Estimation of unknown function of a class of integral inequalities and applications in fractional integral equations

It is well known that differential equations, integral equations and integral-differential equations have gained considerable importance and attention due to their applications in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physical, mechanics, chemistry, aerodynamics, and the electrodynamics of complex mediums, etc. Gronwall-Bellman inequality is an important tool in the study of existence, uniqueness, boundedness, and other qualitative properties of solutions of differential equations and integral equation. In this paper, we discuss we establish a class of retarded iterated integral inequalities, which includes a nonconstant term outside the integrals. By integral inequality technique, the upper bound of the embedded unknown function is estimated explicitly. The derived result can be applied in the study of solutions of fractional integral equations.