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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 x˙(t)+a(t)x(h(t))=00,a(t)≥,h(t)≤t,t≥t0, and x˙(t)+∑k=1mak(t)x(hk(t))=0,ak(t)≥0,hk(t)≤t, and t ≥ t0, in the critical case including equations with several unbounded delays, without the usual assumption that the parameters a, h, ak, and hk 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, John Wiley & Sons, vol. 2010(1).
    • Handle: RePEc:wly:jnlaaa:v:2010:y:2010:i:1:n:417869
    DOI: 10.1155/2010/417869
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    References listed on IDEAS

    as
    1. Azizollah Babakhani, 2010. "Positive Solutions for System of Nonlinear Fractional Differential Equations in Two Dimensions with Delay," Abstract and Applied Analysis, Hindawi, vol. 2010, pages 1-16, May.
    2. Azizollah Babakhani, 2010. "Positive Solutions for System of Nonlinear Fractional Differential Equations in Two Dimensions with Delay," Abstract and Applied Analysis, John Wiley & Sons, vol. 2010(1).
    3. Ravi P. Agarwal & Donal O’Regan & Patricia J. Y. Wong, 1999. "Positive Solutions of Differential, Difference and Integral Equations," Springer Books, Springer, number 978-94-015-9171-3, March.
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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. 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).
    3. Zhenlai Han & Tongxing Li & Chenghui Zhang & Ying Sun, 2011. "Oscillation Criteria for Certain Second‐Order Nonlinear Neutral Differential Equations of Mixed Type," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).
    4. 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).
    5. 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).
    6. 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).
    7. 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).
    8. 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).
    9. 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).
    10. 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).

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