Generalized Bihari-Type Inequality via <img src=image/13443173_01.gif>-Hilfer Fractional Operators and Applications to Nonlocal Cauchy-Type Systems

This paper develops a generalized Bihari-type integral inequality within the framework of <img src=image/13443173_02.gif>-Hilfer fractional operators and demonstrates its applicability to the analysis of nonlocal Cauchy-type fractional differential systems. The proposed inequality extends the classical Bihari inequality to fractional integrals defined with respect to an increasing function <img src=image/13443173_02.gif>, thereby encompassing the Riemann–Liouville and Caputo cases as special instances. This generalization offers a unified perspective on several well-known inequalities and provides a powerful tool for investigating nonlinear fractional systems with memory and hereditary properties. Building on this theoretical foundation, we study a class of nonlinear coupled nonlocal Cauchy-type systems governed by <img src=image/13443173_02.gif>-Hilfer derivatives. Using fixed point theory, we establish rigorous existence and uniqueness results and further demonstrate that the solutions exhibit Mittag-Leffler–Ulam–Hyers (ML-UH) stability. Specifically, we prove that small perturbations in system parameters or initial data lead to deviations that remain uniformly bounded by explicit Mittag-Leffler functions, ensuring solution robustness. Numerical simulations are provided to illustrate the theoretical findings, clearly visualizing solution trajectories and their stability under perturbations, thereby confirming the accuracy of the stability estimates. The main contribution of this work lies in extending the Bihari inequality to the <img src=image/13443173_02.gif>-Hilfer setting and applying it to stability analysis of nonlocal fractional problems—a topic that, to the best of our knowledge, has not been fully addressed in the literature. This study introduces a novel methodological tool for fractional calculus and highlights its significance in analyzing qualitative properties of fractional-order systems. The results also offer potential applications in engineering, physics, and control theory, where fractional models are increasingly used to describe anomalous diffusion, viscoelasticity, and long-memory processes. The limitations of this work primarily concern the Lipschitz assumptions imposed on the nonlinear terms and the restriction to deterministic systems. Future research may extend the analysis to more general nonlinearities of Osgood type, impulsive or stochastic fractional systems, or to tempered or distributed-order fractional operators. Overall, this research strengthens the theoretical foundation for studying nonlocal fractional systems and provides practical criteria for ensuring stability in complex dynamical models.