Skip to main content
SpringerLink
Log in

On Exact Sufficient Oscillation Conditions for Solutions of Linear Differential and Difference Equations of the First Order With Aftereffect

  • Published:
Russian Mathematics Aims and scope Submit manuscript

Abstract

We obtain new unimprovable effective oscillation conditions for all solutions of linear first-order differential and difference equations with several delays. We show that known results of the kind are consequences of the new results. We reveal the reasons for the impossibility to obtain oscillation conditions for equations with several delays, as sharp as the conditions for the equation with one delay, in the case when only known approaches are used.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Myshkis, A. D. “On Solutions of Linear Homogeneous Differential Equations of the First Order of Stable Type With a Retarded Argument”, Mat. Sb. (N. S.) 28, No. 3, 641–658 (1951) [in Russian].

    MathSciNet  MATH  Google Scholar 

  2. Ladas, G. “Sharp Conditions for Oscillations Caused by Delays”, Appl. Anal. 9, No. 2, 93–98 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  3. Koplatadze, R. G., Chanturiya, T. A. “Oscillating and Monotone Solutions of First-Order Differential EquationsWith Deviating Argument”, Differ. Uravn. 18, No. 8, 1463–1465 (1982) [in Russian].

    MATH  Google Scholar 

  4. Ladas, G., Lakshmikantham, V., Papadakis, J. S. “Oscillations of Higher-Order Retarded Differential Equations Generated by the Retarded Argument”, in Proc. Conf., Park City, Utah, 1972, “Delay and Functional Differential Equations and Their Applications (Academic Press, N. Y., 1972), p. 219–231.

    Google Scholar 

  5. Tramov, M. I. “Conditions for the Oscillation of the Solutions of First Order Differential Equations With Retarded Argument”, Izv. Vyssh. Uchebn. Zaved.Mat., No. 3, 92–96 (1975) [in Russian].

    MathSciNet  MATH  Google Scholar 

  6. Ladde, G. S., Lakshmikantham, V., Zhang, B. G. Oscillation Theory of Differential Equations With Deviating Arguments (Marcel Dekker, New York, 1987).

    MATH  Google Scholar 

  7. Azbelev, N. V., Maksimov, V. P., Rakhmatullina, L. F. Introduction to the Theory of Linear Functional-Differential Equations (Nauka, Moscow, 1991;World Federation Publishers Company, Atlanta, 1995).

    MATH  Google Scholar 

  8. Chudinov, K. “Note on Oscillation Conditions for First-Order Delay Differential Equations”, Electron. J. Qual. Theory Diff. Eq., Paper No. 2, 1–10 (2016).

    MathSciNet  MATH  Google Scholar 

  9. Zhang, B. G., Tian, Ch. J. “Nonexistence and Existence of Positive Solutions for Difference EquationsWith Unbounded Delay”, Comput. Math. Appl. 36, No. 1, 1–8 (1998).

    Article  MathSciNet  Google Scholar 

  10. Chatzarakis, G. E., Koplatadze, R., Stavroulakis, I. P. “Optimal Oscillation Criteria for FirstOrder Difference EquationsWith Delay Argument”, Pacific J.Math. 235, No. 1, 15–33 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  11. Chatzarakis, G. E., Koplatadze, R., Stavroulakis, I. P. “Oscillation Criteria of First Order Linear Difference EquationsWith Delay Argument”, Nonlinear Anal. 68, No. 4, 994–1005 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  12. Fukagai, N., Kusano, T. “Oscillation Theory of FirstOrder Functional-Differential EquationsWithDeviating Arguments”, Ann. Mat. Pura Appl. 136, No. 4, 95–117 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  13. Grammatikopoulos, M. K., Koplatadze, R., Stavroulakis, I. P. “On the Oscillation of Solutions of First Order Differential EquationsWith Retarded Arguments”, GeorgianMath. J. 10, No. 1, 63–76 (2003).

    MATH  Google Scholar 

  14. Li, B. “Oscillation of First Order Delay Differential Equations”, Proc. Amer.Math. Soc. 124, No. 12, 3729–3737 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  15. Stavroulakis, I. P. “Oscillation Criteria for Delay and Difference Equations With Non-Monotone Arguments”, Appl.Math. Comput. 226, 661–672 (2014).

    MathSciNet  MATH  Google Scholar 

  16. Stavroulakis, I. P. “Oscillations of Delay and Difference Equations With Variable Coefficients and Arguments”, in Differential and Difference EquationsWith Applications, Springer Proceedings in Mathematics & Statistics 164 (Springer Proc.Math. Stat., 2016), pp. 169–189.

    Chapter  Google Scholar 

  17. Koplatadze, R., Pinelas, S. “Oscillation Criteria for First-Order Linear Difference Equation With Several Delay Arguments”, Nelı¯nı¯ ıňı¯ Koliv. 17, No. 2, 248–267 (2014).

    MATH  Google Scholar 

  18. Tang, X. H., Yu, J. S. “Oscillation of Delay Difference Equation”, Comput. Math. Appl. 37, No. 7, 11–20 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  19. Chatzarakis, G. E., Pinelas, S., Stavroulakis, I. P. “Oscillations of Difference Equations With Several Deviated Arguments”, Aequationes Math. 88, No. 1–2, 105–123 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  20. Braverman, E., Chatzarakis, G. E., Stavroulakis, I. P. “Iterative Oscillation Tests for Difference Equations With Several Non-Monotone Arguments”, J. Diff. Equat. Appl. 21, No. 9, 854–874 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  21. Braverman, E., Chatzarakis, G. E., Stavroulakis, I. P. Corrigendum to Braverman, E., Chatzarakis, G. E., Stavroulakis, I. P. “Iterative Oscillation Tests for Difference Equations With Several Non-Monotone Arguments”, J. Diff. Equat. Appl. 21, No. 9, 854–874 (2015), J. Diff. Equat. Appl. (2017). DOI: 10.1080/10236198.2017.1318866.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Perm National Research Polytechnic University Komsomol’skii, pr. 29, Perm, 614990, Russia

    K. M. Chudinov

Authors
  1. K. M. Chudinov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. M. Chudinov.

Additional information

Original Russian Text © K.M. Chudinov, 2018, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2018, No. 5, pp. 93–98.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chudinov, K.M. On Exact Sufficient Oscillation Conditions for Solutions of Linear Differential and Difference Equations of the First Order With Aftereffect. Russ Math. 62, 79–84 (2018). https://doi.org/10.3103/S1066369X18050110

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X18050110

Keywords

Search

Navigation