A Multiple Iterated Integral Inequality and Applications

Abstract

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 $w_1(u(s)),w_2(u(s)),\dots , w_n (u(s))$ of the unknown function $u$ with different nonlinear functions $w_1,w_2,\dots ,w_n$, 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.