  In this paper, we prove existence results of a one-dimensional periodic
solution to equations with the fractional Laplacian of order $s\in(1/2,1)$,
singular nonlinearity, and gradient term under various situations, including
nonlocal contra-part of classical Lienard vector equations, as well other
nonlocal versions of classical results know only in the context of second-order
ODE. Our proofs are based on degree theory and Perron's method, so before that
we need to establish a variety of priori estimates under different assumptions
on the nonlinearities appearing in the equations. Besides, we obtain also
multiplicity results in a regime where a priori bounds are lost and bifurcation
from infinity occurs.
