A Multiple Iterated Integral Inequality and Applications

We establish new multiple iterated Volterra-Fredholm type integral inequalities, where the composite function w(u(s)) of the unknown function u with nonlinear function w in integral functions in [Ma, QH, Pečarić, J: Estimates on solutions of some new nonlinear retarded Volterra-Fredholm type integral inequalities. Nonlinear Anal. 69 (2008) 393–407] is changed into the composite functions w1(u(s)),w2(u(s)),…, wn (u(s)) of the unknown function u with different nonlinear functions w1,w2,…,wn, respectively. By adopting novel analysis techniques, the upper bounds of the embedded unknown functions are estimated explicitly. The derived results can be applied in the study of solutions of ordinary differential equations and integral equations