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On the Critical Case in Oscillation for Differential Equations with a Single Delay and with Several Delays

Author

Listed:
  • Jaromír Baštinec
  • Leonid Berezansky
  • Josef Diblík
  • Zdeněk Šmarda

Abstract

New nonoscillation and oscillation criteria are derived for scalar delay differential equations ̇ 𠑥 ( 𠑡 ) + 𠑎 ( 𠑡 ) 𠑥 ( ℎ ( 𠑡 ) ) = 0 , 𠑎 ( 𠑡 ) ≥ 0 , ℎ ( 𠑡 ) ≤ 𠑡 , 𠑡 ≥ 𠑡 0 , and ∑ ̇ 𠑥 ( 𠑡 ) + 𠑚 𠑘 = 1 𠑎 𠑘 ( 𠑡 ) 𠑥 ( ℎ 𠑘 ( 𠑡 ) ) = 0 , 𠑎 𠑘 ( 𠑡 ) ≥ 0 , ℎ 𠑘 ( 𠑡 ) ≤ 𠑡 , and 𠑡 ≥ 𠑡 0 , in the critical case including equations with several unbounded delays, without the usual assumption that the parameters 𠑎 , ℎ , 𠑎 𠑘 , and ℎ 𠑘 of the equations are continuous functions. These conditions improve and extend some known oscillation results in the critical case for delay differential equations.

Suggested Citation

  • Jaromír Baštinec & Leonid Berezansky & Josef Diblík & Zdeněk Šmarda, 2010. "On the Critical Case in Oscillation for Differential Equations with a Single Delay and with Several Delays," Abstract and Applied Analysis, Hindawi, vol. 2010, pages 1-20, September.
    • Handle: RePEc:hin:jnlaaa:417869
    DOI: 10.1155/2010/417869
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    Citations

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    Cited by:

    1. Tongxing Li & Zhenlai Han & Chenghui Zhang & Hua Li, 2011. "Oscillation Criteria for Second‐Order Superlinear Neutral Differential Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).
    2. M. De la Sen & A. Ibeas & R. Nistal, 2014. "Approximate Solutions by Truncated Taylor Series Expansions of Nonlinear Differential Equations and Related Shadowing Property with Applications," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    3. Josef Rebenda & Zdeněk Šmarda, 2013. "Stability of a Functional Differential System with a Finite Number of Delays," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    4. S. Hristova & A. Golev, 2012. "Monotone‐Iterative Method for the Initial Value Problem with Initial Time Difference for Differential Equations with “Maxima”," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    5. Zeqing Liu & Shin Min Kang & Young Chel Kwun, 2012. "Positive Solutions and Iterative Approximations for a Nonlinear Two‐Dimensional Difference System with Multiple Delays," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    6. J. Baštinec & L. Berezansky & J. Diblík & Z. Šmarda, 2011. "A Final Result on the Oscillation of Solutions of the Linear Discrete Delayed Equation Δx(n) = −p(n)x(n − k) with a Positive Coefficient," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).
    7. J. Diblík & I. Hlavičková, 2012. "Asymptotic Upper and Lower Estimates of a Class of Positive Solutions of a Discrete Linear Equation with a Single Delay," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    8. Zdeněk Šmarda & Josef Rebenda, 2012. "Asymptotic Behaviour of a Two‐Dimensional Differential System with a Finite Number of Nonconstant Delays under the Conditions of Instability," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    9. J. Baštinec & J. Diblík & Z. Šmarda, 2011. "An Explicit Criterion for the Existence of Positive Solutions of the Linear Delayed Equation x˙(t)=−c(t)x(t−τ(t))," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).

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