clarke's theorem on the Lipschitz inverse, and prove that the class of such mappings are open. Moreover, we also present a quantitative form for Lipschitz implicit function theorem.