New Findings of Gronwall–Bellman–Bihari Type Integral Inequalities with Applications to Fractional and Composite Nonlinear Systems

This paper is dedicated to the investigation of new generalizations of the classical Gronwall–Bellman–Bihari integral inequalities, which are fundamental tools in the qualitative and quantitative analysis of differential, integral, and integro-differential equations. We establish two primary, novel theorems. The first theorem presents a significant generalization for inequalities involving composite nonlinear functions and iterated integrals. This result provides an explicit bound for an unknown function $u\left(t\right)$ satisfying an inequality of the form $\mathsf{\Phi }\left(u\right(t\left)\right)\le a\left(t\right)+{\int }_{t_0}^t f\left(s\right)\mathsf{\Psi }\left(u\right(s\left)\right)ds+{\int }_{t_0}^t g\left(s\right)\mathsf{\Omega }({\int }_{t_0}^s h(\tau \left)K\right(u\left(\tau \right)\left)d\tau \right)ds$. The proof is achieved by defining a novel auxiliary function and applying a rigorous comparison principle. The second main theorem establishes a new bound for a class of fractional integral inequalities involving the Riemann–Liouville fractional integral operator ${\mathcal{I}}^{\alpha }$ and a non-constant coefficient function $b\left(t\right)$ in the form $u\left(t\right)\le a\left(t\right)+b\left(t\right){\mathcal{I}}^{\alpha }\left[\omega \right(u\left(s\right)\left)\right]$. This result extends several recent findings in the field of fractional calculus. The mathematical derivations are detailed, and the assumptions on the involved functions are made explicit. To illustrate the utility and potency of our main results, we present two applications. The first application demonstrates how our first theorem can be used to establish uniqueness and boundedness for solutions to a complex class of nonlinear integro-differential equations. The second application utilizes our fractional inequality theorem to analyze the qualitative behavior (specifically, the boundedness of solutions) for a generalized class of fractional integral equations. These new inequalities provide a powerful analytical framework for studying complex dynamical systems that were not adequately covered by existing results.