Abstract
In this paper, we study a nonlinear fractional boundary value problem involving fractional derivative with nonsingular Mittag-Leffler kernels and an anti-periodic boundary conditions. Some results of the existence of solutions of the problem will be given under different assumptions on the function Theta and the Lyapunov-type inequality will be obtained in the case Theta(t,z(t)) = eta(t)z(t). An application of this inequality to an eigenvalue problem is also given.