Comparison theorems and necessary/sufficient conditions for existence of nonoscillatory solutions of forced impulsive delay differential equations - Cheng, Sui Sun; Shao, Yuan Huang (Ukrains’kyi Matematychnyi Zhurnal, 2012)
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.