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On the asymptotic behavior of a class of third order nonlinear neutral differential equations

  • Research Article
  • Published:
Central European Journal of Mathematics

Abstract

The objective of this paper is to study asymptotic properties of the third-order neutral differential equation

$$ \left[ {a\left( t \right)\left( {\left[ {x\left( t \right) + p\left( t \right)x\left( {\sigma \left( t \right)} \right)} \right]^{\prime \prime } } \right)^\gamma } \right]^\prime + q\left( t \right)f\left( {x\left[ {\tau \left( t \right)} \right]} \right) = 0, t \geqslant t_0 . \left( E \right) $$

. We will establish two kinds of sufficient conditions which ensure that either all nonoscillatory solutions of (E) converge to zero or all solutions of (E) are oscillatory. Some examples are considered to illustrate the main results.

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References

  1. Akın-Bohner E., Došlá Z., Lawrence B., Oscillatory properties for three-dimensional dynamic systems, Nonlinear Anal., 2008, 69(2), 483–494

    Article  MATH  MathSciNet  Google Scholar 

  2. Baculíková B., Džurina J., Oscillation of third-order neutral differential equations, Math. Comput. Modelling, 2010, 52(1–2), 215–226

    Article  Google Scholar 

  3. Baculíková B., Elabbasy E.M., Saker S.H., Džurina J., Oscillation criteria for third-order nonlinear differential equations, Math. Slovaca, 2008, 58(2), 201–220

    Article  MATH  MathSciNet  Google Scholar 

  4. Baĭnov D.D., Mishev D.P., Oscillation Theory for Neutral Differential Equations with Delay, Adam Hilger, Bristol, 1991

    MATH  Google Scholar 

  5. Džurina J., Asymptotic properties of third order delay differential equations, Czechoslovak Math. J., 1995, 45(120)(3), 443–448

    MATH  MathSciNet  Google Scholar 

  6. Džurina J., Asymptotic properties of the third order delay differential equations, Nonlinear Anal., 1996, 26(1), 33–39

    Article  MATH  MathSciNet  Google Scholar 

  7. Džurina J., Oscillation theorems for neutral differential equations of higher order, Czechoslovak Math. J., 2004, 54(129)(1), 107–117

    Article  MATH  MathSciNet  Google Scholar 

  8. Džurina J., Baculíková B., Oscillation of third-order functional differential equations, Electron. J. Qual. Theory Differ. Equ., 2010, No. 43

  9. Erbe L.H., Kong Q., Zhang B.G., Oscillation Theory for Functional-Differential Equations, Monogr. Textbooks Pure Appl. Math., 190, Marcel Dekker, New York, 1995

    Google Scholar 

  10. Grace S.R., Agarwal R.P., Pavani R., Thandapani E., On the oscillation of certain third order nonlinear functional differential equations, Appl. Math. Comput., 2008, 202(1), 102–112

    Article  MATH  MathSciNet  Google Scholar 

  11. Greguš M., Third Order Linear Differential Equations, Math. Appl. (East European Ser.), Reidel, Dordrecht, 1987

    Google Scholar 

  12. Györi I., Ladas G., Oscillation Theory of Delay Differential Equations, Oxford Math. Monogr., Clarendon Press, Oxford, 1991

    Google Scholar 

  13. Hassan T.S., Oscillation of third order nonlinear delay dynamic equations on time scales, Math. Comput. Modelling, 2009, 49(7–8), 1573–1586

    Article  MATH  MathSciNet  Google Scholar 

  14. Kiguradze I.T., Chanturia T.A., Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Math. Appl. (Soviet Ser.), 89, Kluwer Acad. Publ., Dordrecht, 1993

    Google Scholar 

  15. Kusano T., Naito M., Comparison theorems for functional-differential equations with deviating arguments, J. Math. Soc. Japan, 1981, 33(3), 509–533

    Article  MATH  MathSciNet  Google Scholar 

  16. Lacková D., The asymptotic properties of the solutions of the nth order functional neutral differential equations, Appl. Math. Comput., 2003, 146(2–3), 385–392

    Article  MATH  MathSciNet  Google Scholar 

  17. Ladde G.S., Lakshmikantham V., Zhang B.G., Oscillation Theory of Differential Equations with Deviating Arguments, Monogr. Textbooks Pure Appl. Math., 110, Marcel Dekker, New York, 1987

    Google Scholar 

  18. Lazer A.C., The behavior of solutions of the differential equation y’" + p(x)y’ + q(x)y = 0, Pacific J. Math., 1966, 17(3), 435–466

    MATH  MathSciNet  Google Scholar 

  19. Parhi N., Das P., Asymptotic property of solutions of a class of third-order differential equations, Proc. Amer. Math. Soc., 1990, 110(2), 387–393

    MATH  MathSciNet  Google Scholar 

  20. Parhi N., Padhi S., On asymptotic behavior of delay-differential equations of third order, Nonlinear Anal., 1998, 34(3), 391–403

    Article  MATH  MathSciNet  Google Scholar 

  21. Parhi N., Padhi S., Asymptotic behaviour of solutions of third order delay-differential equations, Indian J. Pure Appl. Math., 2002, 33(10), 1609–1620

    MATH  MathSciNet  Google Scholar 

  22. Philos Ch.G., On the existence of nonoscillatory solutions tending to zero at 1 for differential equations with positive delays, Arch. Math. (Basel), 1981, 36(2), 168–178

    MATH  MathSciNet  Google Scholar 

  23. Saker S.H., Oscillation criteria of third-order nonlinear delay differential equations, Math. Slovaca, 2006, 56(4), 433–450

    MATH  MathSciNet  Google Scholar 

  24. Saker S.H., Džurina J., On the oscillation of certain class of third-order nonlinear delay differential equations, Math. Bohemica, 2010, 135(3), 225–237

    Google Scholar 

  25. Tanaka K., Asymptotic analysis of odd order ordinary differential equations, Hiroshima Math. J., 1980, 10(2), 391–408

    MATH  MathSciNet  Google Scholar 

  26. Tiryaki A., Aktas M.F., Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping, J. Math. Anal. Appl., 2007, 325(1), 54–68

    Article  MATH  MathSciNet  Google Scholar 

  27. Zhong J., Ouyang Z., Zou S., Oscillation criteria for a class of third-order nonlinear neutral differential equations, J. Appl. Anal. (in press)

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Electrical Engineering and Informatics, Technical University of Košice, Košice, Slovakia

    Blanka Baculíková & Jozef Džurina

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  1. Blanka Baculíková
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  2. Jozef Džurina
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Correspondence to Blanka Baculíková.

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Baculíková, B., Džurina, J. On the asymptotic behavior of a class of third order nonlinear neutral differential equations. centr.eur.j.math. 8, 1091–1103 (2010). https://doi.org/10.2478/s11533-010-0072-x

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