Existence and asymptotic properties for the solutions to nonlinear SFDEs driven by G-Brownian motion with infinite delay

Abstract: The aim of this paper is to present the analysis for the solutions of nonlinear stochastic functional differential equation driven by G-Brownian motion with infinite delay (G-SFDEwID). Under some useful assumptions, we have proved that the G-SFDEwID admits a unique local solution. The mentioned theory has been further generalized to show that G-SFDEwID admits a unique strong global solution. The asymptotic properties, mean square boundedness and convergence of solutions with different initial data have been derived. We have assessed that the solution map $X_t$ is mean square bounded and two solution maps from different initial data are convergent. In addition, the exponential estimate for the solution has been studied.