Oscillations of second-order nonlinear ordinary differential equations with impulses. (English) Zbl 0944.34026

Here, the second-order nonlinear differential equation with impulses of the form \[ \begin{aligned} &[r(t)x'(t)]'+f(t,x(t))=0,\quad t\geq t_0,\;t\neq t_k,\;k=1,2,\dots ,\\ &x(t_k^{+})=g_k(x(t_k)),\;x'(t_k^{+})=h_k(x'(t_k)),\;\;k=1,2,\dots ,\\ &x(t_0^{+})=x_0,\;\;x'(t_0^{+})=x'_0, \end{aligned} \] with \(0\leq t_0<t_1<\dots <t_k<\dots \), and \(\lim_{k\to\infty}t_k=+\infty\) is studied. The authors derive sufficient conditions for every solution to (1) to be oscillatory. Results of the paper generalize and improve those of Y. S. Chen and W. Z. Feng [J. Math. Anal. Appl. 120, No. 1, 150-169 (1997; Zbl 0877.34014)].


34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34A37 Ordinary differential equations with impulses


Zbl 0877.34014
Full Text: DOI Link


[1] Chen, Y.S.; Feng, W.Z., Oscillations of second order nonlinear ODE with impulses, J. math. anal. appl., 120, 150-169, (1997) · Zbl 0877.34014
[2] Butler, G.J., Integral averages and the oscillation of second order ordinary differential equation, SIAM J. math. anal., 11, 190-200, (1980) · Zbl 0424.34033
[3] Lakshmikantham, V.; Bainov, D.D.; Simienov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore
[4] Travis, C.C., A note on second order nonlinear oscillations, Math. jap., 18, 261-264, (1973) · Zbl 0297.34032
[5] Wong, J.S.W., On second order nonlinear oscillations, Funkcial. ekavac., 11, 207-234, (1968) · Zbl 0157.14802
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.