  We apply the averaging method to a coupled system consisting of two evolution
equations which has a slow component driven by fractional Brownian motion (FBM)
with the Hurst parameter $H_1> \frac12$ and a fast component driven by additive
FBM with the Hurst parameter $ H_2\in(1-H_1,1)$. The main purpose is to show
that the slow component of such a couple system can be described by a
stochastic evolution equation with averaged coefficients. Our first result
provides a pathwise mild solution for the system of mixed stochastic evolution
equations. Our main result deals with an averaging procedure which proves that
the slow component converges almost surely to the solution of the corresponding
averaged equation using the approach of time discretization. To do this we
generate a stationary solution by a exponentially attracting random fixed point
of the random dynamical system generated by the fast component.
