A Univalence Theorem for Nonlinear Mappings: An Elementary Approach

This note is to extend a well-known theorem due to Gale and Nikaido on the univalence of nonlinear mappings. Our approach is based on a simple elimination method of variables, and the key proposition used is the implicit function theorem. In terms of the condition on signs of principal minors, our result is more general than that of Gale and Nikaido since the sign of a minor can be positive or negative. Besides we require the sign condition only for the leading principal minors. On the other hand, the domain of mappings we can deal with has to be unbounded for all but one variable. In addition, the value of each principal minor must be in a finite range. Some remarks are given in the final section.

研究ノート (Note)