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((-infinity. 0 vertical bar; R-d) which denotes the family of bounded continuous R-d-value functions phi defined on (-infinity. 0 vertical bar with norm parallel to phi parallel to = sup(-infinity < 0 <= 0) vertical bar phi (0)vertical bar 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. (c) 2007 Elsevier B.V. All rights reserved.