On Characterizations of P- and P ₀-Properties in Nonsmooth Functions

For a Fréchet differentiable function f from a closed rectangle Q in Rⁿ into Rⁿ, a result of Gale and Nikaido essentially asserts that f is a P-function on Q if the Jacobian matrix Jf(x) is a P-matrix for all x ∈ Q, and a result of Moré and Rheinboldt asserts that f is a P₀-function on Q if and only if Jf(x) is a P₀-matrix for all x ∈ Q. In this article, we generalize these results to nonsmooth functions that admit H-differentials. Specialized to a locally Lipschitzian function f from Rⁿ into itself, our results say that f is a P-function if the (Clarke) generalized Jacobian ∂f(x) consists of P-matrices at all x, and when f is semismooth, f is a P₀-function if and only if the Bouligand differential ∂_Bf(x) consists of P₀-matrices at all x.