Opuscula Mathematica
Opuscula Math. 42, no. 6 (2022), 769-791

New oscillation conditions for first-order linear retarded difference equations with non-monotone arguments

Emad R. Attia
Bassant M. El-Matary
George E. Chatzarakis
Abstract. In this paper, we study the oscillatory behavior of the solutions of a first-order difference equation with non-monotone retarded argument and nonnegative coefficients, based on an iterative procedure. We establish some oscillation criteria, involving \(\lim \sup\), which achieve a marked improvement on several known conditions in the literature. Two examples, numerically solved in MAPLE software, are also given to illustrate the applicability and strength of the obtained conditions.
Keywords: oscillation, difference equations, non-monotone argument.
Mathematics Subject Classification: 39A10, 39A21.
Full text (pdf)
References ▼
  1. [1] R.P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, New York, 1992.
  2. [2] E.R. Attia, Oscillation tests for first-order linear differential equations with non-monotone delays, Adv. Difference Equ. 2021 (2021), Article no. 41. https://doi.org/10.1186/s13662-020-03209-4
  3. [3] E.R. Attia, G.E. Chatzarakis, Oscillation tests for difference equations with non-monotone retarded arguments, Appl. Math. Lett. 123 (2022), 107551. https://doi.org/10.1016/j.aml.2021.107551
  4. [4] E.R. Attia, B.M. El-Matary, New aspects for the oscillation of first-order difference equations with deviating arguments, Opuscula Math. 42 (2022), 393-413. https://doi.org/10.7494/OpMath.2022.42.3.393
  5. [5] E. Braverman, B. Karpuz, On oscillation of differential and difference equations with non-monotone delays, Appl. Math. Comput. 218 (2011), 3880-3887. https://doi.org/10.1016/j.amc.2011.09.035
  6. [6] E. Braverman, G.E. Chatzarakis, I.P. Stavroulakis, Iterative oscillation tests for difference equations with several non-monotone arguments, J. Difference Equ. Appl. 21 (2015), 854-874. https://doi.org/10.1080/10236198.2015.1051480
  7. [7] G.E. Chatzarakis, Sufficient oscillation conditions for deviating difference equations, Filomat 33 (2019), 3291-3305. https://doi.org/10.2298/FIL1911291C
  8. [8] G.E. Chatzarakis, I. Jadlovská, Oscillations in deviating difference equations using an iterative technique, J. Inequal. Appl. 2017 (2017), Article no. 173. https://doi.org/10.1186/s13660-017-1450-8
  9. [9] G.E. Chatzarakis, I. Jadlovská, Improved iterative oscillation tests for first-order deviating difference equations, Int. J. Difference Equ. 12 (2017), 185-210.
  10. [10] G.E. Chatzarakis, I. Jadlovská, Oscillations of deviating difference equations using an iterative method, Mediterr. J. Math. 16 (2019), Article no. 16. https://doi.org/10.1007/s00009-018-1288-4
  11. [11] G.E. Chatzarakis, L. Shaikhet, Oscillation criteria for difference equations with non-monotone arguments, Adv. Difference Equ. 2017 (2017), Article no. 62. https://doi.org/10.1186/s13662-017-1119-0
  12. [12] G.E. Chatzarakis, S.R. Grace, I. Jadlovská, Oscillation tests for linear difference equations with non-monotone arguments, Tatra Mt. Math. Publ. 79 (2021), 81-100. https://doi.org/10.2478/tmmp-2021-0021
  13. [13] G.E. Chatzarakis, R. Koplatadze, I.P. Stavroulakis, Oscillation criteria of first order linear difference equations with delay argument, Nonlinear Anal. 68 (2008), 994-1005. https://doi.org/10.1016/j.na.2006.11.055
  14. [14] G.E. Chatzarakis, R. Koplatadze, I.P. Stavroulakis, Optimal oscillation criteria of first order linear difference equations with delay argument, Pacific J. Math. 235 (2008), 15-33.
