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Stability to vector Liénard equation with constant deviating argument

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Abstract

In this paper, we consider the vector Liénard equation with the constant deviating argument, τ>0,

$$X''(t) + F\bigl(X(t),X'(t) \bigr)X'(t) + H\bigl(X(t - \tau)\bigr) = P(t) $$

in two cases: (i) P(.)≡0, (ii) P(.)≠0. Based on the Lyapunov–Krasovskii functional approach, the asymptotic stability of the zero solution and the boundedness of all solutions are discussed for these cases. We give an example to illustrate the theoretical analysis made in this work and to show the effectiveness of the method utilized here.

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References

  1. Ahmad, S., Mohana Rao M, R.: Theory of Ordinary Differential Equations. With Applications in Biology and Engineering. East–West Press, New Delhi (1999)

    Google Scholar 

  2. Bellman, R.: Introduction to matrix analysis. Reprint of the second (1970) edition. With a foreword by Gene Golub. Classics in Applied Mathematics, 19. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1997). Reprint of the second (1970) edition. With a foreword by Gene Golub

  3. Burton, T.A.: On the equation x″+f(x)h(x′)x′+g(x)=e(t). Ann. Mat. Pura Appl. (4) 85, 277–285 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  4. Burton, T.A.: Stability and Periodic Solutions of Ordinary and Functional Differential Equations. Academic Press, Orlando (1985)

    MATH  Google Scholar 

  5. Burton, T.A., Zhang, B.: Boundedness, periodicity, and convergence of solutions in a retarded Liénard equation. Ann. Mat. Pura Appl. (4) 165, 351–368 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Caldeira-Saraiva, F.: The boundedness of solutions of a Liénard equation arising in the theory of ship rolling. IMA J. Appl. Math. 36(2), 129–139 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cantarelli, G.: On the stability of the origin of a non-autonomous Liénard equation. Boll. Unione Mat. Ital., A 10(7), 563–573 (1996)

    MathSciNet  MATH  Google Scholar 

  8. Èl’sgol’ts, L.È.: Introduction to the Theory of Differential Equations with Deviating Arguments. Holden-Day, Inc., San Francisco (1966). Translated from the Russian by Robert J. McLaughlin

    Google Scholar 

  9. Èl’sgol’ts, L.È., Norkin, S.B.: Introduction to the Theory and Application of Differential Equations with Deviating Arguments. Mathematics in Science and Engineering, vol. 105. Academic Press, New York (1973). Translated from the Russian by John L. Casti

    MATH  Google Scholar 

  10. Gao, S.Z., Zhao, L.Q.: Global asymptotic stability of generalized Liénard equation. Chin. Sci. Bull. 40(2), 105–109 (1995)

    MathSciNet  MATH  Google Scholar 

  11. Hale, J.: Sufficient conditions for stability and instability of autonomous functional-differential equations. J. Differ. Equ. 1, 452–482 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hara, T., Yoneyama, T.: On the global center of generalized Liénard equation and its application to stability problems. Funkc. Ekvacioj 28(2), 171–192 (1985)

    MathSciNet  MATH  Google Scholar 

  13. Hara, T., Yoneyama, T.: On the global center of generalized Liénard equation and its application to stability problems. Funkc. Ekvacioj 31(2), 221–225 (1988)

    MathSciNet  MATH  Google Scholar 

  14. Heidel, J.W.: Global asymptotic stability of a generalized Liénard equation. SIAM J. Appl. Math. 19(3), 629–636 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  15. Heidel, J.W.: A Liapunov function for a generalized Liénard equation. J. Math. Anal. Appl. 39, 192–197 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  16. Huang, L.H., Yu, J.S.: On boundedness of solutions of generalized Liénard’s system and its application. Ann. Differ. Equ. 9(3), 311–318 (1993)

    MathSciNet  MATH  Google Scholar 

  17. Jitsuro, S., Yusuke, A.: Global asymptotic stability of non-autonomous systems of Lienard type. J. Math. Anal. Appl. 289(2), 673–690 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kato, J.: On a boundedness condition for solutions of a generalized Liénard equation. J. Differ. Equ. 65(2), 269–286 (1986)

