A Note on Qualitative Economics for Univalence of Nonlinear Mappings

概要

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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 discontinuousmappings should be interesting.