Okayama Economic Review

Published by the Economic Association of Okayama University
Permalink : http://escholarship.lib.okayama-u.ac.jp/41533

Fujimoto, Takao

Ranade, Ravindra R.

Abstract

This note is aimed at presenting an easy and simple proposition on the univalence of a given nonlinear differentiable mapping whose Jacobian matrix has sign-regularity. First the notion of sign-regularity of Jacobian matrix on a domain is defined. We classifY the sign patterns into four categories: plus, minus, zero, and the rest. The plus sign is given to the (i, j) entry of the Jacobian matrix when the i-th component function is always increasing with respect to the j-th coordinate variable, the negative sign when the function is always decreasing, and the sign of zero when the function does not include the j-th coordinate variable. Otherwise, the sign is set as an asterisk *. Our proof is simple and elementary by use ofthe mean value theorem. In the final section, we give a list of our future research topics, some of which are under way. Especially a generalization to discontinuous
mappings should be interesting.

Note

研究ノート (Note)

ISSN

0386-3069

NCID

AN00032897