Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-28T08:01:38.824Z Has data issue: false hasContentIssue false
  • English
  • Français

Oscillation of symplectic dynamic systems

Published online by Cambridge University Press:  17 February 2009

Martin Bohner  and
Ondřej Došlý
Martin Bohner
Affiliation:
University of Missouri-Rolla, Department of Mathematics and Statistics, 115 Rolla Building, Rolla, Missouri 65409-0020, USA; e-mail: bohner@umr.edu.
Ondřej Došlý
Affiliation:
Masaryk University Brno, Department of Mathematics, Faculty of Science, Janáčkovo nám. 2a, CZ–66295 Brno, Czech Republic; e-mail: dosly@math.muni.cz.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We investigate oscillatory properties of a perturbed symplectic dynamic system on a time scale that is unbounded above. The unperturbed system is supposed to be nonoscillatory, and we give conditions on the perturbation matrix, which guarantee that the perturbed system becomes oscillatory. Examples illustrating the general results are given as well.

Type
Research Article
Information
The ANZIAM Journal , Volume 46 , Issue 1 , July 2004 , pp. 17 - 32
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Bohner, M., “Linear Hamiltonian difference systems: disconjugacy and Jacobi-type conditions”, J. Math. Anal. Appl. 199 (1996) 804–826.CrossRefGoogle Scholar
[2]Bohner, M. and Došlý, O., “Disconjugacy and transformations for symplectic systems”, Rocky Mountain J. Math. 27 (1997) 707–743.CrossRefGoogle Scholar
[3]Bohner, M., Došlý, O. and Hilscher, R., “Linear Hamiltonian dynamic systems on time scales: Sturmian property of the principal solution”, Nonlinear Anal. 47 (2001) 849–860.CrossRefGoogle Scholar
[4]Bohner, M., Došlý, O. and Kratz, W., “An oscillation theorem for discrete eigenvalue problems”, Rocky Mountain J. Math. 33 (2002) 1233–1260.Google Scholar
[5]Bohner, M. and Peterson, A., Dynamic equations on time scales: an introduction with applications (Birkhäuser, Boston, 2001).CrossRefGoogle Scholar
[6]Bohner, M. and Peterson, A., Advances in dynamic equations on time scales (Birkhäuser, Boston, 2003).CrossRefGoogle Scholar
[7]Chen, S. and Erbe, L., “Oscillation results for second order scalar and matrix difference equations”, Comput. Math. Appl. 28 (1994) 55–69.CrossRefGoogle Scholar
[8]Došlý, O., “On the existence of conjugate points for linear differential systems”, Math. Slovaca 40 (1990) 87–99.Google Scholar
[9]Došlý, O., “Oscillation and spectral properties of a class of singular differential operators”, Math. Nachr. 188 (1997) 49–68.CrossRefGoogle Scholar
[10]Došlý, O., “Oscillation criteria for higher order Sturm-Liouville difference equations”, J. Differ. Equations Appl. 4 (1998) 425–450.CrossRefGoogle Scholar
[11]Došlý, O., “Principal and nonprincipal solutions of symplectic dynamic systems on time scales”, in Proceedings of the 6th Colloquium on the Qualitative Theory of Differential Equations (Szeged, 1999), Electron. J. Qual. Theory Differ. Equ., No. 5, (electronic), (Szeged, 2000) pp. 14.Google Scholar
[12]Došlý, O. and Hilscher, R., “Disconjugacy, transformations and quadratic functionals for symplectic dynamic systems on time scales”, J. Differ. Equations Appl. 7 (2001) 265–295.Google Scholar
[13]Erbe, L. and Yan, P., “On the discrete Riccati equation and its applications to discrete Hamiltonian systems”, Rocky Mountain J. Math. 25 (1995) 167–178.CrossRefGoogle Scholar
[14]Hilger, S., “Analysis on measure chains —a unified approach to continuous and discrete calculus”, Results Math. 18 (1990) 18–56.CrossRefGoogle Scholar
[15]Hilscher, R., “Discrete spectral criteria for a certain class of singular differential and difference operators”, Comput. Math. Appl. 42 (2001) 465–476.CrossRefGoogle Scholar
[16]Hilscher, R., “Reid roundabout theorem for symplectic dynamic systems on time scales”, Appl. Math. Optim. 43 (2001) 129–146.CrossRefGoogle Scholar
[17]Peterson, A., “Oscillation of second order linear matrix difference equations”, J. Differential Equations 89 (1991) 65–88.CrossRefGoogle Scholar