Generalized Integral Inequalities for Fractional Delay Systems: A Unified Framework Based on Mittag-Leffler Functions

This paper addresses the fundamental problem of integral inequality theory for coupled fractional differential equations and delay systems. We establish a unified theoretical framework based on multi-parameter Mittag-Leffler functions, providing the first precise expressions and optimal bound estimates for generalized Gronwall-Bellman-type inequalities that simultaneously incorporate Caputo fractional derivatives and distributed delay terms. The core innovations are: (1) proving the compositional properties of fractional-delay coupled operators and establishing the corresponding convolution kernel theory; (2) utilizing Laplace transforms and monotone operator theory to provide a complete characterization of the existence, uniqueness, and asymptotic behavior of solutions for such systems. Theoretical results show that when the fractional parameter and the delay distribution measure satisfies specific conditions, system solutions satisfy exponential decay estimates. Numerical validation confirms the precision of the theoretical bounds with relative errors less than 3%. This theory provides rigorous mathematical tools for the stability analysis of fractional control systems and memory-type biological models.