Oscillation of second order linear differential equations with impulses. (English) Zbl 1130.34314

We use the associated Riccati techniques and the equivalence transformation to discuss the oscillation and the nonoscillation of the second order linear ordinary differential equation with impulses of the form
\[ \begin{cases} (a(t)x'(t))'+p(t)x(t)=0, & t\geq t_0,\;t\neq t_k,\\ x(t^+_k)=b_kx(t_k),\quad x'(t^+_k)=c_kx'(t_k), & k=1,2,\dots.\end{cases} \]
Some examples are also given which show that the oscillation of impulsive differential equations can be caused by impulsive perturbations, though the corresponding classical equation admits a nonoscillatory solution.


34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34A37 Ordinary differential equations with impulses
34A30 Linear ordinary differential equations and systems
Full Text: DOI


[1] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002
[2] Buther, G.J., Integral averages and the oscillation of second order ordinary differential equations, SIAM J. math. anal., 11, 190-200, (1980) · Zbl 0424.34033
[3] Hille, E., Non-oscillation theorems, Trans. amer. math. soc., 64, 234-252, (1948) · Zbl 0031.35402
[4] Moore, R.A., The behavior of solutions of a linear differential equation of second order, Pacific J. math., 5, 125-145, (1955) · Zbl 0064.08401
[5] Leighton, W., On self-adjoint differential equations of second order, J. London math. soc., 27, 37-43, (1952) · Zbl 0048.06503
[6] Swanson, G.A., Comparison and oscillation theory of linear differential equations, (1968), Academic Press New York, London · Zbl 0191.09904
[7] Chen, Y.-S.; Feng, W.-Z., Oscillations of second order nonlinear ODE with impulses, J. math. anal. appl., 210, 150-169, (1997) · Zbl 0877.34014
[8] Cooke, C.H.; Kroll, J., The existence of periodic solutions to certain impulsive differential equations, Comput. math. appl., 44, 667-676, (2002) · Zbl 1054.34014
[9] He, Z.; Ge, W., Oscillations of second-order nonlinear impulsive ordinary differential equations, J. comput. appl. math., 158, 397-406, (2003) · Zbl 1042.34063
[10] Luo, J., Second-order quasilinear oscillation with impulsives, Comput. math. appl., 46, 279-291, (2003) · Zbl 1063.34004
[11] Li, H.-J.; Yeh, C.-C., Oscillation and nonoscillation criteria for second order linear differential equations, Math. nachr., 194, 171-184, (1998) · Zbl 0912.34032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.