  Backward stochastic differential equations (BSDEs) belong nowadays to the
most frequently studied equations in stochastic analysis and computational
stochastics. In this paper we prove that Picard iterations of BSDEs with
globally Lipschitz continuous nonlinearities converge exponentially fast to the
solution. Our main result in this paper is to establish a fundamental lemma to
prove the global existence and uniqueness of an adapted solution to a singular
backward stochastic nonlinear Volterra integral equation (for short singular
BSVIE) of order $\alpha \in (\frac{1}{2},1)$ under a weaker condition than
Lipschitz one in a Hilbert space.
