Nonlinear Dynamics

, Volume 57, Issue 1–2, pp 97–106 | Cite as

On the stability and boundedness of solutions to third order nonlinear differential equations with retarded argument

Original Paper

Abstract

This paper investigates stability and boundedness of solutions to third order nonlinear differential equation with retarded argument:
$$\begin{array}{l}x'''(t)+\varphi\bigl(x(t-r),x'(t-r),x''(t-r)\bigr)x''(t)\\\qquad{}+\psi\bigl(x'(t-r)\bigr)+h\bigl(x(t-r)\bigr)\\\quad=p\bigl(t,x(t),x(t-r),x'(t),x'(t-r),x''(t)\bigr).\end{array}$$
By the use of the Lyapunov functional, sufficient conditions for stability and boundedness of solutions to the considered equations are obtained. Examples are introduced throughout the paper for illustrations.

Keywords

Stability Boundedness Lyapunov functional Third order nonlinear differential 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Burton, T.A.: Stability and Periodic Solutions of Ordinary and Functional-Differential Equations. Mathematics in Science and Engineering, vol. 178. Academic Press, Orlando (1985) MATHGoogle Scholar
  2. 2.
    Burton, T.A.: Volterra Integral and Differential Equations. Mathematics in Science and Engineering, vol. 202, 2nd edn. Elsevier, Amsterdam (2005) MATHGoogle Scholar
  3. 3.
    Burton, T.A., Zhang, S.N.: Unified boundedness, periodicity, and stability in ordinary and functional-differential equations. Ann. Mater. Pura Appl. 145(4), 129–158 (1986). doi: 10.1007/BF01790540 MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Èl’sgol’ts, L.È.: Introduction to the Theory of Differential Equations with Deviating Arguments. Holden-Day, San Francisco (1966). Translated from the Russian by Robert J. McLaughlin Google Scholar
  5. 5.
    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). A Subsidiary of Harcourt Brace Jovanovich, Publishers. Translated from the Russian by John L. Casti MATHGoogle Scholar
  6. 6.
    Gopalsamy, K.: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Mathematics and Its Applications, vol. 74. Kluwer Academic, Dordrecht (1992) MATHGoogle Scholar
  7. 7.
    Hale, J.: Theory of Functional Differential Equations. Springer, New York (1977) MATHGoogle Scholar
  8. 8.
    Hale, J., Verduyn Lunel, S.M.: Introduction to Functional-Differential Equations. Applied Mathematical Sciences, vol. 99. Springer, New York (1993) MATHGoogle Scholar
  9. 9.
    Kolmanovskii, V., Myshkis, A.: Introduction to the Theory and Applications of Functional Differential Equations. Kluwer Academic, Dordrecht (1999) MATHGoogle Scholar
  10. 10.
    Kolmanovskii, V.B., Nosov, V.R.: Stability of Functional-Differential Equations. Mathematics in Science and Engineering, vol. 180. Academic Press, London (1986). A subsidiary Harcourt Brace Jovanovich, Publishers MATHGoogle Scholar
  11. 11.
    Krasovskii, N.N.: Stability of Motion. Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay. Stanford University Press, Stanford (1963). Translated by J.L. Brenner MATHGoogle Scholar
  12. 12.
    Lyapunov, A.M.: Stability of Motion. Mathematics in Science and Engineering, vol. 30. Academic Press, New York (1966) MATHGoogle Scholar
  13. 13.
    Palusinski, O., Stern, P., Wall, E., Moe, M.: Comments on “An energy metric algorithm for the generation of Liapunov functions”. IEEE Trans. Autom. Control 14(1), 110–111 (1969) CrossRefGoogle Scholar
  14. 14.
    Reissig, R., Sansone, G., Conti, R.: Non-Linear Differential Equations of Higher Order. Noordhoff International Publishing, Leyden (1974). Translated from the German MATHGoogle Scholar
  15. 15.
    Sadek, A.I.: Stability and boundedness of a kind of third-order delay differential system. Appl. Math. Lett. 16(5), 657–662 (2003). doi: 10.1016/S0893-9659(03)00063-6 MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Sinha, A.S.C.: On stability of solutions of some third and fourth order delay-differential equations. Inf. Control 23, 165–172 (1973). doi: 10.1016/S0019-9958(73)90651-7 MATHCrossRefGoogle Scholar
  17. 17.
    Tejumola, H.O., Tchegnani, B.: Stability, boundedness and existence of periodic solutions of some third and fourth order nonlinear delay differential equations. J. Niger. Math. Soc. 19, 9–19 (2000) MathSciNetGoogle Scholar
  18. 18.
    Tunç, C.: Uniform ultimate boundedness of the solutions of third-order nonlinear differential equations. Kuwait J. Sci. Eng. 32(1), 39–48 (2005) Google Scholar
  19. 19.
    Tunç, C.: New results about stability and boundedness of solutions of certain non-linear third-order delay differential equations. Arab. J. Sci. Eng. 31(2A), 185–196 (2006) MathSciNetGoogle Scholar
  20. 20.
    Tunç, C.: On stability of solutions of certain fourth-order delay differential equations. Appl. Math. Mech. (English ed.) 27(8), 1141–1148 (2006). Chinese translation appears in Appl. Math. Mech. 27(8), 994–1000 (2006) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    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) MATHCrossRefGoogle Scholar
  22. 22.
    Tunç, C.: On asymptotic stability of solutions to third order nonlinear differential equations with retarded argument. Commun. Appl. Anal. 11(3–4), 515–527 (2007) MATHMathSciNetGoogle Scholar
  23. 23.
    Tunç, C.: Stability and boundedness of solutions of nonlinear differential equations of third-order with delay. Differ. Uravn. Protsessy Upr. 3, 1–13 (2007) Google Scholar
  24. 24.
    Tunç, C.: On the boundedness of solutions of third-order delay differential equations. Differ. Equ. (Differ. Uravn.) 44(4), 464–472 (2008) MATHCrossRefGoogle Scholar
  25. 25.
    Tunç, C.: On the stability of solutions to a certain fourth-order delay differential equation. Nonlinear Dyn. 51(1–2), 71–81 (2008) Google Scholar
  26. 26.
    Yoshizawa, T.: Stability theory by Liapunov’s second method. The Mathematical Society of Japan, Tokyo (1966) Google Scholar
  27. 27.
    Zhu, Y.F.: On stability, boundedness and existence of periodic solution of a kind of third order nonlinear delay differential system. Ann. Differ. Equ. 8(2), 249–259 (1992) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Arts and SciencesYüzüncü Yıl UniversityVanTurkey

Personalised recommendations