Averaging Principle for Stochastic Delay Differential Equations with Hadamard Fractional Integral

In this work, we investigate the existence and uniqueness of solutions for a class of stochastic differential equations (SDEs) incorporating time delay and Hadamard-type fractional integrals. The Hadamard fractional integral, which features a logarithmic kernel, introduces distinct analytical challenges compared to traditional fractional operators, leading to novel theoretical insights. Furthermore, we extend our analysis to derive an averaging principle for the considered equations, which provides a rigorous connection between the delayed fractional system and its averaged counterpart by employing the well-known Hölder and Gronwall inequalities. To demonstrate the applicability of our theoretical findings, we present an example involving a nonlinear stochastic system with delayed feedback and Hadamard fractional integration.