Invertibility of nonsmooth mappings

Let $F : \mathbb{R}^N \to \mathbb{R}^N$ be a locally Lipschitz continuous function. We prove that $F$ is a global homeomorphism or only injective, under suitable assumptions on the subdifferential $\partial F(x)$. We use variational methods, nonsmooth inverse function theorem and extensions of the Hadamard–Levy Theorem. We also address questions on the Markus–Yamabe conjecture.

Keywords

injectivity, invertibility, homeomorphism, Lipschitz continuous functions, Markus–Yamabe Conjecture

2010 Mathematics Subject Classification

26A16, 26B10, 37E30, 49J40, 49J52

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M. Montenegro has been supported by CNPq.

A. Presoto has been supported by FAPESP.

Received 24 February 2016

Received revised 1 March 2017

Accepted 15 March 2017

Published 26 September 2017