  This paper is concerned with a class of mean-field type coupled
forward-backward stochastic differential equations (MF-FBSDEs, for short), in
which the coupling appears in integral terms, terminal terms, and initial
terms. Inspired by various mean-field type linear-quadratic (MF-LQ,for short)
optimal control problems, we proposed a type of randomized
domination-monotonicity conditions, under which and the usual Lipschitz
condition, we obtain a well-posedness result on MF-FBSDEs in the sense of
square integrability including the unique solvability, an estimate of the
solution, and the related continuous dependence property of the solution on the
coefficients.The result of MF-FBSDEs in turn extends MF-LQ problems in the
literature to a general situation where the initial states or the terminal
states are also controlled at the same time, and gives explicit expressions of
the related unique optimal controls.
