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**A class of globally univalent differentiable mappings.**
*(English)*
Zbl 0734.26007

Let \(F'(x)\) be the Jacobian matrix of a differentiable map F from \({\mathbb{R}}^ n\) into \({\mathbb{R}}^ n\). We say that \(F'(x)\) is almost P (almost N) everywhere if for every x in \({\mathbb{R}}^ n\) all principal minors of \(F'(x)\) are positive (negative) except the determinant which is negative (positive). Theorem 3. Assume that F is of class \(C^ 1\) and that for each x the Jacobian matrix \(F'(x)\) is almost P. Then F is univalent on \({\mathbb{R}}^ n\). Theorem 4 is obtained from Theorem 3 by replacing P with N. Both theorems remain valid if the domain of F is any rectangular region \(\Omega\) of \({\mathbb{R}}^ n\) instead of \({\mathbb{R}}^ n\). They are a contribution to the following problem stated by D. Gale and H. Nikaido [Math. Ann. 159, 81-93 (1965; Zbl 0158.049)]: Let F: \(\Omega\to {\mathbb{R}}^ n\) be continuously differentiable and assume that \(F'(x)\) has all principal minors non-vanishing on \(\Omega\). Is F globally one-to-one? This problem has already been solved for \(n=3\) by G. Ravindran [J. Orissa Math. Soc. 4, No.2, 129-140 (1985; Zbl 0644.26012)].

Reviewer: S.Marcus (Bucureşti)

### MSC:

26B10 | Implicit function theorems, Jacobians, transformations with several variables |

47H10 | Fixed-point theorems |