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Title: Global asymptotic stability for half-linear differential systems with coefficients of indefinite sign (English)
Author: Sugie, Jitsuro
Author: Onitsuka, Masakazu
Language: English
Journal: Archivum Mathematicum
ISSN: 0044-8753 (print)
ISSN: 1212-5059 (online)
Volume: 44
Issue: 4
Year: 2008
Pages: 317-334
Summary lang: English
Category: math
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. (English)
Keyword: global asymptotic stability
Keyword: half-linear differential systems
Keyword: growth conditions
Keyword: eigenvalue
MSC: 34D05
MSC: 34D23
MSC: 37B25
MSC: 37B55
idZBL: Zbl 1212.34156
idMR: MR2493428
Date available: 2009-01-29T09:15:36Z
Last updated: 2013-09-19
Stable URL:
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