Modified Gronwall–Bellman-Type Retarded Integral Inequalities and Applications

We establish new nonlinear retarded integral inequalities of Gronwall–Bihari type involving time-dependent kernels, variable delays, nonconstant free terms, and iterated Volterra-type contributions. The first class of results provides explicit estimates for inequalities with kernels depending on both the observation time and the integration variable. A second class treats composite nonlinearities through successive comparison transformations and extends known retarded inequalities to the presence of an additional iterated kernel. The obtained estimates are then applied to nonlinear Volterra integral equations and neutral functional differential equations. For the Volterra problem, sufficient conditions for uniqueness, global existence, and boundedness are derived under Osgood-type assumptions. For neutral functional differential equations, the retarded estimates are combined with Driver’s continuation framework to obtain a global existence criterion. A delayed Liénard system is included as an illustrative application. These results extend several classical Gronwall–Bihari and retarded integral inequalities and provide a flexible tool for the qualitative analysis of nonlinear integral and functional differential equations.