Malliavin differentiability and regularity of densities in semi-linear stochastic delay equations driven by weighted fractional Brownian motion

In this work, we will show the existence and uniqueness of the solution to the semi linear stochastic differential equations driven by weighted fractional Brownian motion with delay. We also prove smoothness of the density of the solution with respect to Lebesgue's measure on R^d for d ≥ 1.