Remarks on the Periodic Boundary Value Problems for Nonlinear Second   Order Ordinary Differential Equations
  This paper is devoted to study the existence of solutions and the monotone method of second-order periodic boundary value problems when the lower and upper solutions $\alpha$ and $\beta$ violate the boundary conditions $ \alpha(0)=\alpha(2\pi)$ and $\beta(0)=\beta(2\pi)$. We present several comparison results. We show that the method of lower and upper solutions coupled with the monotone iterative technique is valid to obtain constuctive proof of existence of solutions. 
Faouzi Haddouchi and Slimane Benaicha
math.CA