Opuscula Math. 41, no. 5 (2021), 613-627
https://doi.org/10.7494/OpMath.2021.41.5.613

 
Opuscula Mathematica

Oscillation criteria for linear difference equations with several variable delays

Vasileios Benekas
Ábel Garab
Ardak Kashkynbayev
Ioannis P. Stavroulakis

Abstract. We obtain new sufficient criteria for the oscillation of all solutions of linear delay difference equations with several (variable) finite delays. Our results relax numerous well-known limes inferior-type oscillation criteria from the literature by letting the limes inferior be replaced by the limes superior under some additional assumptions related to slow variation. On the other hand, our findings generalize an oscillation criterion recently given for the case of a constant, single delay.

Keywords: oscillation, difference equations, several delays, non-monotone argument, slowly varying function.

Mathematics Subject Classification: 39A21, 39A10.

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  1. R.P. Agarwal, M. Bohner, S.R. Grace, D. O'Regan, Discrete Oscillation Theory, Hindawi Publishing Corporation, New York, 2005.
  2. J.M. Ash, P. Erdős, L.A. Rubel, Very slowly varying functions, Aequationes Math. 10 (1974), 1-9.
  3. V. Benekas, A. Kashkynbayev, I.P. Stavroulakis, A sharp oscillation criterion for a difference equation with constant delay, Adv. Difference Equ. 2020, Paper no. 566, 9 pp.
  4. G.E. Chatzarakis, R. Koplatadze, I.P. Stavroulakis, Optimal oscillation criteria for first order difference equations with delay argument, Pac. J. Appl. Math. 235 (2008), no. 1, 15-33.
  5. G.E. Chatzarakis, S. Pinelas, I.P. Stavroulakis, Oscillations of difference equations with several deviated arguments, Aequationes Math. 88 (2014), 105-123.
  6. G.E. Chatzarakis, S. Pinelas, I.P. Stavroulakis, Correction to: Oscillations of difference equations with several deviated arguments, Aequationes Math. 92 (2018), 1195-1200.
  7. Á. Garab, A sharp oscillation criterion for a linear differential equation with variable delay, Symmetry 11 (2019), Article no. 1332.
  8. Á. Garab, I.P. Stavroulakis, Oscillation criteria for first order linear delay differential equations with several variable delays, Appl. Math. Lett. 106 (2020), 106366, 9 pp.
  9. Á. Garab, M. Pituk, I.P. Stavroulakis, A sharp oscillation criterion for a linear delay differential equation, Appl. Math. Lett. 93 (2019), 58-65.
  10. I. Győri, G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford, 1991.
  11. B. Karpuz, I.P. Stavroulakis, Oscillation and nonoscillation of difference equations with several delays, Mediterr. J. Math. 18 (2021), no. 1, Paper no. 3, 15 pp.
  12. R. Koplatadze, S. Pinelas, Oscillation criteria for first-order linear difference equations with several delay arguments, Nelīnīĭnī Koliv. 17 (2014), no. 2, 248-267; reprinted in J. Math. Sci. (N.Y.) 208 (2015), no. 5, 571-592.
  13. R.G. Koplatadze, T.A. Chanturija, On the oscillatory and monotonic solutions of first order differential equations with deviating arguments, Differentsial'nye Uravneniya 18 (1982), 1463-1465.
  14. M. Pituk, Oscillation of a linear delay differential equation with slowly varying coefficient, Appl. Math. Lett. 73 (2017), 29-36.
  15. E. Seneta, Regularly Varying Functions, Lecture Notes in Mathematics, vol. 508, Springer-Verlag, Berlin-New York, 1976.
  16. I.P. Stavroulakis, Oscillations of delay and difference equations with variable coefficients and arguments, [in:] Differential and Difference Equations with Applications, Springer Proc. Math. Stat., vol. 164, Springer, 2016, pp. 169-189.
  17. X.H. Tang, J.S. Yu, Oscillation of delay difference equation, Comput. Math. Appl. 37 (1999), no. 7, 11-20.
  18. W. Yan, Q. Meng, J.R. Yan, Oscillation criteria for difference equation of variable delays, Dyn. Contin. Discret. Impuls. Syst. Ser. A Math. Anal. 13A (2006), Part 2, suppl., 641-647.
  • Communicated by Josef Diblík.
  • Received: 2021-08-04.
  • Accepted: 2021-08-20.
  • Published online: 2021-09-30.
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Cite this article as:
Vasileios Benekas, Ábel Garab, Ardak Kashkynbayev, Ioannis P. Stavroulakis, Oscillation criteria for linear difference equations with several variable delays, Opuscula Math. 41, no. 5 (2021), 613-627, https://doi.org/10.7494/OpMath.2021.41.5.613

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