  15. [15] G.E. Chatzarakis, Ch.G. Philos, I.P. Stavroulakis, On the oscillation of the solutions to~linear difference equations with variable delay, Electron. J. Differential Equations 2008 (2008), no. 50, 1-15.
  16. [16] G.E. Chatzarakis, Ch.G. Philos, I.P. Stavroulakis, An oscillation criterion for linear difference equations with general delay argument, Port. Math. 66 (2009), 513-533. https://doi.org/10.4171/PM/1853
  17. [17] G.E. Chatzarakis, I.K. Purnaras, I.P. Stavroulakis, Oscillation of retarded difference equations with a non-monotone argument, J. Difference Equ. Appl. 23 (2017), 1354-1377. https://doi.org/10.1080/10236198.2017.1332053
  18. [18] L.H. Erbe, B.G. Zhang, Oscillation of discrete analogues of delay equations, Differential Integral Equations 2 (1989), 300-309.
  19. [19] B. Karpuz, Sharp oscillation and nonoscillation tests for linear difference equations, J. Difference Equ. Appl. 23 (2017), 1929-1942. https://doi.org/10.1080/10236198.2017.1379515
  20. [20] G. Ladas, Ch.G. Philos, Y.G. Sficas, Sharp conditions for the oscillation of delay difference equations, J. Appl. Math. Simulation 2 (1989), 101-111. https://doi.org/10.1155/S1048953389000080
  21. [21] J. Shen, I.P. Stavroulakis, Oscillation criteria for delay difference equations, Electron. J. Differential Equations 2001 (2001), 1-15.
  22. [22] I.P. Stavroulakis, Oscillations of delay difference equations, Comput. Math. Appl. 29 (1995), 83-88. http://doi.org/10.1016/0898-1221(95)00020-Y
  23. [23] I.P. Stavroulakis, Oscillation criteria for first order delay difference equations, Mediterr. J. Math. 1 (2004), 231-240. https://doi.org/10.1007/s00009-004-0013-7
  24. [24] I.P. Stavroulakis, Oscillation criteria for delay and difference equations with non-monotone arguments, Appl. Math. Comput. 226 (2014), 661-672. http://doi.org/10.1016/j.amc.2013.10.041
  25. [25] B.G. Zhang, C.J. Tian, Nonexistence and existence of positive solutions for difference equations with unbounded delay, Comput. Math. Appl. 36 (1998), 1-8. http://doi.org/10.1016/S0898-1221(98)00103-5
About the authors ▼
Emad R. Attia (corresponding author)
https://orcid.org/0000-0002-7978-5386
Prince Sattam Bin Abdulaziz University, College of Sciences and Humanities in Alkharj, Department of Mathematics, Alkharj 11942, Saudi Arabia; Damietta University, Faculty of Science, Department of Mathematics, New Damietta 34517, Egypt
Bassant M. El-Matary
https://orcid.org/0000-0003-4525-156X
Qassim University, College of Science and Arts, Department of Mathematics, Al-Badaya, Buraidah, Saudi Arabia; Damietta University, Faculty of Science, Department of Mathematics, New Damietta 34517, Egypt
George E. Chatzarakis
https://orcid.org/0000-0002-0477-1895
Department of Electrical and Electronic Engineering Educators, School of Pedagogical and Technological, Education (ASPETE), Athens, Marousi 15122, Athens, Greece
About the article ▼
Communicated by Josef Diblík.
Received: 2021-12-26.
Revised: 2022-08-06.
Accepted: 2022-08-08.
Published online: 2022-11-24.
Cite this article as ▼
Emad R. Attia, Bassant M. El-Matary, George E. Chatzarakis, New oscillation conditions for first-order linear retarded difference equations with non-monotone arguments, Opuscula Math. 42, no. 6 (2022), 769-791, https://doi.org/10.7494/OpMath.2022.42.6.769.
Download this article's citation as: