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Oscillation Criteria for Certain Fourth-Order Nonlinear Delay Differential Equations

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Abstract

In this article, we establish some new criteria for the oscillation of fourth-order nonlinear delay differential equations of the form

$$(r_2(t)(r_1(t)(y''(t))^\alpha)')' + p(t)(y''(t))^\alpha + q(t)f(y(g(t))) = 0$$

provided that the second-order equation

$$(r_2(t)z'(t))') + \frac{p(t)}{r_1(t)}z(t) = 0$$

is nonoscillatory or oscillatory.

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References

  1. Agarwal, R.P., Grace, S.R., O’Regan, D.: Oscillation theory for second order dynamic equations. In: Series Mathematical Analysis Applications, vol. 5. Taylor and Francis, London (2003)

  2. Agarwal R.P., Grace S.R.: The oscillation of higher-order differential equations with deviating arguments. Comput. Math. Appl. 38, 185–199 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Agarwal R.P., Grace S.R., Kguradze I.T., O’Regan D.: Oscillation of functional differential equations. Math. Comput. Model. 41, 417–461 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Agarwal R.P., Grace S.R., O’Regan D.: Oscillation of certain fourth order functional differential equations. Ukrain. Mat. Zh. 59, 291–313 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Agarwal R.P., Grace S.R., Wong P.J.: On the bounded oscillation of certain fourth order functional differential equations. Nonlinear Dyn. Syst. Theory 5, 215–227 (2005)

    MathSciNet  MATH  Google Scholar 

  6. Agarwal R.P., Grace S.R., O’Regan D.: Oscillation Theory for Difference and Functional Differential Equations. Kluwer Academic, Dordrecht (2000)

    Book  MATH  Google Scholar 

  7. Agarwal R.P., Grace S.R., O’Regan D.: Oscillation criteria for certain nth order differential equations with deviating arguments. J. Math. Anal. Appl. 262, 601–622 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Agarwal R.P., Grace S.R., O’Regan D.: Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations. Kluwer Academic, Dordrecht (2002)

    Book  MATH  Google Scholar 

  9. Agarwal R.P., O’Regan D.: Nonlinear generalized quasi-variational inequalities: a fixed point approach. Math. Inequal. Appl. 6, 133–143 (2003)

    MathSciNet  MATH  Google Scholar 

  10. Grace S.R., Lalli B.J.: On oscillation and nonoscillation of general functional-differential equations. J. Math. Anal. Appl. 109, 522–533 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  11. Grace S.R.: Oscillation theorems for second order nonlinear differential equations with damping. Math. Nachr. 141, 117–127 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Grace S.R., Graef J.R., El-Beltagy M.A.: On the oscillation of third order neutral delay dynamic equations on time scales. Comput. Math. Appl. 63, 775–782 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Grace S.R.: Oscillation theorems for nth-order differential equations with deviating arguments. J. Math. Anal. Appl. 101, 268–296 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  14. Grace S.R., Lalli B.J.: A comparison theorem for general nonlinear ordinary differential equations. J. Math. Anal. Appl. 120, 39–43 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gyori I., Ladas G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon Press, Oxford (1991)

    MATH  Google Scholar 

  16. Philos C.G.: On the existence of nonoscillatory solutions tending to zero at \({\infty}\) for differential equations with positive delays. Arch. Math. (Basel) 36, 168–178 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  17. Grace S.R.: Oscillation criteria for third order nonlinear delay differential equations with damping. Opuscula Math. 35, 485–497 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Orman, Giza, 12221, Egypt

    Said R. Grace

  2. School of Mathematical Sciences, University of Jinan, Jinan, 250022, Shandong, People’s Republic of China

    Shurong Sun

  3. Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, 65409-0020, USA

    Elvan Akın

Authors
  1. Said R. Grace
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  2. Shurong Sun
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  3. Elvan Akın
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Corresponding author

Correspondence to Shurong Sun.

Additional information

This research is supported by the Natural Science Foundation of China (61374074, 61374002), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2013AL003).

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Grace, S.R., Sun, S. & Akın, E. Oscillation Criteria for Certain Fourth-Order Nonlinear Delay Differential Equations. Mediterr. J. Math. 13, 2383–2396 (2016). https://doi.org/10.1007/s00009-015-0630-3

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  • DOI: https://doi.org/10.1007/s00009-015-0630-3

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