Skip to main content
SpringerLink
Log in

Oscillatory Behavior of Second-Order Nonlinear Differential Equations with a Nonpositive Neutral Term

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

We shall present new oscillation criteria of second-order nonlinear differential equations with a nonpositive neutral term of the form:

$$\begin{aligned} \left( (a(t)\left( \left( x(t)-p(t)x(\sigma (t) )^{\prime } \right) ^{\gamma } \right) ^{\prime }+q(t)x^{\beta }(\tau (t))=0,\right. \end{aligned}$$

with positive coefficients. The obtained results answer an open problem raised in Li et al. [Adv Differ Equ 35:7, 2015, Remark 4.3 (P2)]. Examples are given to illustrate the main results.

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. Agarwal, R.P., Grace, S.R., O’Regan, D.: Oscillation Theory for Second Order Dynamic Equations. Ser. Math. Anal. Appl. 5. Taylor & Francis Ltd., London (2003)

    Book  Google Scholar 

  2. Agarwal, R.P., Bohner, M., Li, T., Zhang, C.: A new approach in the study of oscillatory behavior of even-order neutral delay differential equations. Appl. Math. Comput. 225, 787–794 (2013)

    MathSciNet  MATH  Google Scholar 

  3. Baculikova, B., Dzurina, J.: Oscillation theorems for second order nonlinear neutral differential equations. Comput. Math. Appl. 62(12), 4472–4478 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Candan, T.: Oscillation of second-order nonlinear neutral dynamic equations on time scales with distributed deviating arguments. Comput. Math. Appl. 62(11), 4118–4125 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dong, J.G.: Oscillation behavior of second order nonlinear neutral differential equations with deviating arguments. Comput. Math. Appl. 59, 3710–3717 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Grace, S.R., Lalli, B.S.: Oscillation of nonlinear second order neutral differential equations. Rat. Math. 3, 77–84 (1987)

    MathSciNet  MATH  Google Scholar 

  7. Hardy, G.H., Littlewood, I.E., Polya, G.: Inequalities. University Press, Cambridge (1959)

    MATH  Google Scholar 

  8. Li, Q., Wang, R., Chen, F., Li, T.: Oscillation of second-order nonlinear delay differential equations with nonpositive neutral coeffcients. Adv. Differ. Equ. 35, 7 (2015)

  9. Li, T., Rogovchenko, Y., Zhang, C.: Oscillation results for second-order nonlinear neutral differential equations. Adv. Differ. Equ. 2013, 336 (2013)

    Article  MathSciNet  Google Scholar 

  10. Li, T., Rogovchenko, Y.: Oscillation of second order neutral differential equations. Math. Nachr. 288(10), 1150–1162 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Qin, H., Shang, N., Lu, Y.: A note on oscillation criteria of second order nonlinear neutral delay differential equations. Comput. Math. Appl. 56, 2987–2992 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Wong, J.S.W.: Necessary and sufficient conditions for oscillation of second order neutral differential equations. J. Math. Anal. Appl. 252, 342–352 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Yang, Q., Yang, l, Zhu, S.: Interval criteria for oscillation of second-order nonlinear neutral differential equations. Comput. Math. Appl. 46.5, 903–918 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Orman, Giza, 12613, Egypt

    Said R. Grace

Authors
  1. Said R. Grace
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Said R. Grace.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grace, S.R. Oscillatory Behavior of Second-Order Nonlinear Differential Equations with a Nonpositive Neutral Term. Mediterr. J. Math. 14, 229 (2017). https://doi.org/10.1007/s00009-017-1026-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-017-1026-3

Keywords

Mathematics Subject Classification

Search

Navigation