Comparison theorems and necessary/sufficient conditions for the existence of nonoscillatory solutions of forced impulsive differential equations with delay - Shao Yuan Huang; Sui Sun Cheng (Український математичний журнал, 2012)
In 1997, A. H. Nasr provided necessary and sufficient conditions for the oscillation of the equation where η > 0, p, and g are continuous functions on [0,∞) such that p(t) ≥ 0, g(t) ≤ t, g′(t) ≥ α > 0, and lim t→∞ g(t) =∞. It is important to note that the condition g′(t) ≥ α > 0 is required. In the paper, we remove this restriction under the superlinear assumption η > 1. In fact, 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 get oscillation criteria for the analyzed equations.