Skip to main content
SpringerLink
Log in

Stability and Boundedness Results for Solutions of Certain Third Order Nonlinear Vector Differential Equations

  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, we investigate the asymptotic stability of the zero solution and boundedness of all solutions of a certain third order nonlinear ordinary vector differential equation. The results are proved using Lyapunov’s second (or direct method). Our results include and improve some well known results existing in the literature.

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. Reissig, R., Sansone, G., and Conti, R., Nonlinear Differential Equations of Higher Order, Noordhoff, Groningen, 1974.

  2. Ezeilo, J. O. C., ‘Periodic solutions of certain third order differential equations’, Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali. Serie 57(8), 1974, 1–2; 1975, 54–60.

  3. Ezeilo, J. O. C., ‘Periodic solutions of certain third order differential equations of the nondissipative type’, Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali. Serie 63(8), 1977, 3–4; 1978, 212–224.

  4. Feng, C. H., ‘On the existence of almost periodic solutions of nonlinear third-order differential equations’, Annals of Differential Equations 9(4), 1993, 420–424.

    MATH  MathSciNet  Google Scholar 

  5. Feng, C. H., ‘The existence of periodic solutions for a third-order nonlinear differential equation’, (Chinese) Gongcheng Shuxue Xuebao 11(2), 1994, 113–117.

  6. Mehri, B., ‘Periodic solution for certain nonlinear third order differential equations’, Indian Journal of Pure and Applied Mathematics 21(3), 1990, 203–210.

    MATH  MathSciNet  Google Scholar 

  7. Mehri, B. and Shadman, D., ‘Periodic solutions of certain third order nonlinear differential equations’, Studia Scientiarum Mathematicarum Hungarica 33(4), 1997, 345–350.

    MathSciNet  MATH  Google Scholar 

  8. Qian, C., ‘On global stability of third-order nonlinear differential equations’, Nonlinear Analysis: Theory, Methods & Applications Ser. A 42(4), 2000, 651–661.

  9. Qian, C., ‘Asymptotic behavior of a third-order nonlinear differential equation’, Journal of Mathematical Analysis and Applications 284(1), 2003, 191–205.

    Article  MATH  MathSciNet  Google Scholar 

  10. Shadman, D. and Mehri, B., ‘On the periodic solutions of certain nonlinear third order differential equations’, Zeitschrift für Angewandte Mathematik und Mechanik 75(2), 1995, 164–166.

    MathSciNet  MATH  Google Scholar 

  11. Tejumola, H. O., ‘Periodic solutions of certain third order differential equations’, Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali. Serie 66(8), 1979, no. 4, 243–249.

  12. Vladimir, V., ‘To existence of the periodic solution of a third-order nonlinear differential equation’, Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 29, 1990, 123–163.

  13. Yoshizawa, T., ‘Stability theory and the existence of periodic solutions and almost periodic solutions’, Applied Mathematical Sciences, Vol. 14, Springer-Verlag, New York-Heidelberg, 1975.

    Google Scholar 

  14. Zhang, X. Z., ‘Stability in the large of a class of third-order nonlinear ordinary differential equations’, Kexue Tongbao (English edn.) 28(7), 1983, 999–1000.

  15. Wang, L. and Wang, M. Q., ‘On the construction of globally asymptotically stable Lyapunov functions for a type of nonlinear third-order system’, Acta Mathematicae Applicatae Sinica 6(3), 1983, 309–323.

    MathSciNet  MATH  Google Scholar 

  16. Abou-El-Ela, A. M. A., ‘Boundedness of the solutions of certain third-order vector differential equations’, Annals of Differential Equations 1(2), 1985, 127–139.

    MATH  MathSciNet  Google Scholar 

  17. Afuwape, A.U., ‘Ultimate boundedness results for a certain system of third-order nonlinear differential equations’, Journal of Mathematical Analysis and Applications 97(1), 1983, 140–150.

    Article  MATH  MathSciNet  Google Scholar 

  18. Afuwape, A. U., ‘ Further ultimate boundedness results for a third-order nonlinear system of differential equations’, Unione Matematica Italiana Bollettino C. Serie VI 4(1), 1985, 347–361.

  19. Edziwy Spolhk, S., ‘Stability of certain third order vector differential equations’, Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 23, 1975, 15–21.

  20. Ezeilo, J. O. C. and Tejumola, H. O., ‘Boundedness and periodicity of solutions of a certain system of third-order non-linear differential equations’, Annali di Matematica Pura ed Applicata 74(4), 1966, 283–316.

    MathSciNet  MATH  Google Scholar 

  21. Feng, C., ‘On the existence of periodic solutions for a certain system of third order nonlinear differential equations’, Annals of Differential Equations 11(3), 1995, 264–269.

    MATH  MathSciNet  Google Scholar 

  22. Kalantarov, V. and Tiryaki, A., ‘On the stability results for third order differential-operator equations’, Turkish Journal of Mathematics 21(2), 1997, 179–186.

    MathSciNet  MATH  Google Scholar 

  23. Meng, F. W., ‘Ultimate boundedness results for a certain system of third order nonlinear differential equations’, Journal of Mathematical Analysis and Applications 177(2), 1993, 496–509.

    Article  MATH  MathSciNet  Google Scholar 

  24. Tiryaki, A., ‘Boundedness and periodicity results for a certain system of third order nonlinear differential equations’, Indian Journal of Pure and Applied Mathematics 30(4), 1999, 361–372.

    MATH  MathSciNet  Google Scholar 

  25. Tunç, C., ‘On the boundedness and periodicity of the solutions of a certain vector differential equation of third-order’, Applied Mathematics and Mechanics (English edn.) 20(2), 1999, 163–170.

  26. Tunç, C., ‘Uniform ultimate boundedness of the solutions of third-order nonlinear differential equations’, Kuwait Journal of Science & Engineering 32(1), 2005, 39–48.

    MathSciNet  Google Scholar 

  27. Tunç, C., ‘On the asymptotic behavior of solutions of certain third-order nonlinear differential equation’, Journal of Applied Mathematics and Stochastic Analysis 2005(1), 2005, 29–35.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Cemil Tunç

  2. Van Technical High School, Yüzüncü Yıl University, 65080, Van-Turkey

    Muzaffer Ateş

Authors
  1. Cemil Tunç
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Muzaffer Ateş
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cemil Tunç.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tunç, C., Ateş, M. Stability and Boundedness Results for Solutions of Certain Third Order Nonlinear Vector Differential Equations. Nonlinear Dyn 45, 273–281 (2006). https://doi.org/10.1007/s11071-006-1437-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-006-1437-3

Keywords

Search

Navigation