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

• Дата: 2012
• Автори: Shao Yuan Huang, Sui Sun Cheng
• Формат: Стаття
• Мова: English
• Опубліковано: Український математичний журнал 2012
• Назва видання: Український математичний журнал
• Теми: Статті
• Онлайн доступ: https://nasplib.isofts.kiev.ua/handle/123456789/164517
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Цитувати: 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. — Т. 64, № 9. — С. 1233-1248. — Бібліогр.: 8 назв. — англ.

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.