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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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