Invertibility of nonsmooth mappings

Let F :RN→RN be a locally Lipschitz continuous function. We prove that F is a global homeomorphism or only injective, under suitable assumptions on the subdifferential ∂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.