Some new gronwall-bellmann type discrete fractional inequalities arising in the theory of discrete fractional calculus

In this paper, we present some new Gronwall-Bellmann type discrete fractional sum inequalities, and based on them present some Volterra-Fredholm type discrete inequalities. These inequalities are of new forms compared with the existing results in the literature, and can be used in the research of boundedness and continuous dependence on the initial value for solutions of fractional difference equations. As for applications, we apply the presented results to research the initial value problem of a certain fractional difference equation.