Remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay

In this paper, we obtain some results on the existence and uniqueness of solutions to stochastic functional differential equations with infinite delay at phase space BC((-∞,0];Rd)BC((-∞,0];Rd) which denotes the family of bounded continuous RdRd-value functions ϕϕ defined on (-∞,0](-∞,0] with norm ∥ϕ∥=sup-∞<θ⩽0|ϕ(θ)| under non-Lipschitz condition with Lipschitz condition being considered as a special case and a weakened linear growth condition. The solution is constructed by the successive approximation.