Univalence for mappings with Leontief type Jacobians

Abstract

In this chapter we prove several results on univalence for mappings with Leontief type Jacobians. The first result is in some sense a sort of converse to Gale-Nikaido's theorem on univalence. Here we prove a result due to Gale-Nikaido and this says that if F and F⁻¹ are differentiable and if F⁻¹ is monotonic increasing then the Jacobian of F is a P-matrix provided the Jacobian matrix of F is of Leontief type. The second result due to Nikaido says that there exists a unique solution to F(x)=0 provided its domain is non-negative orthant and the Jacobian matrix is of Leontief type satisfying certain uniform diagonal dominance property. Then we present related results on M-functions and inverse isotone maps due to More and Rheinboldt. Finally we give some results on the univalence of the composition of maps F and G when their Jacobians are of Leontief type. In particular we show that F o G is a P-function when F and G are maps from R³ to R³ with their Jacobians Leontief type P-matrices throughout. We give an example to show that F o G need not be a P-function in R⁴.