In this paper, by introducing a new singular fractional difference comparison theorem, the existence of maximal and minimal quasi-solutions are proved for the singular fractional difference system with maxima combined with the method of upper and lower solutions and the monotone iterative technique. Finally, we give an example to show the validity of the established results.

In this paper, we give a new method to show a monotonicity result for a function $f$ satisfying $(_{a}\nabla_{h}^{\nu}f)(t)\leq0$ ($(_{a}\nabla_{h,\ast}^{\nu}f)(t)\leq0$) with $\nu\in(0,1]$, which has never been solved in other papers. In addition, we give an example to illustrate one of our main results.