  We study a general class of fully coupled backward-forward stochastic
differential equations of mean-field type (MF-BFSDE). We derive existence and
uniqueness results for such a system under weak monotonicity assumptions and
without the non-degeneracy condition on the forward equation. This is achieved
by suggesting an implicit approximation scheme that is shown to converge to the
solution of the system of MF-BFSDE. We apply these results to derive an
explicit form of open-loop Nash equilibrium strategies for nonzero sum
mean-field linear-quadratic stochastic differential games with random
coefficients. These strategies are valid for any time horizon of the game.
