Oscillation of a first order linear impulsive delay differential equation with continuous and piecewise constant arguments

Abstract

A class of first order linear impulsive delay differential equation with continuous and piecewise constant arguments is studied. Using a connection between impulsive delay differential equations and non-impulsive delay differential equations sufficient conditions for the oscillation of the solutions are obtained.

Keywords

• Oscillation,Delay,Piecewise constant argument,Impulse

References

1. Busenberg, S. and Cooke, K.L. Models of vertically transmitted diseases with sequential- continuous dynamics, In: Nonlinear Phenomena in Mathematical Sciences, V. Lakshmikantham (Ed.), Academic Press, 179-187, 1982.
2. Dai, L. and Singh, M.C. On oscillatory motion of spring mass systems subject to piesewise constant forces, J. Sound Vib. 173, 217-233, 1994.
3. Dai, L. and Singh, M.C. An analytical and numerical method for solving linear and nonlinear vibration problems, Int. J. Solids Struct. 34, 2709-2731, 1997.
4. Cooke, K.L. and Wiener, J. Retarded dierential equations with piecewise constant delays, J. Math. Anal. Appl. 99, 265-297, 1984.
5. Aftabizadeh, A.R. and Wiener, J. Oscillatory properties of first order linear functional differential equations, Applicable Anal. 20, 165-187, 1985.
6. Aftabizadeh, A.R., Wiener, J. and Ming Xu, J. Oscillatory and periodic solutions of delay differential equations with piecewise constant argument, Proc. of American Math. Soc. 99, 673-679, 1987.
7. Shen, J.H. and Stavroulakis, I.P. Oscillatory and nonoscillatory delay equations with piece- wise constant argument, J. Math. Anal. Appl. 248, 385-401, 2000.
8. Wiener, J. Generalized Solutions of Functional Differential Equations, World Scientic, Singapore, 1994.

9. Karakoç, F., Bereketoglu H. and Seyhan, G. Oscillatory and periodic solutions of impulsive differential equations with piecewise constant argument, Acta Appl. Math. 110, 499-510, 2010.
10. Bereketoglu, H., Seyhan G. and Ogun, A. Advanced impulsive differential equations with piecewise constant arguments, Math. Model. Anal. 15, 175-187, 2010.
11. Karakoç, F., Ogun Unal, A. and Bereketoglu, H. Oscillation of nonlinear impulsive differential equations with piecewise constant arguments, E. J. Qualitative Theory of Di. Equ. No. 49, 1-12, 2013.
12. Gyori, I. and Ladas, G. Oscillation Theory of Delay Differential Equations, Oxford University Press, 1991.
13. Akhmet, M. Nonlinear Hybrid Continuous/Discrete-Time Models, Atlantis Press, Springer, 2011.
14. Gopalsamy, K., Gyori I. and Ladas, G. Oscillations of a class of delay equations with continuous and piecewise constant arguments, Funkcialaj Ekvacioj, 32, 395-406, 1989.
15. Agarwal, R.P., Karakoç, F. and Zafer, A. A survey on oscillation of impulsive ordinary differential equations, Advances in Difference Equations, Volume 2010, Article ID 354841, 52 pages, doi:10.1155/2010/354841.
16. Yan J. and Kou, C. Oscillation of solutions of impulsive delay differential equations, J. Math. Anal. Appl. 254, 358-370, 2001.
17. Agarwal, R.P. and Karakoç, F. A survey on oscillation of impulsive delay differential equa- tions, Comput. Math. Appl. 60, 1648-1685, 2010.
18. Chiu, K.S. and Pinto, M. Periodic solutions of differential equations with a general piecewise constant argument and applications, E. J. Qualitative Theory of Di. Equ., No. 46, 1-19, 2010.
19. Chiu, K.S. and Jeng, J.C. Stability of oscillatory solutions of differential equations with general piecewise constant arguments of mixed type, Mathematische Nachrichten 288(10), 1085-1097, 2015.