  The existence, uniqueness and stability of periodic traveling waves for the
fractional Benjamin-Bona-Mahony equation is considered. In our approach, we
give sufficient conditions to prove a uniqueness result for the single-lobe
solution obtained by a constrained minimization problem. The spectral stability
is then shown by determining that the associated linearized operator around the
wave restricted to the orthogonal of the tangent space related to the momentum
and mass at the periodic wave has no negative eigenvalues. We propose the
Petviashvili's method to investigate the spectral stability of the periodic
waves for the fractional Benjamin-Bona-Mahony equation, numerically. Some
remarks concerning the orbital stability of periodic traveling waves are also
presented.
