---

Discontinuity, Nonlinearity, and Complexity 8(4) (2019) 403--418 | DOI:10.5890/DNC.2019.12.005

V. Muthulakshmi, R. Manjuram

Department of Mathematics, Periyar University, Salem-636011, Tamilnadu, India

Download Full Text PDF

 

Abstract

This paper is devoted to the study of the oscillatory behavior of damped second order nonlinear impulsive differential equations with variable delay. The results obtained here extend and complement to some known results in the literature.

Acknowledgments

This work was partialy supported by UGC-Special Assistance Programme (No.F.510/7/DRS-1/2016 (SAP-1)) and R. Manjuram was supported by University Grants Commission, New Delhi 110 002, India (Grant No. F1-17.1/2013-14/RGNF-2013-14-SCTAM-38915/(SA-III/Website)).

References

1. [1]	Bainov, D.D. and Simeonov, P.S. (1993), Impulsive Differential Equations: Periodic Solution and Applications, Longman, London.

2. [2]	Deo, S.G. and Pandit, S.G. (1982), Differential Systems Involving Impulses, Springer, New York.

3. [3]	Lakshmikantham, V., Bainov, D.D., and Simieonov, P.S. (1989), Theory of Impulsive Differential Equations, World Scientific Publishers, Singapore.

4. [4]	Driver, R.D. (1977), Ordinary and Delay Differential Equations, Springer- Verlag, New York.

5. [5]	Gy¨ori, I. and Ladas, G. (1991), Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford.

6. [6]	El- Sayed, M.A. (1993), An oscillation criteria for forced second order linear differential equations, Proceedings of the American Mathematical Society, 118, 813-817.

7. [7]	Guo, Z., Zhou, X., and Ge, W. (2011), Interval oscillation criteria for second-order forced impulsive differential equations with mixed nonlinearities, Journal of Mathematical Analysis and Applications, 381, 187-201.

8. [8]	Liu, X. and Xu, Z. (2007), Oscillation of a forced super-linear second order differential equation with impulses, Computers and Mathematics with Applications, 53, 1740-1749.

9. [9]	Liu, X. and Xu, Z. (2009), Oscillation criteria for a forced mixed type Emden-Fowler equation with impulses, Applied Mathematics and Computation, 215, 283-291.

10. [10]	Muthulakshmi, V. and Thandapani, E. (2011), Interval criteria for oscillation of second order impulsive differential equations with mixed nonlinearities, Electronic Journal of Differential Equations, 2011(40), 1-14.

11. [11]	Özbekler, A. and Zafer, A. (2009), Interval criteria for the forced oscillation of super-half-linear differential equations under impulse effects, Mathematical and Computer Modelling, 50, 59-65.

12. [12]	Huang, M. and Feng, W. (2010), Forced oscillations for second order delay differential equations with impulses, Computers and Mathematics with Applications, 59, 18-30.

13. [13]	Zhou, X., Guo, Z., and Wang, W.S. (2014), Interval oscillation criteria for nonlinear impulsive delay differential equations with damping term, Applied Mathematics and Computation, 249, 32-44.

14. [14]	Zhou, X. andWang,W.S. (2016), Interval oscillation criteria for nonlinear impulsive differential equations with variable delay, Electronic Journal of Qualitative Theory of Differential Equations, 1(101), 1-18.

15. [15]	Zhou, X.,Wang, W.S., and Chen, R. (2018), Interval oscillation criteria for forced mixed nonlinear impulsive differential equations with variable delay, Applied Mathematics Letters, 75, 89-95.

16. [16]	Philos, Ch.G. (1989), Oscillation theorems for linear differential equations of second order, Archiv der Mathematik, 53, 482-492.

17. [17]	Xing, L. and Zheng, Z. (2008), New oscillation criteria for forced second order half-linear differential equations with damping, Applied Mathematics and Computation, 198, 481-486.

18. [18]	Beckenbach, E.F. and Bellman, R. (1961), Inequalities, Springer, Berlin.

19. [19]	Hardy, G.H., Littlewood, J.E., and P´olya, G. (1934), Inequalities, Cambridge University Press, Cambridge.

20. [20]	Kong, Q. (1999), Interval criteria for oscillation of second order linear ordinary differential equations, Journal of Mathematical Analysis and Applications, 229, 258-270.