  In this paper, a maximum principle for the one-dimensional sub-diffusion
equation with Atangana-Baleanu fractional derivative is formulated and proved.
The proof of the maximum principle is based on an extremum principle for the
Atangana-Baleanu fractional derivative that is given in the paper, too. The
maximum principle is then applied to show that the initial-boundary-value
problem for the linear and nonlinear time-fractional diffusion equations
possesses at most one classical solution and this solution continuously depends
on the initial and boundary conditions.
