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Uniform ultimate boundedness and asymptotic behaviour of third order nonlinear delay differential equation

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Abstract

In this paper, Lyapunov second method was employed to obtain criteria for uniform ultimate boundedness and asymptotic behaviour of solutions of nonlinear delay differential equations (DDE) of the third order. The results obtained in this investigation include and extend some well known results on third order nonlinear DDE in the literature. For illustration, an example is also given.

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References

  1. Ademola, A.T., Arawomo, P.O.: Uniform stability and boundedness of solutions of nonlinear delay differential equations of third order. Math. J. Okayama Univ. 55, 157–166 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Ademola, A.T., Arawomo, P.O., Ogunlaran, O.M., Oyekan, E.A.: Uniform stability, boundedness and asymptotic behaviour of solutions of some third order nonlinear delay differential equations. Differ. Equ. Control Process. N4, 43–66 (2013)

    Google Scholar 

  3. Afuwape, A.U., Adesina, O.A.: On the bounds for mean-values of solutions to certain third order nonlinear differential equations. Fasciculi Mathematici 36, 5–14 (2005)

    MathSciNet  MATH  Google Scholar 

  4. Afuwape, A.U., Omeike, M.O.: Convergence of solutions for certain non-homogeneous third order differential equations. Kragujevac J. Math. 31, 5–16 (2008)

    MathSciNet  MATH  Google Scholar 

  5. Afuwape, A.U., Omeike, M.O.: On the stability and boundedness of solutions of a kind of third order delay differential equations. Appl. Math. Comput. 200(1), 444–451 (2008)

    MathSciNet  MATH  Google Scholar 

  6. Burton, T.A.: Stability and Periodic Solutions of Ordinary and Functional Differential Equations. Mathematics in Science and Engineering, vol. 178. Academic Press Inc, Orlando (1985)

    MATH  Google Scholar 

  7. Burton, T.A.: Volterra Integral and Differential Equations, vol. 202, 2nd edn. Mathematics in Science and Engineering. Elsevier (2005)

  8. Chukwu, E.N.: On boundedness of solutions of third order differential equations. Ann. Mat. Pura. Appl. 104(4), 123–149 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chukwu, E.N.: On the boundedness and the existence of a periodic solution of some nonlinear third order delay differential equation. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 64(5), 440–447 (1978)

    MathSciNet  MATH  Google Scholar 

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

  11. Ezeilo, J.O.C.: A boundedness theorem for a certain third order differential equation. Proc. Lond. Math. Soc. (3) 13, 99–124 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ezeilo, J.O.C.: A generalization of some boundedness results by Reissig and Tejumola. J. Math. Anal. Appl. 41, 411–419 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ezeilo, J.O.C.: A note on a boundedness theorem for some third order differential equations. J. Lond. Math. Soc. 36, 439–444 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ezeilo, J.O.C., Tejumola, H.O.: Boundedness theorems for certain third order differential equations. Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (10) 55, 194–201 (1973)

    MathSciNet  MATH  Google Scholar 

  15. Hale, J.: Theory of Functional Differential Equations. Springer-Verlag, New York (1977)

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  17. Kolmanovskii, V.B., Nosov, V.R.: Stability of Functional-Differential Equations. Mathematics in Science and Engineering, vol. 180. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London (1986)

  18. Krasovskii, N.N.: Stability of Motion. Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay. Translated by J.L. Brenner, Stanford University Press, Stanford, California (1963)

  19. Ogundare, B.S.: On the convergence of solutions of certain third order nonlinear differential equation. Math. Sci. Res. J. 9(11), 304–312 (2005)

    MathSciNet  MATH  Google Scholar 

  20. Oudjedi, L., Beldjerd, D., Remili, M.: On the stability of solutions for non-autonomous delay differential equations of third-order. Differ. Equ. Control Process. 1, 22–34 (2014)

