Some new generalized Gronwall–Bellman type discrete fractional inequalities

•Some new Gronwall–Bellman type discrete fractional inequalities are established.•Explicit bounds for unknown functions in fractional inequalities are derived.•Properties of solutions of one certain fractional difference equation are researched.

In this paper, we present some new generalized Gronwall–Bellman type discrete fractional inequalities including fractional sum inequalities and fractional difference inequalities. These inequalities are of new forms compared with the existing results so far in the literature, and can be used to provide explicit bounds for unknown functions concerned. For illustrating the effectiveness of the presented results, we apply them to research qualitative and quantitative properties for the solutions of an IVP with respect to a fractional difference equation.