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

Two fixed point theorems in complete random normed modules and their applications to backward stochastic equations