  In this paper we are concerned with the construction of periodic solutions of
the nonlocal problem $(-\Delta)^s u= f(u)$ in $\mathbb{R}$, where $(-\Delta)^s$
stands for the $s$-Laplacian, $s\in (0,1)$. We introduce a suitable framework
which allows to reduce the search for such solutions to the resolution of a
boundary value problem in a suitable Hilbert space, thereby making it possible
to reach for the usual tools of nonlinear analysis, like bifurcation theory or
variational methods. We obtain some existence theorems which are lately
enlightened with the analysis of some examples.
