  In this paper, we consider a system of forward-backward stochastic
differential equations (FBSDEs) with monotone functionals. We show the
existence and uniqueness of such a system by the method of continuation
similarly to Peng and Wu (1999) for classical FBSDEs and obtain estimates under
conditional probability. As applications, we prove the well-posedness result
for a mean field FBSDE with conditional law and show the existence of a
decoupling function. In addition, we show that mean field games with common
noise are uniquely solvable under a linear controlled process with convex and
weak-monotone cost functions and prove that the optimal control is in a
feedback form depending only on the current state and conditional law.
