Mean square stability with general decay rate of nonlinear neutral stochastic function differential equations in the $G$-framework

Few results seem to be known about the stability with general decay rate of nonlinear neutral stochastic function differential equations driven by $G$-Brownain motion ( $G$-NSFDEs in short). This paper focuses on the $G$-NSFDEs, and the coefficients of these considered $G$-NSFDEs can be allowed to be nonlinear. It is first proved the existence and uniqueness of the global solution of a $G$-NSFDE. It is then obtained the trivial solution of the $G$-NSFDE is mean square stable with general decay rate (including the trivial solution of the $G$-NSFDE is mean square exponentially stable and the trivial solution of the $G$-NSFDE is mean square polynomially stable) by $G$-Lyapunov functions technique. In this paper, auxiliary functions are used to dominate the Lyapunov function and the diffusion operator. Finally, an example is presented to illustrate the obtained theory.