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Keywords:
global asymptotic stability; half-linear differential systems; growth conditions; eigenvalue
global asymptotic stability; half-linear differential systems; growth conditions; eigenvalue
Summary:
This paper is concerned with the global asymptotic stability of the zero solution of the half-linear differential system
\[ x^{\prime } = -\,e(t)x + f(t)\phi _{p^*}\!(y)\,,\quad y^{\prime } = -\,g(t)\phi _p(x) - h(t)y\,, \]
where $p > 1$, $p^* > 1$ ($1/p + 1/p^* = 1$), and $\phi _q(z) = |z|^{q-2}z$ for $q = p$ or $q = p^*$. The coefficients are not assumed to be positive. This system includes the linear differential system $\mathbf{x}^{\prime } = A(t)\mathbf{x}$ with $A(t)$ 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 and the real part of every eigenvalue of $A(t)$ is not always negative for $t$ sufficiently large. Some suitable examples are included to illustrate our results.
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