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

In 1997, A. H. Nasr provided necessary and sufficient conditions for the oscillation of the equation\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {x}^{\prime\prime}(t)+p(t){{\left| {x\left( {g(t)} \right)} \right|}^{\eta }}\operatorname{sgn}\left( {x\left( {g(t)} \right)} \right)=e(t), $$\end{document}where η > 0, p, and g are continuous functions on [0,∞) such that p(t) ≥ 0, g(t) ≤ t, g′(t) ≥ α > 0, and limt→∞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.