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.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 9, pp. 1233–1248, September, 2012.
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Huang, S.Y., Cheng, S.S. Comparison theorems and necessary/sufficient conditions for the existence of nonoscillatory solutions of forced impulsive differential equations with delay. Ukr Math J 64, 1403–1420 (2013). https://doi.org/10.1007/s11253-013-0724-7
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DOI: https://doi.org/10.1007/s11253-013-0724-7