MiS Preprint Repository

We study the existence of periodic and homoclinic solutions for a class of non-autonomous second order advanced-delayed differential equations of the type $$\ddot{u}(t)+f_0(t,u(t))=\sum _{i=1}^N[f_i(t,u(t+{\tau }_i)-u(t))-f_i(t-{\tau }_i,u(t)-u(t-{\tau }_i))].$$ We prove, under some growth conditions on the non-linearities, the existence of non-constant periodic solutions with period any given positive integer. Using very simple arguments, the existence of a non-trivial homoclinic solution is also established. This homoclinic is obtained as the limit of subharmonics. An application to the existence of periodic and homoclinic travelling waves in an infinite lattice of partciles with $N$-nearest-neighbour interaction and on-site potential is given.