In this article, we derive the exact performance expressions of the system performance in terms of outage probability and ergodic capacity for multi-user of a downlink non-orthogonal multiple access (NOMA) systems with a full-duplex decode-and-forward relay R over the Nakagami-m fading channels. For the considered FDR-NOMA system, we evaluate the impacts of realistic parameters of imperfect residual self-interference (RSI) cancellation, successive interference cancellation (SIC), and channel state information (CSI). The results of the Monte-Carlo simulation are provided to validate the accuracy of numerical analysis and show the potential to double the capacity of the FDR-NOMA system compared to the half-duplex NOMA (HDR-NOMA). Furthermore, we investigate the optimal position of the relay R to reach maximum capacity.