A multiple iterated integral inequality and applications (Q2336343)

Summary: 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 [\textit{Q.-H. Ma} and \textit{J. Pečarić}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 2, 393--407 (2008; Zbl 1151.26330)] is changed into the composite functions \(w_1(u(s)), w_2(u(s)), \ldots, w_n(u(s))\) of the unknown function \(u\) with different nonlinear functions \(w_1, w_2, \ldots, w_n\), 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.