A note on existence of (anti-)periodic and heteroclinic solutions for a class of second-order odes

We discuss the existence of anti-periodic solutions to the following second-order differential equation q̈=u(t,q) by using fixed point theory together with the Green’s function for the anti-periodic boundary value problem in the first part (Section 2) of the paper. Then in the next part (Section 3), we construct a kind of heteroclinic solution to some special cases of the equation above. Our method is variational in nature and is inspired by the ideas of Rabinowitz and Stredulinsky in [P. Rabinowitz, E. Stredulinsky, On some results of Moser and of Bangert, AIHP Anal. Nonlin. 21 (2004) 673–688].