Algebraic Univalence Theorems for Nonsmooth Functions

A well known univalence result due to D. Gale and H. Nikaido Ž1965, 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 functionŽas 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. QiŽ1993, Math. Oper. Res. 18, 227 244..