A new result on averaging principle for Caputo-type fractional delay stochastic differential equations with Brownian motion

In this paper, we mainly explore the averaging principle of Caputo-type fractional delay stochastic differential equations with Brownian motion. Firstly, the solutions of this considered system are derived with the aid of the Picard iteration technique along with the Laplace transformation and its inverse. Secondly, we obtain the unique result by using the contradiction method. In addition, the averaging principle is discussed by means of the Burkholder-Davis-Gundy inequality, Jensen inequality, Holder inequality and Gronwall-Bellman inequality under some hypotheses. Finally, an example with numerical simulations is carried out to prove the relevant theories.