Consider a surjective mapping $F \colon \mathbb R^n \to \mathbb R^n$, $F\in C^1$, $\dfrac{\partial F_i}{\partial x_j} > 0$, and $\det \left(\!\left( \dfrac{\partial F_i}{\partial x_j} \right)\!\right) \neq 0$.

Does the inverse mapping $x = F(y) \colon \mathbb R^n \to \mathbb R^n$ exist? Are there general results concerning this issue?

For instance, if $n = 2$, and we suppose that two roots $a$ and $b$ of $F(x) = y$ exist then $\det J(a) > 0$ and $\det J(b) < 0$ because the vectors $\nabla F_i$ are oriented with respect to each other in different ways at the roots, so $\det J(x)$ takes zero which leads to contradiction (one may depict the isolines of zero level of functions $F_1$ and $F_2$).