Non-Trivial Periodic Solutions for a Class of Second Order Differential Equations with Large Delay

We provide a result on the existence of a positive periodic solution for the following class of delay equations

$$ \theta ''(t)-\theta (t)+f(\theta (t-r))=0. $$

In particular, we find an infinite family of disjoint intervals having the following property: if the delay is within one of these intervals, then the equation admits a non-trivial and even \(2r\)-periodic solution. Furthermore, the length of these intervals is constant and depends on the size of the term \(|f'(\eta )|\), where \(\eta \) is the unique positive equilibrium point of the equation. Consequently, we can find periodic solutions for arbitrarily large delays.