Global univalent results in R² and R³

Abstract

In this chapter we prove global univalent results obtained by Gale Nikaido and Schramm when F is a map in R². Some extensions are indicated in R³. However the following two interesting open problems posed by Gale-Nikaido remain unanswered : (i) Suppose F is a differentiable map from a rectangular region μ ⊂ R³ to R³. Suppose the Jacobian is non-vanishing and every entry in the Jacobian is non-negative. Is F one-one? (ii) Suppose F is a differentiable map from a rectangular region μ ⊂ R³ to R³. Suppose every principal minor of the Jacobian is non-vanishing for every x ε μ. Is F one-one? Gale and Nikaido have shown that both (i) and (ii) together imply that F is one-one in any rectangular region in R³. We have shown that (i) together with the assumption that the diagonal entries are identically zero will imply that F is one-one in any open convex region in R³-this result supplements the result obtained by Gale and Nikaido. We can weaken our assumptions in rectangular regions in R³ using Garcia-Zangwill's result given in the previous chapter.