Skip to main content
SpringerLink
Log in

New stability and boundedness results of Liénard type equations with multiple deviating arguments

  • Differential Equations
  • Published:
Journal of Contemporary Mathematical Analysis Aims and scope Submit manuscript

Abstract

The paper considers a Liénard type equation withmultiple variable deviating arguments. Some sufficient conditions, under which the solution of this equation is asymptotically stable and bounded by means of the Lyapunov functional approach, are found. An example showing the effectiveness of the result is given.

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. S. Ahmad, M. Rama Mohana Rao, Theory of Ordinary Differential Equations. With Applications in Biology and Engineering (Affiliated East-West Press Pvt. Ltd., New Delhi, 1999).

    Google Scholar 

  2. L. É. Él’sgol’ts, Introduction to the Theory of Differential Equations with Deviating Arguments. Translated from Russian (Robert J. McLaughlin Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1966).

  3. L. É. Él’sgol’ts, S. B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments Translated from Russian by John L. Casti. Mathematics in Science and Engineering, 105 (Academic Press [A Subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973).

  4. J. K. Hale, “Sufficient Conditions for Stability and Instability of Autonomous Functional-Differential Equations”, J. Differential Equations, 1 452–482 (1965).

    Article  MATH  MathSciNet  Google Scholar 

  5. J. K. Hale, Theory of Functional Differential Equations (Springer-Verlag, New York-Heidelberg, 1977).

    MATH  Google Scholar 

  6. J. W. Heidel, “Global Asymptotic Stability of a Generalized Liénard Equation”, SIAM J. Appl. Math., 19 629–636 (1970).

    Article  MATH  MathSciNet  Google Scholar 

  7. V. Kolmanovskii, A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations (Kluwer Academic Publishers, Dordrecht, 1999).

    MATH  Google Scholar 

  8. N. N. Krasovskii, Stability of Motion. Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay (Stanford University Press, Stanford, Calif., 1963).

    MATH  Google Scholar 

  9. B. Liu, L. Huang, “Boundedness of Solutions for a Class of Liénard Equations with a Deviating Argument”, Appl.Math. Lett., 21(2) 109–112 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  10. Xin-Ge Liu, Mei-Lan Tang, Ralph R. Martin, “Periodic Solutions for a Kind of Liщnard Equation”, J. Comput. Appl.Math., 219(1) 263–275 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  11. W. S. Luk, “Some Results Concerning the Boundedness of Solutions of Liénard Equations with Delay”, SIAM J. Appl.Math., 30(4) 768–774 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  12. aA. M. Lyapunov, Stability of Motion (Academic Press, London, 1966).

    MATH  Google Scholar 

  13. I. A. Malyseva, “Boundedness of Solutions of a Liénard Differential Equation”, Differetial’niye Uravneniya, 15(8) 1420–1426 (1979).

    MathSciNet  Google Scholar 

  14. A. S. C. Sinha, “On Stability of Solutions of Some Third and Fourth Order Delay-Differential Equations”, Information and Control, 23 165–172 (1973).

    Article  MATH  MathSciNet  Google Scholar 

  15. J. Sugie, Y. Amano, “Global Asymptotic Stability of Non-Autonomous Systems of Liщnard Type”, J.Math. Anal. Appl, 289(2) 673–690 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  16. C. Tunc, “Some Stability and Boundedness Results to Nonlinear Differential Equations of Liщnard Type with Finite Delay”, J. Comput. Anal. Appl., 11(4) 711–727 (2009).

    MATH  MathSciNet  Google Scholar 

  17. C. Tunc, E. Tunc, “On the Asymptotic Behavior of Solutions of Certain Second-Order Differential Equations”, J. Franklin Inst., Engineering and Applied Mathematics, 344(5), 391–398 (2007).

    Article  MathSciNet  Google Scholar 

  18. W. R. Utz, “Boundedness and Periodicity of Solutions of the Generalized Lińard Equation”, Ann. Mat. Pura Appl., 42(4) 313–324 (1956).

    MATH  MathSciNet  Google Scholar 

  19. Y. Yu, C. Zhao, “Boundedness of Solutions for a Liénard Equation with Multiple Deviating Arguments”, Electron. J. Equations 14, 5 pp.?? (2009).

  20. T. Yoshizawa, “Stability Theory by Liapunov’s Second Method”, Publications of the Mathematical Society of Japan, 9 (TheMathematical Society of Japan, Tokyo, 1966).

    Google Scholar 

  21. B. Zhang, “Necessary and Sufficient Conditions for Boundedness and Oscillation in the Retarded Liénard Equation”, J.Math. Anal. Appl., 200(2) 453–473 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  22. L. Zhao, “Boundedness and Convergence for the Non-Liénard type Differential Equation”, ActaMath. Sci. Ser. B Engl. Ed., 27(2) 338–346 (2007).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Yuzuncu Yil University, Van, Turkey

    Cemil Tunc

Authors
  1. Cemil Tunc
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cemil Tunc.

Additional information

Original Russian Text © Cemil Tunc, 2010, published in Izvestiya NAN Armenii. Matematika, 2010, No. 4, pp. 47–56.

About this article

Cite this article

Tunc, C. New stability and boundedness results of Liénard type equations with multiple deviating arguments. J. Contemp. Mathemat. Anal. 45, 214–220 (2010). https://doi.org/10.3103/S1068362310040047

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1068362310040047

MSC2010 numbers

Key words

Search

Navigation