Periodic solutions for one-dimensional nonlinear nonlocal problem with drift including singular nonlinearities

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