Existence of the solutions of backward–forward SDE's with continuous monotone coefficients

We seek an alternative approach to produce the solution of a certain class of BFSDE's without employing the classical time restriction. In the literature there are various results about this problem, none of them implying the other. The previous methods always assume to have globally Lipschitz coefficients. Here, under some particular choices for the coefficients, we show that if one of them satisfies a uniform growth condition and they are accordingly monotone, one can find a solution (not necessarily unique), without even resorting to the Lipschitz property. Finally we provide some examples to show this is a new class of equations.