Algebraic univalence theorems for nonsmooth functions

A well known univalence result due to D. Gale and H. Nikaido (1965, Math. A nn. 159, 81-93) asserts that if the Jacobian matrix of a differentiable fun ction from a closed rectangle K in R-n into R-n is a P-matrix at each point of K, then f is one-to-one on K. In this paper, by introducing the concept s of H-differentiability and ii-differential of a function las a set of mat rices), we generalize the Gale-Nikaido result to nonsmooth functions. Our r esults further extend those of other authors valid for compact rectangles. We show that our results are applicable when the ii-differential is any one of the following: the Jacobian matrix of a differentiable function, the ge neralized Jacobian of a locally Lipschitzian function, the Bouligand subdif furential of a semismooth function, and the C-differential of L. Qi (1993, Math. Oper. Res. 18, 227-244). (C) 2000 Academic Press.