Applying Khasminskii-type criteria to stochastic Hilfer equations with time-lag under averaging principle

Abstract We examine a specific class of Hilfer fractional stochastic differential equations characterized by temporal lags. By relaxing the constraints to local Lipschitz conditions and utilizing Khasminskii-type criteria, we establish the fundamental existence and uniqueness of the solution set. A significant portion of this study is dedicated to demonstrating the Averaging Principle (AP) under generalized averaging conditions. Through the application of key integral inequalities (Burkholder-Davis-Gundy(BDG), Jensen, and H $$\ddot{o}$$ lder), we show that our results encompass a broader range of stochastic systems compared to those restricted by monotone conditions. The theoretical framework is shown to be more versatile than existing models, with the accuracy of the limit behavior confirmed via numerical simulations as $$\epsilon$$ vanishes.