Fixed point technique for a class of backward stochastic differential equations

We establish a new theorem on the existence and uniqueness of the adapted solution to backward stochastic differential equations under some weaker conditions than the Lipschitz one. The extension is based on Athanassov's condition for ordinary differential equations. In order to prove the existence of the solutions we use a fixed point technique based on Schauder's fixed point theorem. Also, we study some regularity properties of the solution for this class of stochastic differential equations.