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

We obtain several new comparison results on the distance between zeros and local extrema of solutions for the second order delay differential equation x''(t) + p(t)x(t - tau(t)) = 0, t > swhere tau : R ->[0, +infinity), p : R -> 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) <= 0, t is an element of R.

Automated summary

In this paper, we investigate the distance between zeros and local extrema of solutions for the second order delay differential equation. We obtain new comparison results and calculate upper bounds on the semicycle length to guarantee the boundedness or convergence to zero of oscillatory solutions. We also analyze the classification of solutions in the case of p(t) <= 0, t is an element of R.