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

This paper investigates the existence and uniqueness theorem of solutions to neutral stochastic differential equations with infinite delay ( short for INSFDEs) at a space BC((-infinity, 0]; R-d). Under the uniform Lipschitz condition, linear growth condition is weaken to obtain the moment estimate of the solution for INSFDEs. Furthermore, the existence, uniqueness theorem of the solution for INSFDEs is derived, and the estimate for the error between approximate solution and exact solution is given. On the other hand, under the linear growth condition, the uniform Lipschitz condition is replaced by the local Lipschitz condition, the existence, uniqueness theorem is also valid for INSFDEs on [t(0), T]. Moreover, the existence, uniqueness theorem still holds on interval [t(0), infinity), where t(0) is an element of R is an arbitrary real number. (C) 2009 Elsevier Inc. All rights reserved.