On the convergence of the continuous version of the Moudafi's viscosity approximation method

We study the asymptotic behavior of trajectories of the continuous dynamical system (CDS) associated to the the discrete viscosity approximation method for fixed point problem of nonexpansive mapping (DDS) which was introduced by Moudafi in 2000 [A. Moudafi, Viscosity approximation methods for fixed points problems, J. Math. Anal. Appl. 241 (2000), 46-55]. We establish that the trajectories $x(t)$ of the system (CDS) and the sequences $(x_n)$ generated by the the discrete process (DDS) have a very similar asymptotic behaviors.