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))|^η sgn (x(g(t))) = e(t), where η > 0, p, and g are continuous functions on [0, ∞) such that p(t) ≥ 0, g(t) ≤ t, g’(t) ≥ α > 0, and _t → ∞ g(t) = ∞ It is important to note that the condition g’(t) ≥ α > 0 is required. In this paper, we remove this restriction under the superlinear assumption η > 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.