Periodic boundary value problems and periodic solutions of second order FDE with upper and lower solutions (Q874800)

Consider the periodic boundary value problem \[ \begin{gathered} y''(t)= f(t, y(t), g(w(t))),\quad t\in (0,T),\\ y(0)= y(T),\;y'(0)= y'(T),\end{gathered}\tag{\(*\)} \] where \([0, T]\subset [a,b]\), \(T> 0\), \(f\in C([0, T]\times \mathbb{R}^2,\mathbb{R})\), \(w\in C([0, T], [a,b])\), and the problem of existence of a \(T\)-periodic solution of the functional differential equation (FDE) \[ y''(t)= f(t, y(t), y(t- \tau(t))),\quad t\in\mathbb{R},\tag{\(**\)} \] where \(f\) and \(\tau\) are continuous functions which are \(T\)-periodic in \(t\). The authors apply the method of lower and upper solutions in reversed order to establish the existence of at least one solution of \((*)\) and \((**)\). For this purpose they prove an anti-maximum principle which guarantees the construction of monotone sequences converging uniformly to extremal solutions of \((*)\) and \((**)\).