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 even2r-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.