This paper is concerned with the global asymptotic stability of the zero solution of the half-linear differential system $$x^{\text{'}}=-e\left(t\right)x+f\left(t\right){\phi }_{p^{*}}\left(y\right),y^{\text{'}}=-g\left(t\right){\phi }_p\left(x\right)-h\left(t\right)y,$$ where $p>1$, $p^{*}>1$ ($1/p+1/p^{*}=1$), and ${\phi }_q\left(z\right)={\left|z\right|}^{q-2}z$ for $q=p$ or $q=p^{*}$. The coefficients are not assumed to be positive. This system includes the linear differential system ${𝐱}^{\text{'}}=A\left(t\right)𝐱$ with $A\left(t\right)$ being a $2\times 2$ matrix as a special case. Our results are new even in the linear case ($p=p^{*}=2$). Our results also answer the question whether the zero solution of the linear system is asymptotically stable even when Coppel’s condition does not hold...

We study the structure of a differentiable autonomous system on the plane with non-positive divergence outside a bounded set. It is shown that under certain conditions such a system has a global attractor. The main result here can be seen as an improvement of the results of Olech and Meisters in [7,9] concerning the global asymptotic stability conjecture of Markus and Yamabe and the Jacobian Conjecture.