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.
@article{119771,
author = {Jitsuro Sugie and Masakazu Onitsuka},
title = {Global asymptotic stability for half-linear differential systems with coefficients of indefinite sign},
journal = {Archivum Mathematicum},
volume = {044},
year = {2008},
pages = {317-334},
zbl = {1212.34156},
mrnumber = {2493428},
language = {en},
url = {http://dml.mathdoc.fr/item/119771}
}
Sugie, Jitsuro; Onitsuka, Masakazu. Global asymptotic stability for half-linear differential systems with coefficients of indefinite sign. Archivum Mathematicum, Tome 044 (2008) pp. 317-334. http://gdmltest.u-ga.fr/item/119771/
Oscillation Theory for Second Order Linear, Half-linear, Superlinear and Sublinear Dynamic Equations, Kluwer Academic Publishers, Dordrecht-Boston-London, 2002. (2002) | MR 2091751 | Zbl 1073.34002
On the second order half-linear differential equation, Studia Sci. Math. Hungar. 3 (1968), 411–437. (1968) | MR 0267190 | Zbl 0167.37403
Stability and Asymptotic Behavior of Differential Equations, Heath, Boston, 1965. (1965) | MR 0190463 | Zbl 0154.09301
Slowly varying system $\dot{x}=A(t)x$, IEEE Trans. Automat. Control AC-14 (1969), 780–781. (1969) | Article | MR 0276562
Stability of linear systems with parametric excitation, Trans. ASME Ser. E J. Appl. Mech. 37 (1970), 228–230. (1970) | Article | MR 0274876 | Zbl 0215.44602
Half-linear Differential Equations, North-Holland Mathematics Studies 202, Elsevier, Amsterdam, 2005. | MR 2158903
Oscillation and nonoscillation theorems for some nonlinear ordinary differential equations, Ordinary and Partial Differential Equations (Dundee, 1982), Lecture Notes in Math. 964, Springer-Verlag, Berlin-Heidelberg-New York. | MR 0693113 | Zbl 0528.34034
A half-linear second order differential equation, Qualitative Theory of Differential Equations, Vol I, II (Szeged, 1979) (Farkas, M., ed.), Colloq. Math. Soc. János Bolyai 30, North-Holland, Amsterdam-New York, 1981, pp. 153–180. (1981) | MR 0680591 | Zbl 0511.34006
Asymptotic behaviour of autonomous half-linear differential systems on the plane, Studia Sci. Math. Hungar. 19 (1984), 447–464. (1984) | MR 0874513 | Zbl 0629.34066
On the asymptotic stability for a two-dimensional linear nonautonomous differential system, Nonlinear Anal. 25 (1995), 991–1002. (1995) | Article | MR 1350721 | Zbl 0844.34050
Integral conditions on the asymptotic stability for the damped linear oscillator with small damping, Proc. Amer. Math. Soc. 124 (1996), 415–422. (1996) | Article | MR 1317039 | Zbl 0844.34051
A necessary and sufficient condition for the asymptotic stability of the damped oscillator, J. Differential Equations 119 (1995), 209–223. (1995) | Article | MR 1334491 | Zbl 0831.34052
Asymptotic stability for the equilibrium of the damped oscillator, Differential Integral Equations 6 (1993), 835–848. (1993) | MR 1222304
Nonoscillation theory for second order half-linear differential equations in the framework of regular variation, Results Math. 43 (2003), 129–149. (2003) | Article | MR 1962855 | Zbl 1047.34034
Stability by Liapunov’s Direct Method, with Applications, Mathematics in Science and Engineering 4, Academic Press, New-York-London, 1961. (1961) | MR 0132876
Sturmian comparison theorem for half-linear second-order differential equations, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), 1193–1204. (1995) | MR 1362999 | Zbl 0873.34020
Global stability criteria for differential systems, Osaka Math. J. 12 (1960), 305–317. (1960) | MR 0126019 | Zbl 0096.28802
On some analogs of Sturm’s and Kneser’s theorems for nonlinear systems, J. Math. Anal. Appl. 53 (1976), 418–425. (1976) | Article | MR 0402184 | Zbl 0327.34027
Asymptotic stability for intermittently controlled nonlinear oscillators, SIAM J. Math. Anal. 25 (1994), 815–835. (1994) | Article | MR 1271312 | Zbl 0809.34067
Stability Theory by Liapunov’s Direct Method, Applied Mathematical Sciences 22, Springer-Verlag, New York-Heidelberg-Berlin, 1977. (1977) | MR 0450715 | Zbl 0364.34022
Asymptotic behavior of solutions of nonautonomous half-linear differential systems, Studia Sci. Math. Hungar. 44 (2007), 159–189. (2007) | MR 2325518 | Zbl 1174.34042
Comparison theorems for oscillation of second-order half-linear differential equations, Acta Math. Hungar. 111 (2006), 165–179. (2006) | Article | MR 2188979 | Zbl 1116.34030
On a criterion of instability in the sense of Lyapunov of the solutions of a linear system of ordinary differential equations, Dokl. Akad. Nauk 84 (1952), 201–204. (1952) | MR 0050749
Stability Theory by Liapunov’s Second Method, Math. Society Japan, Tokyo (1966). (1966) | MR 0208086
Mathematical Methods for the Study of Automatic Control Systems, Pergamon Press, New-York-Oxford-London-Paris, 1962. (1962) | MR 0151695 | Zbl 0103.06001