Fundamental global univalence results of Gale-Nikaido-Inada

Abstract

In this chapter we will derive fundamental global univalence results of Gale-Nikaido-Inada for differentiable maps in rectangular regions whose Jacobian matrices are either P-matrices or N-matrices. Using mean-value theorem of differential calculus we will obtain global univalence results for differentiable maps whose Jacobian matrices are quasi-positive definite matrices, in convex regions. This result is due to Gale and Nikaido. Finally we present two new results on univalent mappings in R³.

Let F be a differentiable map from μ ⊂ Rⁿ to Rⁿ. We are interested in finding suitable conditions such that F is univalent throughout μ. Non-vanishing of the Jacobian matrices will not suffice as we shall see below. There are at least two approaches to the problem under consideration. One approach places topological assumptions on the map and the other places further conditions on the Jacobian matrices. We will study the former in the next chapter and the latter in the present chapter.