    Article  MATH  Google Scholar 

  19. Kato, J.: A simple boundedness theorem for a Liénard equation with damping. Ann. Pol. Math. 51, 183–188 (1990)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  21. Krasovskiì, N.N.: 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 

  22. Li, H.Q.: Necessary and sufficient conditions for complete stability of the zero solution of the Liénard equation. Acta Math. Sin. 31(2), 209–214 (1988)

    MATH  Google Scholar 

  23. Liu, B., Huang, L.: Boundedness of solutions for a class of retarded Liénard equation. J. Math. Anal. Appl. 286(2), 422–434 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Liu, B., Huang, L.: Boundedness of solutions for a class of Liénard equations with a deviating argument. Appl. Math. Lett. 21(2), 109–112 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Liu, C.J., Xu, S.L.: Boundedness of solutions of Liénard equations. J. Qingdao Univ. Nat. Sci. Ed. 11(3), 12–16 (1998)

    MathSciNet  Google Scholar 

  26. Liu, Z.R.: Conditions for the global stability of the Liénard equation. Acta Math. Sin. 38(5), 614–620 (1995)

    MATH  Google Scholar 

  27. Long, W., Zhang, H.-X.: Boundedness of solutions to a retarded Liénard equation. Electron. J. Qual. Theory Differ. Equ. 24, 9 (2010)

    MathSciNet  Google Scholar 

  28. Luk, W.S.: Some results concerning the boundedness of solutions of Liénard equations with delay. SIAM J. Appl. Math. 30(4), 768–774 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  29. Malyseva, I.A.: Boundedness of solutions of a Liénard differential equation. Differ. Uravn. 15(8), 1420–1426 (1979)

    MathSciNet  Google Scholar 

  30. Muresan, M.: Boundedness of solutions for Liénard type equations. Mathematica 40(63), 243–257 (1998)

    MathSciNet  Google Scholar 

  31. Nápoles Valdés, J.E.: Boundedness and global asymptotic stability of the forced Liénard equation. Rev. Unión Mat. Argent. 41(4), 47–59 (2000). 2001

    MATH  Google Scholar 

  32. Sugie, J.: On the boundedness of solutions of the generalized Liénard equation without the signum condition. Nonlinear Anal. 11(12), 1391–1397 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sugie, J., Amano, Y.: Global asymptotic stability of non-autonomous systems of Liénard type. J. Math. Anal. Appl. 289(2), 673–690 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  34. Sugie, J., Chen, D.L., Matsunaga, H.: On global asymptotic stability of systems of Liénard type. J. Math. Anal. Appl. 219(1), 140–164 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. Tunç, C.: On the stability of solutions to a certain fourth-order delay differential equation. Nonlinear Dyn. 51(1–2), 71–81 (2008)

    MATH  Google Scholar 

  36. Tunç, C.: On the stability and boundedness of solutions to third order nonlinear differential equations with retarded argument. Nonlinear Dyn. 57(1–2), 97–106 (2009)

    Article  MATH  Google Scholar 

  37. Tunç, C.: Stability and bounded of solutions to non-autonomous delay differential equations of third order. Nonlinear Dyn. 62(4), 945–953 (2010)

    Article  MATH  Google Scholar 

  38. Tunç, C.: Some new stability and boundedness results of solutions of Liénard type equations with deviating argument. Nonlinear Anal. Hybrid Syst. 4(1), 85–91 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  39. Tunç, C.: A note on boundedness of solutions to a class of non-autonomous differential equations of second order. Appl. Anal. Discrete Math. 4(2), 361–372 (2010)

    Article  MathSciNet  Google Scholar 

  40. Tunç, C.: New stability and boundedness results of Liénard type equations with multiple deviating arguments. Izv. Akad. Nauk Arm. SSR, Mat. 45(4), 47–56 (2010)

    Google Scholar 

  41. Tunç, C.: Boundedness results for solutions of certain nonlinear differential equations of second order. J. Indones. Math. Soc. 16(2), 115–128 (2010)

