Journal of Applied Mathematics and Computing

, Volume 59, Issue 1–2, pp 257–270 | Cite as

On oscillatory first order nonlinear neutral differential equations with nonlinear impulses

  • Shyam S. SantraEmail author
  • Arun K. Tripathy
Original Research


In this work, we study the oscillatory behaviour of solutions of a class of first order impulsive neutral delay differential equations of the form
$$\begin{aligned} {\left\{ \begin{array}{ll} \bigl (y(t)-p(t)y(t-\tau )\bigr )' + q(t)G\bigl (y(t-\sigma )\bigr )=0,\;t\ne t_k,\;t \ge t_0 \\ y(t^+_k)=I_k\bigl (y(t_k)\bigr ), \;k \in {\mathbb {N}} \\ y(t^+_k-\tau )=I_k\bigl (y(t_k-\tau )\bigr ), \;k \in {\mathbb {N}} \end{array}\right. } \end{aligned}$$
for different ranges of the neutral coefficient p. Finally, two illustrative examples are included to show the effectiveness and feasibility of the main results.


Oscillation Nonoscillation Neutral Impulsive differential equations Delay 

Mathematics Subject Classification




This work is supported by the Department of Science and Technology (DST), New Delhi, India, through the bank instruction Order No. DST/INSPIRE Fellowship/2014/140, dated Sept. 15, 2014.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no competing interests.


  1. 1.
    Agarwal, R.P., Karakoc, F.: A survey on oscillation of impulsive delay differential equations. Comput. Math. Appl. 60, 1648–1685 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bainov, D.D., Simeonov, P.S.: Impulsive Differential Equations: Asymptotic Properties of the Solutions. Series on Advances in Mathematics for Applied Sciences, vol. 28. World Scientific, Singapore (1995)zbMATHGoogle Scholar
  3. 3.
    Berezansky, L., Braverman, E.: Oscillation of a linear delay impulsive differential equations. Commun. Appl. Nonlinear Anal. 3, 61–77 (1996)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Diblik, J., Svoboda, Z., Smarda, Z.: Retract principle for neutral functional differential equation. Nonlinear Anal. Theory Methods Appl. 71(12), 1393–1400 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Diblik, J.: Positive solutions of nonlinear delayed differential equations with impulses. Appl. Math. Lett. 72, 16–22 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gopalsamy, K., Zhang, B.G.: On delay differential equations with impulses. J. Math. Anal. Appl. 139, 110–122 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Graef, J.R., Shen, J.H., Stavroulakis, I.P.: Oscillation of impulsive neutral delay differential equations. J. Math. Anal. Appl. 268, 310–333 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hale, J.K.: Theory of Functional Differential Equations. Spinger, New Yerk (1977)CrossRefzbMATHGoogle Scholar
  9. 9.
    Karpuz, B., Ocalan, O.: Oscillation criteria for a first-order forced differential equations under impulsive effects. Adv. Dyn. Syst. Appl. 7(2), 205–218 (2012)MathSciNetGoogle Scholar
  10. 10.
    Lakshmikantham, V., Bainov, D.D., Simieonov, P.S.: Oscillation Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)CrossRefGoogle Scholar
  11. 11.
    Luo, Z., Jing, Z.: Periodic boundary value problem for first-order impulsive functional differential equations. Comput. Math. Appl. 55, 2094–2107 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Pandian, S., Purushothaman, G.: Oscillation of impulsive neutral differential equation with several positive and negative coefficients. J. Math. Comput. Sci. 2, 241–254 (2012)MathSciNetGoogle Scholar
  13. 13.
    Shen, J.H., Wang, Z.C.: Oscillation and asympotic behaviour of solutions of delay differential equations with impulses. Ann. Differ. Eqs. 10(1), 61–68 (1994)Google Scholar
  14. 14.
    Shen, J.H.: The existence of nonoscillatory solutions of delay differential eqations with impulses. Appl. Math. Comput. 77, 153–165 (1996)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Shen, J., Zou, Z.: Oscillation criteria for first order impulsive differential equations with positive and negative coefficients. J. Comput. Appl. Math. 217, 28–37 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Tripathy, A.K.: Oscillation criteria for a class of first order neutral impulsive differential–difference equations. J. Appl. Anal. Comput. 4, 89–101 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Tripathy, A.K., Santra, S.S.: Necessary and sufficient conditions for oscillation of a class of first order impulsive differential equations. Funct. Differ. Equ. 22(3–4), 149–167 (2015)MathSciNetGoogle Scholar
  18. 18.
    Tripathy, A.K., Santra, S.S., Pinelas, S.: Necessary and sufficient condition for asymptotic behaviour of solutions of a class of first order impulsive systems. Adv. Dyn. Syst. Appl. 11(2), 135–145 (2016)MathSciNetGoogle Scholar
  19. 19.
    Tripathy, A.K., Santra, S.S.: Pulsatile constant and charecterization of first order neutral impulsive differential equations. Commun. Appl. Anal. 20, 65–76 (2016)zbMATHGoogle Scholar
  20. 20.
    Yu, J., Yan, J.: Positive solutions and asymptotic behavior of delay differential equations with nonlinear impulses. J. Math. Anal. Appl. 207, 388–396 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  1. 1.Department of MathematicsSambalpur UniversitySambalpurIndia

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