  In this paper, we study the backward stochastic differential equations driven
by G-Brownian motion with double mean reflections, which means that the
constraints are made on the law of the solution. Making full use of the
backward Skorokhod problem with two nonlinear reflecting boundaries and the
fixed-point theory, the existence and uniqueness of solutions are established.
We also consider the case where the coefficients satisfy a non-Lipschitz
condition using the Picard iteration argument only for the Y component.
Moreover, some basic properties including a new version of comparison theorem
and connection with a deterministic optimization problem are also obtained.
