2018
Том 70
№ 1

# Comparison theorems and necessary/sufficient conditions for existence of nonoscillatory solutions of forced impulsive delay differential equations

Abstract

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.

English version (Springer): Ukrainian Mathematical Journal 64 (2012), no. 9, pp 1403-1420.

Citation Example: Cheng Sui Sun, Shao Yuan Huang Comparison theorems and necessary/sufficient conditions for existence of nonoscillatory solutions of forced impulsive delay differential equations // Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1233-1248.

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