Ukrainian Mathematical Journal

, Volume 65, Issue 2, pp 249–259 | Cite as

Oscillation of solutions of the second-order linear functional-difference equations

  • O. V. Karpenko
  • V. I. Kravets’
  • O. M. Stanzhyts’kyi

We establish conditions required for the oscillation behavior of solutions of linear functional-difference equations and discrete linear difference equations of the second order in the case where the corresponding solutions of their differential analogs are oscillating on a segment.


Initial Data Difference Equation Oscillation Behavior Initial Function Linear Difference Equation 
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  1. 1.
    A. N. Shiryaev, Foundations of Stochastic Financial Mathematics [in Russian], FAZIS, Moscow (1998).Google Scholar
  2. 2.
    G. Ladas, “Explicit conditions for the oscillation of difference equations,” J. Math. Anal. Appl., 153, 276–287 (1990).CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Ö. Öcalan, “Linearized oscillation of nonlinear difference equations with advanced arguments,” Arch. Mat., 45, 203–212 (2009).Google Scholar
  4. 4.
    Ö. Öcalan, “Oscillation of nonlinear difference equations with several coefficients,” Commun. Math. Anal., 4, No. 1, 35–44 (2008).zbMATHMathSciNetGoogle Scholar
  5. 5.
    M. Bohner and A. Peterson, Dynamical Equations on Time Scales. An Introduction with Applications, Birkhäuser, Boston (2003).Google Scholar
  6. 6.
    K. Messer, “A second-order self-adjoint dynamic equation on time scale,” Dynam. Syst. Appl., 8, No. 8, 451–460 (2002).Google Scholar
  7. 7.
    L. Grüne, Asymptotic Behavior of Dynamical and Control Systems Under Perturbation and Discretization, Springer, Berlin (2002).CrossRefzbMATHGoogle Scholar
  8. 8.
    B. M. Garay and K. Lee, “Attractors under discretization with variable stepsize,” Discrete Contin. Dynam. Syst, 13, No. 3, 827–841 (2005).CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    L. Grüne, “Attraction rates, robustness, and discretization of attractors,” SIAM J. Numer. Anal., 41, No. 6, 2096–2113 (2003).CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    I. Karafyllis and L. Grüne, “Feedback stabilization methods for the numerical solution of systems of ordinary differential equations,” Discrete Contin. Dynam. Syst., Ser. B, 16, No. 1, 283–317 (2011).CrossRefzbMATHGoogle Scholar
  11. 11.
    O. M. Stanzhyts’kyi and A. M. Tkachuk, “On the relationship between the properties of solutions of difference equations and the corresponding differential equations,” Ukr. Mat. Zh., 57, No. 7, 989–996 (2005); English translation: Ukr. Math. J., 57, No. 7, 1167–1176 (2005).Google Scholar
  12. 12.
    M. A. Skalkina, “On the oscillations of solutions of finite-difference equations,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 6, 138–144 (1959).Google Scholar
  13. 13.
    A. M. Ateivi, Oscillating Properties of the Solutions of Differential Equations and Their Stability [in Ukrainian], Candidate-Degree Thesis (Physics and Mathematics), Kyiv (1997).Google Scholar
  14. 14.
    A. M. Samoilenko, M. O. Perestyuk, and I. O. Parasyuk, Differential Equations. A Textbook [in Ukrainian], Lybid’, Kiev (2003).Google Scholar
  15. 15.
    D. I. Martynyuk, Lectures on the Qualitative Theory of Difference Equations [in Russian], Naukova Dumka, Kiev (1972).Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • O. V. Karpenko
    • 1
  • V. I. Kravets’
    • 2
  • O. M. Stanzhyts’kyi
    • 3
  1. 1.“Kyiv Polytechnic Institute” Ukrainian National Technical UniversityKyivUkraine
  2. 2.Tavriya State Agrotechnical UniversityKyivUkraine
  3. 3.Shevchenko Kyiv National UniversityKyivUkraine

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