    MATH  Google Scholar 

  21. Qian, C.: Asymptotic behaviour of a third order nonlinear differential equation. J. Math. Anal. Appl. 284(1), 191–205 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Reissig, R., Sansone, G., Conti, R.: Non-linear Differential Equations of Higher Order. Translated from the German. Noordhoff International Publishing, Leyden (1974)

  23. Remili, M., Beldjerd, D.: On the asymptotic behavior of the solutions of third order delay differential equations. Rend. Circ. Mat. Palermo 63, 447–455 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Remili, M., Oudjedi, L.D.: Stability and boundedness of the solutions of non autonomous third order differential equations with delay. Acta Univ. Palacki. Olomuc. Fac. rer. nat. Mathematica 53(2), 139–147 (2014)

    MathSciNet  MATH  Google Scholar 

  25. Remili, M., Oudjedi, L.D.: Uniform stability and boundedness of a kind of third order delay differential equations. Bull. Comput. Appl. Math. 2(1), 25–35 (2014)

    MATH  Google Scholar 

  26. Rouche, N., Habets, N., Laloy, M.: Stability Theory by Liapunov’s Direct Method. Applied Mathematical Sciences, vol. 22. Springer-Verlag, New York (1977)

    Book  MATH  Google Scholar 

  27. Sinha, A.S.C.: On stability of solutions of some third and fourth order delay-differential equations. Inf Control 23, 165–172 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  28. Swick, K.: On the boundedness and the stability of solutions of some non-autonomous differential equations of the third order. J. Lond. Math. Soc. 44, 347–359 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  29. Swick, K.E.: Asymptotic behaviour of the solutions of certain third order differential equations. SIAM J. Appl. 19(1), 96–102 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  30. Tejumola, H.O.: A note on the boundedness of solutions of some nonlinear differential equations of the third order. Ghana J. Sci. 11(2), 117–118 (1970)

    Google Scholar 

  31. Tunç, C.: Boundedness in third order nonlinear differential equations with bounded delay. Bol. Mat. 16(1), 1–10 (2009)

    MathSciNet  MATH  Google Scholar 

  32. Tunç, C.: New results about stability and boundedness of solutions of certain non-linear third-order delay differential equations. Arab. J. Sci. Eng. Sect. A Sci. 31(2), 185–196 (2006)

    MathSciNet  MATH  Google Scholar 

  33. Tunç, C.: On asymptotic stability of solutions to third order nonlinear differential equations with retarded argument. Commun. Appl. Anal. 11(4), 515–528 (2007)

    MathSciNet  MATH  Google Scholar 

  34. Tunç, C.: On some qualitative behaviors of solutions to a kind of third order nonlinear delay differential equations. Electron. J. Qual. Theory Differ. Equ. 12, 1–19 (2010)

    MathSciNet  MATH  Google Scholar 

  35. Tunç, E.: On the convergence of solutions of certain third order differential equations. Discrete Dyn. Nat. Soc. 2009, 1–12 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tunç, C.: On the stability and boundedness of solutions of nonlinear third order differential equations with delay. Filomat 24(3), 1–10 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Tunç, C.: Stability criteria for certain third order nonlinear delay differential equations. Port. Math. 66(1), 71–80 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Yoshizawa, T.: Stability Theory by Liapunov’s Second Method. The Mathematical Society of Japan, Tokyo (1966)

    MATH  Google Scholar 

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  1. Department of Mathematics, University of Oran, 31000, Oran, Algeria

    Moussadek Remili & Lynda Damerdji Oudjedi

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  1. Moussadek Remili
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  2. Lynda Damerdji Oudjedi
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Correspondence to Moussadek Remili.

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Remili, M., Oudjedi, L.D. Uniform ultimate boundedness and asymptotic behaviour of third order nonlinear delay differential equation. Afr. Mat. 27, 1227–1237 (2016). https://doi.org/10.1007/s13370-016-0405-4

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