The solvability of forward-backward stochastic differential equations (FBSD
Es for short) has been studied extensively in recent years. To guarantee th
e existence and uniqueness of adapted solutions, many different conditions,
some quite restrictive, have been imposed. In this paper we propose a new
notion: the approximate solvability of FBSDEs, based on the method of optim
al control introduced in our primary work [15]. The approximate solvability
of a class of FBSDEs is shown under mild conditions; and a general scheme
for constructing approximate adapted solutions is proposed.