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

Jitsuro Sugie; Masakazu Onitsuka

Archivum Mathematicum (2008)

  • Volume: 044, Issue: 4, page 317-334
  • ISSN: 0044-8753

Abstract

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This paper is concerned with the global asymptotic stability of the zero solution of the half-linear differential system x ' = - e ( t ) x + f ( t ) φ p * ( y ) , y ' = - g ( t ) φ p ( x ) - h ( t ) y , where p > 1 , p * > 1 ( 1 / p + 1 / p * = 1 ), and φ 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 𝐱 ' = A ( t ) 𝐱 with A ( t ) being a 2 × 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.

How to cite

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Sugie, Jitsuro, and Onitsuka, Masakazu. "Global asymptotic stability for half-linear differential systems with coefficients of indefinite sign." Archivum Mathematicum 044.4 (2008): 317-334. <http://eudml.org/doc/250459>.

@article{Sugie2008,
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 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.},
author = {Sugie, Jitsuro, Onitsuka, Masakazu},
journal = {Archivum Mathematicum},
keywords = {global asymptotic stability; half-linear differential systems; growth conditions; eigenvalue; global asymptotic stability; half-linear differential system; growth condition; eigenvalue},
language = {eng},
number = {4},
pages = {317-334},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Global asymptotic stability for half-linear differential systems with coefficients of indefinite sign},
url = {http://eudml.org/doc/250459},
volume = {044},
year = {2008},
}

TY - JOUR
AU - Sugie, Jitsuro
AU - Onitsuka, Masakazu
TI - Global asymptotic stability for half-linear differential systems with coefficients of indefinite sign
JO - Archivum Mathematicum
PY - 2008
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 044
IS - 4
SP - 317
EP - 334
AB - 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.
LA - eng
KW - global asymptotic stability; half-linear differential systems; growth conditions; eigenvalue; global asymptotic stability; half-linear differential system; growth condition; eigenvalue
UR - http://eudml.org/doc/250459
ER -

References

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  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. (2002) Zbl1073.34002MR2091751
  2. Bihari, I., On the second order half-linear differential equation, Studia Sci. Math. Hungar. 3 (1968), 411–437. (1968) Zbl0167.37403MR0267190
  3. Coppel, W. A., Stability and Asymptotic Behavior of Differential Equations, Heath, Boston, 1965. (1965) Zbl0154.09301MR0190463
  4. Desoer, C. A., 10.1109/TAC.1969.1099336, IEEE Trans. Automat. Control AC-14 (1969), 780–781. (1969) MR0276562DOI10.1109/TAC.1969.1099336
  5. Dickerson, J. R., 10.1115/1.3408450, Trans. ASME Ser. E J. Appl. Mech. 37 (1970), 228–230. (1970) Zbl0215.44602MR0274876DOI10.1115/1.3408450
  6. Došlý, O., Řehák, P., Half-linear Differential Equations, North-Holland Mathematics Studies 202, Elsevier, Amsterdam, 2005. MR2158903
  7. Elbert, Á., 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. Zbl0528.34034MR0693113
  8. Elbert, Á., 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) Zbl0511.34006MR0680591
  9. Elbert, Á., Asymptotic behaviour of autonomous half-linear differential systems on the plane, Studia Sci. Math. Hungar. 19 (1984), 447–464. (1984) Zbl0629.34066MR0874513
  10. Hatvani, L., 10.1016/0362-546X(95)00093-B, Nonlinear Anal. 25 (1995), 991–1002. (1995) Zbl0844.34050MR1350721DOI10.1016/0362-546X(95)00093-B
  11. Hatvani, L., 10.1090/S0002-9939-96-03266-2, Proc. Amer. Math. Soc. 124 (1996), 415–422. (1996) Zbl0844.34051MR1317039DOI10.1090/S0002-9939-96-03266-2
  12. Hatvani, L., Krisztin, T., Totik, V., 10.1006/jdeq.1995.1087, J. Differential Equations 119 (1995), 209–223. (1995) Zbl0831.34052MR1334491DOI10.1006/jdeq.1995.1087
  13. Hatvani, L., Totik, V., Asymptotic stability for the equilibrium of the damped oscillator, Differential Integral Equations 6 (1993), 835–848. (1993) MR1222304
  14. Jaroš, J., Kusano, T., Tanigawa, T., 10.1007/BF03322729, Results Math. 43 (2003), 129–149. (2003) Zbl1047.34034MR1962855DOI10.1007/BF03322729
  15. LaSalle, J. P., Lefschetz, S., Stability by Liapunov’s Direct Method, with Applications, Mathematics in Science and Engineering 4, Academic Press, New-York-London, 1961. (1961) MR0132876
  16. Li, H.-J., Yeh, C.-C., Sturmian comparison theorem for half-linear second-order differential equations, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), 1193–1204. (1995) Zbl0873.34020MR1362999
  17. Markus, L., Yamabe, H., Global stability criteria for differential systems, Osaka Math. J. 12 (1960), 305–317. (1960) Zbl0096.28802MR0126019
  18. Mirzov, J. D., 10.1016/0022-247X(76)90120-7, J. Math. Anal. Appl. 53 (1976), 418–425. (1976) Zbl0327.34027MR0402184DOI10.1016/0022-247X(76)90120-7
  19. Pucci, P., Serrin, J., 10.1137/S0036141092240679, SIAM J. Math. Anal. 25 (1994), 815–835. (1994) Zbl0809.34067MR1271312DOI10.1137/S0036141092240679
  20. Rouche, N., Habets, P., Laloy, M., Stability Theory by Liapunov’s Direct Method, Applied Mathematical Sciences 22, Springer-Verlag, New York-Heidelberg-Berlin, 1977. (1977) Zbl0364.34022MR0450715
  21. Sugie, J., Onitsuka, M., Yamaguchi, A., Asymptotic behavior of solutions of nonautonomous half-linear differential systems, Studia Sci. Math. Hungar. 44 (2007), 159–189. (2007) Zbl1174.34042MR2325518
  22. Sugie, J., Yamaoka, N., 10.1007/s10474-006-0029-5, Acta Math. Hungar. 111 (2006), 165–179. (2006) Zbl1116.34030MR2188979DOI10.1007/s10474-006-0029-5
  23. Vinograd, R. E., 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) MR0050749
  24. Yoshizawa, T., Stability Theory by Liapunov’s Second Method, Math. Society Japan, Tokyo (1966). (1966) MR0208086
  25. Zubov, V. I., Mathematical Methods for the Study of Automatic Control Systems, Pergamon Press, New-York-Oxford-London-Paris, 1962. (1962) Zbl0103.06001MR0151695

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