Estimation of unknown function of weakly singular iterated integral inequality

A class of nonlinear weakly singular iterated integral inequalities is given. There is a nonconstant term outside the integrals in the integral inequality. The upper bound of the embedded unknown function of the inequalities is estimated explicitly by adopting novel analysis techniques, such as:discrete Jensen inequality, H\"older integral inequality, the techniques of change of variable, and the method of amplification. The derived results can be applied in the study of qualitative properties of solutions of fractional integral equations.