Journal of Applied Mathematics and Computing

, Volume 59, Issue 1–2, pp 257–270

# On oscillatory first order nonlinear neutral differential equations with nonlinear impulses

• Shyam S. Santra
• Arun K. Tripathy
Original Research

## Abstract

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.

## Keywords

Oscillation Nonoscillation Neutral Impulsive differential equations Delay

34K

## Notes

### Acknowledgements

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.

### Conflict of interest

The authors declare that they have no competing interests.

## References

1. 1.
Agarwal, R.P., Karakoc, F.: A survey on oscillation of impulsive delay differential equations. Comput. Math. Appl. 60, 1648–1685 (2010)
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)
3. 3.
Berezansky, L., Braverman, E.: Oscillation of a linear delay impulsive differential equations. Commun. Appl. Nonlinear Anal. 3, 61–77 (1996)
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)
5. 5.
Diblik, J.: Positive solutions of nonlinear delayed differential equations with impulses. Appl. Math. Lett. 72, 16–22 (2017)
6. 6.
Gopalsamy, K., Zhang, B.G.: On delay differential equations with impulses. J. Math. Anal. Appl. 139, 110–122 (1989)
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)
8. 8.
Hale, J.K.: Theory of Functional Differential Equations. Spinger, New Yerk (1977)
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)
10. 10.
Lakshmikantham, V., Bainov, D.D., Simieonov, P.S.: Oscillation Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)
11. 11.
Luo, Z., Jing, Z.: Periodic boundary value problem for first-order impulsive functional differential equations. Comput. Math. Appl. 55, 2094–2107 (2008)
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)
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)
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)
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)
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)
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)
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)
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)