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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Rights and permissions
Copyright information
© 1983 Springer-Verlag
About this chapter
Cite this chapter
Parthasarathy, T. (1983). Global univalent results in R2 and R3 . In: On Global Univalence Theorems. Lecture Notes in Mathematics, vol 977. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0065572
Download citation
DOI: https://doi.org/10.1007/BFb0065572
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-11988-3
Online ISBN: 978-3-540-39462-4
eBook Packages: Springer Book Archive