Distribution dependent stochastic differential equations driven by fractional Brownian motion and standard Brownian motion

In this paper, we study a class of distribution dependent stochastic differential equations driven simultaneously by fractional Brownian motion with Hurst $0< H<1/2$ and standard Brownian motion. We first establish the existence and uniqueness theorem for solutions of the considered equations under some non-Lipschitz conditions by Carathéodory approximation technique. Subsequently, we show that the solutions of distribution dependent stochastic differential equations can be approximated by the solutions of the associated averaged distribution dependent stochastic differential equations in the sense of mean square convergence under certain averaging conditions.