Semicycles and correlated asymptotics of oscillatory solutions to second-order delay differential equations

TL;DR

This paper develops new comparison results for second-order delay differential equations, providing bounds on oscillatory solutions' zeros and extrema, and classifies solutions based on coefficient sign and bounds.

Contribution

It introduces novel bounds on the distance between zeros and extrema, extending oscillation theory for delay differential equations with sign-changing coefficients.

Findings

Upper bounds on semicycle length for bounded solutions

Conditions for solutions to tend to zero

Classification of oscillatory solutions with non-positive coefficients

Abstract

We obtain several new comparison results on the distance between zeros and local extrema of solutions for the second order delay differential equation \begin{equation*} x^{\prime \prime }(t)+p(t)x(t-\tau (t))=0,~~t\geq s\text{ }\ \end{equation*} where $\tau :\mathbb{R}\rightarrow \lbrack 0,+\infty )$, $p:\mathbb{R}% \rightarrow \mathbb{R}$ are Lebesgue measurable and uniformly essentially bounded, including the case of a sign-changing coefficient. We are thus able to calculate upper bounds on the semicycle length, which guarantee that an oscillatory solution is bounded or even tends to zero. Using the estimates of the distance between zeros and extrema, we investigate the classification of solutions in the case $p(t)\leq 0,t\in \mathbb{R}.$