Explicit bounds derived by some new inequalities and applications in fractional integral equations

In this paper, we present some new Gronwall-type inequalities. Explicit bounds for the unknown functions concerned are derived based on these inequalities and the properties of the modified Riemann-Liouville fractional derivative. The inequalities established are of new forms compared with the existing results so far in the literature. For illustrating the validity of the inequalities established, we apply them to research the boundedness, quantitative property, and continuous dependence on the initial value for the solution to a certain fractional integral equation.