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Sharp conditions for the oscillation of delay difference equations

Author

Listed:
  • G. Ladas
  • Ch. G. Philos
  • Y. G. Sficas

Abstract

Suppose that { p n } is a nonnegative sequence of real numbers and let k be a positive integer. We prove that lim n → ∞ inf  [ 1 k ∑ i = n − k n − 1 p i ] > k k ( k + 1 ) k + 1 is a sufficient condition for the oscillation of all solutions of the delay difference equation A n + 1 − A n + p n A n − k = 0 ,    n = 0 , 1 , 2 , … . This result is sharp in that the lower bound k k / ( k + 1 ) k + 1 in the condition cannot be improved. Some results on difference inequalities and the existence of positive solutions are also presented.

Suggested Citation

  • G. Ladas & Ch. G. Philos & Y. G. Sficas, 1989. "Sharp conditions for the oscillation of delay difference equations," International Journal of Stochastic Analysis, Hindawi, vol. 2, pages 1-11, January.
    • Handle: RePEc:hin:jnijsa:150178
    DOI: 10.1155/S1048953389000080
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    Cited by:

    1. Karpuz, Başak & Özsavaş, Büşra, 2021. "An improved product type oscillation test for partial difference equations," Applied Mathematics and Computation, Elsevier, vol. 391(C).
    2. Quanxin Zhang, 2014. "A Comparison Theorem for Oscillation of the Even‐Order Nonlinear Neutral Difference Equation," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    3. Bolat, Yaşar, 2009. "Oscillation of higher order neutral type nonlinear difference equations with forcing terms," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 2973-2980.
    4. J. Diblík & I. Hlavičková, 2011. "Asymptotic Behavior of Solutions of Delayed Difference Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).
    5. 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).
    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. S. H. Saker, 2010. "Global Stability and Oscillation of a Discrete Annual Plants Model," Abstract and Applied Analysis, John Wiley & Sons, vol. 2010(1).

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