    MathSciNet  MATH  Google Scholar 

  42. Tunç, C.: Stability and boundedness of solutions of non-autonomous differential equations of second order. J. Comput. Anal. Appl. 13(6), 1067–1074 (2011)

    MathSciNet  MATH  Google Scholar 

  43. Tunç, C.: On the stability and boundedness of solutions of a class of Liénard equations with multiple deviating arguments. Vietnam J. Math. 39(2), 177–190 (2011)

    MathSciNet  MATH  Google Scholar 

  44. Tunç, C.: Uniformly stability and boundedness of solutions of second order nonlinear delay differential equations. Appl. Comput. Math. 10(3), 449–462 (2011)

    MathSciNet  MATH  Google Scholar 

  45. Tunç, C., Ateş, M.: Stability and boundedness results for solutions of certain third order nonlinear vector differential equations. Nonlinear Dyn. 45(3–4), 273–281 (2006)

    Article  MATH  Google Scholar 

  46. Tunç, C., Tunç, E.: On the asymptotic behavior of solutions of certain second-order differential equations. J. Franklin Inst. 344(5), 391–398 (2007)

    Article  MathSciNet  Google Scholar 

  47. Yang, Q.G.: Boundedness and global asymptotic behavior of solutions to the Liénard equation. J. Syst. Sci. Math. Sci. 19(2), 211–216 (1999)

    MATH  Google Scholar 

  48. Ye, G.-R., Ding, H.-S., Wu, X.-L.: Uniform boundedness of solutions for a class of Liénard equations. Electron. J. Differ. Equ. 97, 5 pp. (2009)

  49. Yoshizawa, T.: Stability Theory by Liapunov’s Second Method. Publications of the Mathematical Society of Japan, no. 9. The Mathematical Society of Japan, Tokyo (1966)

  50. Zhang, B.: On the retarded Liénard equation. Proc. Am. Math. Soc. 115(3), 779–785 (1992)

    MATH  Google Scholar 

  51. Zhang, B.: Boundedness and stability of solutions of the retarded Liénard equation with negative damping. Nonlinear Anal. 20(3), 303–313 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  52. Zhang, X.S., Yan, W.P.: Boundedness and asymptotic stability for a delay Liénard equation. Math. Pract. Theory 30(4), 453–458 (2000)

    MathSciNet  Google Scholar 

  53. Zhou, X., Jiang, W.: Stability and boundedness of retarded Liénard-type equation. Chin. Q. J. Math. 18(1), 7–12 (2003)

    MathSciNet  MATH  Google Scholar 

  54. Zhou, J., Liu, Z.R.: The global asymptotic behavior of solutions for a nonautonomous generalized Liénard system. J. Math. Res. Expo. 21(3), 410–414 (2001)

    MATH  Google Scholar 

  55. Zhou, J., Xiang, L.: On the stability and boundedness of solutions for the retarded Liénard-type equation. Ann. Differ. Equ. 15(4), 460–465 (1999)

    MathSciNet  MATH  Google Scholar 

  56. Wei, J., Huang, Q.: Global existence of periodic solutions of Liénard equations with finite delay. Dyn. Contin. Discrete Impuls. Syst. 6(4), 603–614 (1999)

    MathSciNet  MATH  Google Scholar 

  57. Wiandt, T.: On the boundedness of solutions of the vector Liénard equation. Dyn. Syst. Appl. 7(1), 141–143 (1998)

    MathSciNet  MATH  Google Scholar 

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Acknowledgement

The author of this paper would like to expresses his sincere appreciation to the anonymous referees for their valuable comments and suggestions which have led to an improvement in the presentation of the paper.

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  1. Department of Mathematics, Faculty of Sciences, Yüzüncü Yıl University, 65080, Van, Turkey

    Cemil Tunç

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  1. Cemil Tunç
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Tunç, C. Stability to vector Liénard equation with constant deviating argument. Nonlinear Dyn 73, 1245–1251 (2013). https://doi.org/10.1007/s11071-012-0704-8

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