In this paper, we study linear and nonlinear fractional eigenvalue problems involving the Atangana-Baleanu fractional derivative of the order 1<δ<2. We first estimate the fractional derivative of a function at its extreme points and apply it to obtain a maximum principle for the linear fractional boundary value problem. We then estimate the eigenvalues of the nonlinear eigenvalue problem and obtain necessary conditions to guarantee the existence of eigenfunctions. We also obtain a uniqueness result and a norm estimate of solutions of the linear problem. The obtained maximum principle and results are based on a condition that connects the boundary conditions, the order of the fractional derivative, and the Mittag-Leffler kernel. This condition is different from the ones obtained in previous results with different types of fractional derivatives.