  This paper first proves two fixed point theorems in complete random normed
modules, which are respectively the random generalizations of the classical
Banach's contraction mapping principle and Browder--Kirk's fixed point theorem.
As applications, the first is used to give the existence and uniqueness of
solutions to various kinds of backward stochastic equations under
$L^0$--Lipschitz assumptions and the second is used to establish the existence
of solutions to backward stochastic equations of nonexpansive type.
