Iterative Oscillation Criteria in Deviating Difference Equations

Abstract

The oscillation of the first-order linear difference equations with several non-monotone deviating arguments and nonnegative coefficients is investigated, using an iterative procedure. The conditions obtained by this method achieve a marked improvement on all known conditions in the literature. Examples, numerically solved in MATLAB, are also given to illustrate the applicability and strength of the results.

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References

  1. 1.

    Berezansky, L., Braverman, E.: On existence of positive solutions for linear difference equations with several delays. Adv. Dyn. Syst. Appl. 1, 29–47 (2006)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Braverman, E., Chatzarakis, G.E., Stavroulakis, I.P.: Iterative oscillation tests for difference equations with several non-monotone arguments. J. Differ. Equ. Appl. 21(9), 854–874 (2015)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Chatzarakis, G.E., Horvat-Dmitrović, L., Paš ić, M.: Oscillation tests for difference equations with several non-monotone deviating arguments. Math. Slovaca 68(5), 1083–1096 (2018)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Chatzarakis, G.E., Jadlovská, I.: Oscillations in difference equations with several arguments using an iterative method. Filomat 32(1), 255–273 (2018)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Chatzarakis, G.E., Jadlovská, I.: Difference equations with several non-monotone deviating arguments: iterative oscillation tests. Dyn. Syst. Appl. 27(2), 271–298 (2018)

    Google Scholar 

  6. 6.

    Chatzarakis, G.E., Kusano, T., Stavroulakis, I.P.: Oscillation conditions for difference equations with several variable arguments. Math. Bohem. 140(3), 291–311 (2015)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Chatzarakis, G.E., Manojlovic, J., Pinelas, S., Stavroulakis, I.P.: Oscillation criteria of difference equations with several deviating arguments. Yokohama Math. J. 60, 13–31 (2014)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Chatzarakis, G.E., Pašić, M.: Improved iterative oscillation tests in difference equations with several arguments, 2018. J. Differ. Equ. Appl. 25(1), 64–83 (2019)

    Article  Google Scholar 

  9. 9.

    Chatzarakis, G.E., Pinelas, S., Stavroulakis, I.P.: Oscillations of difference equations with several deviated arguments. Aequat. Math. 88, 105–123 (2014)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Erbe, L.H., Kong, Q.K., Zhang, B.G.: Oscillation Theory for Functional Differential Equations. Marcel Dekker, New York (1995)

    Google Scholar 

  11. 11.

    Li, X., Zhu, D.: Oscillation of advanced difference equations with variable coefficients. Ann. Differ. Equ. 18, 254–263 (2002)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Luo, X.N., Zhou, Y., Li, C.F.: Oscillations of a nonlinear difference equation with several delays. Mathematica Bohemica 128, 309–317 (2003)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Tang, X.H., Yu, J.S.: Oscillation of delay difference equations. Comput. Math. Appl. 37, 11–20 (1999)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Tang, X.H., Zhang, R.Y.: New oscillation criteria for delay difference equations. Comput. Math. Appl. 42, 1319–1330 (2001)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Wang, X.: Oscillation of delay difference equations with several delays. J. Math. Anal. Appl. 286, 664–674 (2003)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Wu, X.P., Wang, L.: Zero-Hopf bifurcation analysis in delayed differential equations with two delays. J. Frankl. Inst. 354, 1484–1513 (2017)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Yan, W., Meng, Q., Yan, J.: Oscillation criteria for difference equation of variable delays. DCDIS Proc. 3, 641–647 (2005)

    Google Scholar 

  18. 18.

    Zhang, B.G., Tian, C.J.: Nonexistence and existence of positive solutions for difference equations with unbounded delay. Comput. Math. Appl. 36, 1–8 (1998)

    MathSciNet  Article  Google Scholar 

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Correspondence to Irena Jadlovská.

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Chatzarakis, G.E., Jadlovská, I. Iterative Oscillation Criteria in Deviating Difference Equations. Mediterr. J. Math. 17, 192 (2020). https://doi.org/10.1007/s00009-020-01635-y

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Keywords

  • Difference equation
  • non-monotone arguments
  • oscillatory solutions
  • nonoscillatory solutions

Mathematics Subject Classification

  • 39A10
  • 39A21