Asymptotic Behavior of Mild Solutions to Stochastic Neutral Functional Differential Equations With Delay and Fractional Brownian Motion

ABSTRACT This paper investigates the existence, uniqueness, continuous dependence, and asymptotic behavior of mild solutions for the nonautonomous neutral stochastic functional differential equations involving fractional Brownian motion and state‐dependent delay. First, by employing estimates on nonlinear terms and operator theory, the existence, uniqueness, and continuous dependence of solutions for the considered system were established. Subsequently, using the Arzelà–Ascoli theorem, the Schauder fixed‐point theorem, and operator‐theoretic methods, we analyze the global existence and asymptotic behavior of mild solutions. Moreover, we study the existence of a global forward‐attracting set in the mean‐square sense. Finally, an illustrative example is provided to demonstrate the applicability of the theoretical results.