# Global asymptotic stability for half-linear differential systems with coefficients of indefinite sign

## Jitsuro Sugie and Masakazu Onitsuka

Address: Department of Mathematics and Computer Science Shimane University, Matsue 690-8504, Japan Department of Mathematics and Computer Science Shimane University, Matsue 690-8504, Japan

E-mail:

jsugie@riko.shimane-u.ac.jp

onitsuka@math.shimane-u.ac.jp

Abstract: 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 Coppelâ€™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.

AMSclassification: primary 34D05; secondary 34D23, 37B25, 37B55.

Keywords: global asymptotic stability, half-linear differential systems, growth conditions, eigenvalue.