Existence, uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay

In this paper, we obtain the existence and uniqueness of solutions to neutral 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)|‖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. Furthermore, we give the continuous dependence of solutions on the initial value by means of the Corollary of Bihari inequality.