A solution to the bidimensional Global Asymptotic Stability Conjecture

If Y : ℜ2 → ℜ2 is а С1 vector field such that Y(0) = 0 and, for all q ∈ ℜ2, all the eigenvalues of DX(q) have negative real part, then the stable manifold of 0 is ℜ2. Let ρ ∈ [0, ∞) and Y : ℜ2 → ℜ2 be a C1 map such that, for all q ∈ ℜ2, the determinant of DY (q) is positive and moreover, for all p ∈ ℜ2, with |p| ≥ ρ, the spectrum of DY (p) is disjoint of the non-negative real half axis. Then Y is injective.