Moment estimate and existence for solutions of stochastic functional differential equations

In this paper, we give the existence–uniqueness theorems and the moment estimates of solutions for a large class of SFDEs. These estimates improve and extend some related results including exponential stability, decay stability and asymptotic behavior. Their corollaries improve and extend the classical Halanay inequality and some of its generalizations. Moreover, the stochastic version of the Wintner theorem in continuous function space is established by the compare principle, which improves and extends the main results of Xu et al. (2008, 2013). When the methods presented are applied to the SFDEs with impulses and SFDEs in Hilbert spaces, we extend the related results of Govindan and Ahmed (2013), Liu et al. (2007, 2010), Vinodkumar (2010) and Xu et al. (2012). Two examples are provided to illustrate the effectiveness of our results.