[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Journal of Applied Nonlinear Dynamics
ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Interval Oscillation of Damped Second-Order Mixed Nonlinear Differential Equation with Variable Delay under Impulse Effects

Journal of Applied Nonlinear Dynamics 9(3) (2020) 493--511 | DOI:10.5890/JAND.2020.09.011

V. Muthulakshmi, R. Manjuram

Department of Mathematics, Periyar University, Salem-636 011, Tamilnadu, India

Download Full Text PDF

 

Abstract

In this paper, we study the oscillatory behavior of damped second-order mixed nonlinear differential equation with variable delay under impulse effects. By using Riccati transformation technique, integral averaging method and some inequalities, we obtain sufficient conditions for oscillation of all solutions. Finally, two examples are presented to illustrate the theoretical results.

Acknowledgments

This work was partially 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]  Khadra, A., Liu, X., and Shen, X. (2003), Application of impulsive synchronization to communication security, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 50, 341-351.
  2. [2]  Liu, X. and Rohlf, K. (1998), Impulsive control of a Lotka-Volterra system, IMA Journal of Mathematical Control and Information, 15, 269-284.
  3. [3]  Stamov, G.T. (2012), Almost Periodic Solutions of Impulsive Differential Equations, Springer-Verlag: Berlin Heidelberg.
  4. [4]  Stamova, I. (2009), Stability Analysis of Impulsive Functional Differential Equations, Walter de Gruyter: Berlin.
  5. [5]  Bainov, D.D. and Simeonov, P.S. (1993), Impulsive Differential Equations: Periodic Solution and Applica- tions, Longman: London.
  6. [6]  Benchohra, M., Henderson, J., and Ntouyas, S. (2006), Impulsive Differential Equations and Inclusions, Hindawi: New York.
  7. [7]  Lakshmikantham, V., Bainov, D.D., and Simeonov, P.S. (1989), Theory of Impulsive Differential Equations, World Scientific: Singapore.
  8. [8]  Samoilenko, A.M. and Perestyuk, N.A. (1995), Impulsive Differential Equations, World Scientific: Singapore.
  9. [9]  Rihan, F.A. and Rihan, B.F. (2015), Numerical modelling of biological systems with memory using delay differential equations, Applied Mathematics & Information Sciences, 9, No.3, 1-14.
  10. [10]  Smith, H. (2011), An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer: New York.
  11. [11]  Sturm, C. (1836), Sur les équations différentielles linéaires du second ordre, Journal de Mathématiques Pures et Appliquées, 1, 106-186.
  12. [12]  Fite, W.B. (1921), Properties of the solutions of certain functional differential equations, Transaction of American Mathematical Society, 22, 311-319.
  13. [13]  Agarwal, R.P., Grace, S.R. and O’Regan, D. (2000), Oscillation Theory for Difference and Functional Dif- ferential Equations, Kluwer Academic Publishers: Dordrecht.
  14. [14]  Erbe, L.H., Kong, Q., and Zhang, B.G. (1995), Oscillation Theory for Functional Differential Equations, Marcel Dekker, Inc: New York.
  15. [15]  Györi, I. and Ladas, G. (1991), Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press: Oxford.
  16. [16]  Saker, S. (2010), Oscillation Theory of Delay Differential and Difference Equations, VDM Verlag Dr. M¨uller Aktiengesellschaft & Co.KG: Germany.
  17. [17]  Gopalsamy, K. and Zhang, B.G. (1989), On delay differential equations with impulses, Journal of Mathemat- ical Analysis and Applications, 139, 110-122.
  18. [18]  Agarwal, R.P. and Karakoc, F. (2010), A survey on oscillation of impulsive delay differential equations, Computers & Mathematics with Applications, 60, 1648-1685.
  19. [19]  Huang, M. and Feng,W. (2010), Forced oscillations for second order delay differential equations with impulses, Computers & Mathematics with Applications, 59, 18-30.
  20. [20]  Li, Q. and Cheung,W-S. (2013), Interval oscillation criteria for second-order forced delay differential equations under impulse effects, Electronic Journal of Differential Equations, 2013, No. 43, 1-11.
  21. [21]  Thandapani, E., Kannan, M., and Pinelas, S. (2016), Interval oscillation criteria for second-order forced impulsive delay differential equations with damping term, SpringerPlus, 5, 1-16.
  22. [22]  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.
  23. [23]  Zhou, X. and Wang, W-S. (2016), Interval oscillation criteria for nonlinear impulsive differential equations with variable delay, Electronic Journal of Qualitative Theory of Differential Equations, 1, No. 101, 1-18.
  24. [24]  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.
  25. [25]  Sun, Y.G. andWong, J.S.W. (2007), Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities, Journal of Mathematical Analysis and Applications, 334, 549-560.
  26. [26]  Agarwal, R.P., Anderson, D.R., and Zafer, A. (2010), Interval oscillation criteria for second-order forced delay dynamic equations with mixed nonlinearities, Computers & Mathematics with Applications, 59, 997-993.
  27. [27]  Beckenbach, E.F. and Bellman, R. (1961), Inequalities, Springer-verlag: Berlin.
  28. [28]  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.
  29. [29]  Philos, Ch. G. (1989), Oscillation theorems for linear differential equations of second order, Archiv der Mathematik, 53, 482-492.
  30. [30]  Kong, Q. (1999), Interval criteria for oscillation of second-order linear ordinary differential equations, Journal of Mathematical Analysis and Applications, 229, 258-270.
  31. [31]  Liu, X. and Xu, Z. (2007), Oscillation of a forced super-linear second order differential equation with impulses, Computers & Mathematics with Applications, 53, 1740-1749.
  32. [32]  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, No. 40, 1-14.

Copyright © 2011-2023 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates