Comparison theorems and necessary/sufficient conditions for existence of nonoscillatory solutions of forced impulsive delay differential equations
In 1997, A. H. Nasr provided necessary and sufficient conditions for the oscillation of the equation $$x''(t) + p(t) |x(g(t))|^{\eta} \text{sgn} (x(g(t))) = e(t),$$ where $\eta > 0$, $p$, and $g$ are continuous functions on $[0, \infty)$ such that $p(t) \geq 0,\;\; g(t) \leq...
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| Date: | 2012 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2012
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/2654 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | In 1997, A. H. Nasr provided necessary and sufficient conditions for the oscillation of the equation
$$x''(t) + p(t) |x(g(t))|^{\eta} \text{sgn} (x(g(t))) = e(t),$$
where $\eta > 0$, $p$, and $g$ are continuous functions on $[0, \infty)$ such that $p(t) \geq 0,\;\; g(t) \leq t,\;\; g'(t) \geq \alpha > 0$, and $\lim_{t \rightarrow \infty} g(t) = \infty$
It is important to note that the condition $g'(t) \geq \alpha > 0$ is required.
In this paper, we remove this restriction under the superlinear assumption $\eta > 0$.
Infact, we can do even better by considering impulsive differential equations with delay and obtain necessary and sufficient conditions
for the existence of nonoscillatory solutions and also a comparison theorem that enables us to apply
known oscillation results for impulsive equations without forcing terms to yield oscillation criteria for our equations. |
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