# Article

 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: http://hdl.handle.net/10338.dmlcz/119771 . Reference: [1] Agarwal, R. P., Grace, S. R., O’Regan, D.: Oscillation Theory for Second Order Linear, Half-linear, Superlinear and Sublinear Dynamic Equations.Kluwer Academic Publishers, Dordrecht-Boston-London, 2002. Zbl 1073.34002, MR 2091751 Reference: [2] Bihari, I.: On the second order half-linear differential equation.Studia Sci. Math. 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