Oscillation of second-order sublinear impulsive differential equations. (English) Zbl 1223.34049
Summary: Oscillation criteria obtained by Kusano and Onose (1973) and by Belohorec (1969) are extended to second-order sublinear impulsive differential equations of Emden-Fowler type:
\[ x''(t) + p(t)|x(\tau(t))|^{\alpha - 1}x(\tau(t)) = 0,\quad t\neq\theta_k; \]
\[ \Delta x'(t)|_{t=\theta_k} + q_k|x(\tau(\theta_k))|^{\alpha-1}x(\tau(\theta_{k})) = 0;\quad \Delta x(t)|_{t=\theta_{k}} = 0,\;(0 < \alpha < 1) \]
by considering the cases \(\tau(t) \leq t\) and \(\tau(t) = t\), respectively. Examples are inserted to show how impulsive perturbations greatly affect the oscillation behavior of the solutions.
\[ x''(t) + p(t)|x(\tau(t))|^{\alpha - 1}x(\tau(t)) = 0,\quad t\neq\theta_k; \]
\[ \Delta x'(t)|_{t=\theta_k} + q_k|x(\tau(\theta_k))|^{\alpha-1}x(\tau(\theta_{k})) = 0;\quad \Delta x(t)|_{t=\theta_{k}} = 0,\;(0 < \alpha < 1) \]
by considering the cases \(\tau(t) \leq t\) and \(\tau(t) = t\), respectively. Examples are inserted to show how impulsive perturbations greatly affect the oscillation behavior of the solutions.
MSC:
| 34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |
| 34A37 | Ordinary differential equations with impulses |
References:
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