An approximate controllability problem and the Jacobian conjecture for G.A.S (Q1355800)

The main result of the paper is the proof that in the known theorem on global asymptotic stability of the origin for the system \(\dot x=f(x)\), with \(f\in C(\mathbb{R}^2,\mathbb{R}^2)\) and \(f(0)=0\), the assumption that there exists a constant \(k>0\) such that \[ |J(x)|\leq k[\text{det }J(x)]^{{1\over 2}},\quad x\in\mathbb{R}^2, \] where \(J\) denotes the Jacobian of \(f\), can be replaced by the weaker condition \[ \lim_{|x|\to\infty} |J(x)|\cdot|x|^{-1}[\text{det }J(x)]^{-{1\over 2}}<\infty. \]