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The discrete Prüfer transformation

Authors: Martin Bohner and Ondrej Doslý
Journal: Proc. Amer. Math. Soc. 129 (2001), 2715-2726
MSC (2000): Primary 39A12; Secondary 39A11, 34K11
Published electronically: February 15, 2001
MathSciNet review: 1838796
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Abstract | References | Similar Articles | Additional Information


The classical Prüfer transformation has proved to be a useful tool in the study of Sturm-Liouville theory. In this paper we introduce the Prüfer transformation for self-adjoint difference equations and use it to obtain oscillation criteria and other results. We then offer an extension of this approach to the case of general symplectic systems on time scales. Time scales have been introduced in order to unify discrete and continuous analysis, and indeed our results cover as special cases both the Prüfer transformation for differential and for difference equations.

References [Enhancements On Off] (What's this?)

  • 1. R. P. Agarwal.
    Difference Equations and Inequalities.
    Marcel Dekker, Inc., New York, 1992. MR 92m:39002
  • 2. R. P. Agarwal and M. Bohner.
    Basic calculus on time scales and some of its applications.
    Results Math., 35(1-2):3-22, 1999. MR 99m:41030
  • 3. C. D. Ahlbrandt and A. C. Peterson.
    Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations.
    Kluwer Academic Publishers, Boston, 1996.
  • 4. D. Anderson.
    Discrete trigonometric matrix functions.
    Panamer. Math. J., 7(1):39-54, 1997. MR 99e:39004
  • 5. B. Aulbach and S. Hilger.
    Linear dynamic processes with inhomogeneous time scale.
    In Nonlinear Dynamics and Quantum Dynamical Systems. Akademie Verlag, Berlin, 1990. MR 91g:58266
  • 6. J. H. Barrett.
    A Prüfer transformation for matrix differential equations.
    Proc. Amer. Math. Soc., 8:510-518, 1957. MR 19:415f
  • 7. M. Bohner and O. Doslý.
    Disconjugacy and transformations for symplectic systems.
    Rocky Mountain J. Math., 27(3):707-743, 1997. MR 99e:39007
  • 8. M. Bohner and O. Doslý.
    Trigonometric transformations of symplectic difference systems.
    J. Differential Equations, 163:113-129, 2000. CMP 2000:11
  • 9. O. Doslý.
    On some properties of trigonometric matrices.
    Cas. Pest. Mat., 112:188-196, 1987. MR 88g:34041
  • 10. O. Doslý and R. Hilscher.
    Disconjugacy, transformations and quadratic functionals for symplectic dynamic systems on time scales,
    2000, J. Differ. Equations Appl., to appear.
  • 11. Á. Elbert.
    A half-linear second order differential equation.
    Colloq. Math. Soc. János Bolyai, 30:158-180, 1979. MR 84g:34008
  • 12. L. Erbe and S. Hilger.
    Sturmian theory on measure chains.
    Differential Equations Dynam. Systems, 1(3):223-244, 1993. MR 95b:39002
  • 13. G. J. Etgen.
    A note on trigonometric matrices.
    Proc. Amer. Math. Soc., 17:1226-1232, 1966. MR 35:4504
  • 14. G. J. Etgen.
    Oscillation properties of certain nonlinear matrix equations of second order.
    Trans. Amer. Math. Soc., 122:289-310, 1966.
  • 15. H. Heuser.
    Gewöhnliche Differentialgleichungen.
    B. G. Teubner, Stuttgart, 1989. MR 90f:34001
  • 16. S. Hilger.
    Analysis on measure chains - a unified approach to continuous and discrete calculus.
    Results Math., 18:18-56, 1990. MR 91m:26027
  • 17. S. Hilger.
    Special functions, Laplace and Fourier transform on measure chains.
    Dynam. Systems Appl., 8(3-4):471-488, 1999.
    Special Issue on ``Discrete and Continuous Hamiltonian Systems'', edited by R. P. Agarwal and M. Bohner. CMP 2000:04
  • 18. W. G. Kelley and A. C. Peterson.
    Difference Equations: An Introduction with Applications.
    Academic Press, San Diego, 1991. MR 93f:39002
  • 19. H. Prüfer.
    Neue Herleitung der Sturm-Liouvilleschen Reihenentwicklung stetiger Funktionen.
    Math. Ann., 95:499-518, 1926.
  • 20. W. T. Reid.
    A Prüfer transformation for differential systems.
    Pacific J. Math., 8:575-584, 1958. MR 20:5913
  • 21. W. T. Reid.
    Generalized polar coordinate transformations for differential systems.
    Rocky Mountain J. Math., 1(2):383-406, 1971. MR 43:6488

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Additional Information

Martin Bohner
Affiliation: Department of Mathematics and Statistics, University of Missouri–Rolla, 313 Rolla Building, Rolla, Missouri 65409-0020

Ondrej Doslý
Affiliation: Mathematical Institute, Czech Academy of Sciences, Žižkova 22, CZ–61662 Brno, Czech Republic

Keywords: Pr\"ufer transformation, Sturm-Liouville difference equations, linear Hamiltonian difference systems, time scales, symplectic systems
Received by editor(s): September 2, 1999
Received by editor(s) in revised form: January 24, 2000
Published electronically: February 15, 2001
Additional Notes: The research of the first author was supported by the University of Missouri Research Board. The research of the second author was supported by the Grant G201/98/0677 of the Czech Grant Agency (Prague).
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2001 American Mathematical Society

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