Oscillation of Second-Order Delay Equations with Positive Feedback

Abstract

We study oscillation of the second-order delay equation \[ x''(t)=x(τ(t)), \] with a nondecreasing delayed argument $\tau (t)\le t$ satisfying $\tau (t)\stackrel{t\to \infty }{\to }\infty$. Set \begin{equation*} a=\liminf_{t\to\infty}(t-τ(t)),\qquad b=\limsup_{t\to\infty}(t-τ(t)). \end{equation*} We describe a continuous strictly decreasing function \begin{equation} β:[0,2/e]\longrightarrow[2/e,\sqrt2],\qquad β(0)=\sqrt2,\qquadβ(2/e)=2/e, \notag \end{equation} such that, if the equation has an eventually positive decreasing solution, then \begin{equation} a\leq2/e,\qquad b\leβ(a). \notag \end{equation} The bound $b\le \beta (a)$ is sharp and interpolates between the classical $\mathrm{limsup}$ threshold and the $\mathrm{liminf}$ autonomous threshold. A detailed spectral analysis gives a simple interpretation of the result and links it directly to the oscillation theory of first-order delay equations.

arXiv comment45 pages, 1 figure