## 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)].

### MSC:

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

### Keywords:

oscillations; nonlinear equations with impulses

Zbl 0877.34014
Full Text:

### References:

 [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
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