SOME QUANTITATIVE RESULTS ON LIPSCHITZ INVERSE AND IMPLICIT FUNCTIONS THEOREMS

Abstract

Let f : Rn → Rn be a Lipschitz mapping with generalized Jacobian

at x0, denoted by ∂f(x0), 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.