  In this paper, we introduce a class of backward stochastic equations (BSEs)
that extend classical BSDEs and include many interesting examples of
generalized BSDEs as well as semimartingale backward equations. We show that a
BSE can be translated into a fixed point problem in a space of random vectors.
This makes it possible to employ general fixed point arguments to find
solutions. For instance, Banach's contraction mapping theorem can be used to
derive general existence and uniqueness results for equations with Lipschitz
coefficients, whereas Schauder-type fixed point arguments can be applied to
non-Lipschitz equations. The approach works equally well for multidimensional
as for one-dimensional equations and leads to results in several interesting
cases such as equations with path-dependent coefficients, anticipating
equations, McKean-Vlasov type equations and equations with coefficients of
superlinear growth.
