Existence and uniqueness results for a class of fractional stochastic neutral differential equations

In this paper, we investigate new results on the existence and uniqueness of mild solutions to stochastic neutral differential equations involving Caputo fractional time derivative operator with Lipschitz coefficients and under some Carathéodory-type conditions on the coefficients through the Picard approximation technique. To do so, we derive a stochastic version of variation of constants formula for Caputo fractional differential systems whose coefficients satisfy standard Lipschitz and non-Lipschitz conditions. The main points are to prove a coincidence between the integral equation and the mild solution by applying Itô’s isometry, martingale representation theorem, and the weighted maximum norm for a class of fractional stochastic neutral differential equations. Finally, examples are provided to support the efficiency of the main theory.