  This paper is devoted to studying the averaging principle for fast-slow
system of rough differential equations driven by mixed fractional Brownian
rough path. The fast component is driven by Brownian motion, while the slow
component is driven by fractional Brownian motion with Hurst index $H ~(1/3 <
H\leq 1/2)$. Combining the fractional calculus approach to rough path theory
and Khasminskii's classical time discretization method, we prove that the slow
component strongly converges to the solution of the corresponding averaged
equation in the $L^1$-sense. The averaging principle for a fast-slow system in
the framework of rough path theory seems new.
