Algebraic univalence theorems for nonsmooth functions

Abstract

A well known univalence result due to D. Gale and H. Nikaido (1965, Math. Ann.159, 81-93) asserts that if the Jacobian matrix of a differentiable function from a closed rectangle K in Rn into Rn is a P-matrix at each point of K, then f is one-to-one on K. In this paper, by introducing the concepts of H-differentiability and H-differential of a function (as a set of matrices), we generalize the Gale-Nikaido result to nonsmooth functions. Our results further extend those of other authors valid for compact rectangles. We show that our results are applicable when the H-differential is any one of the following: the Jacobian matrix of a differentiable function, the generalized Jacobian of a locally Lipschitzian function, the Bouligand subdifferential of a semismooth function, and the C-differential of L. Qi (1993, Math. Oper. Res.18, 227-244). © 2000 Academic Press.