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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 τ(t)≤ t satisfying τ(t)oversett→∞→∞. Set a=liminft→∞(t−τ(t)), b=limsupt→∞(t−τ(t)). We describe a continuous strictly decreasing function β:[0,2/e]⟶[2/e,√2], β(0)=√2, β(2/e)=2/e, notag such that, if the equation has an eventually positive decreasing solution, then a≤2/e, b≤β(a). notag The bound b≤β(a) is sharp and interpolates between the classical limsup threshold and the 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.
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BibTeX
@misc{stavroulakis2026oscillation,
title = {{Oscillation of Second-Order Delay Equations with Positive Feedback}},
author = {John Ioannis Stavroulakis},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31732},
eprint = {2609.31732},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31732},
}
RIS
TY - UNPB
TI - Oscillation of Second-Order Delay Equations with Positive Feedback
AU - John Ioannis Stavroulakis
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.31732
UR - https://arxiv.org/abs/2609.31732
AB - We study oscillation of the second-order delay equation x''(t)=x(τ(t)), with a nondecreasing delayed argument τ(t)≤ t satisfying τ(t)oversett→∞→∞. Set a=liminf_t→∞(t−τ(t)), b=limsup_t→∞(t−τ(t)). We describe a continuous strictly decreasing function β:[0,2/e]⟶[2/e,√2], β(0)=√2, β(2/e)=2/e, notag such that, if the equation has an eventually positive decreasing solution, then a≤2/e, b≤β(a). notag The bound b≤β(a) is sharp and interpolates between the classical limsup threshold and the 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.
ER -