On the convergence of a perturbed one dimensional Mann's process

We consider the perturbed Mann’s iterative process \begin{equation} x_{n+1}=(1-θ_n)x_n+θ_n f(x_n)+r_n, \end{equation} where $f:[0,1]\to [0,1]$ is a continuous function, $\{{\theta }_n\}\in [0,1]$ is a given sequence, and $\{r_n\}$ is the error term. We establish that if the sequence $\{{\theta }_n\}$ converges relatively slowly to $0$ and the error term $r_n$ becomes enough small at infinity, any sequences $\{x_n\}\in [0,1]$ satisfying the process converges to a fixed point of the function $f$. We also study the asymptotic behavior of the trajectories $x(t)$ as $t\to \mathrm{\infty }$ of a continuous version of the the considered. We investigate the similarities between the asymptotic behaviours of the sequences generated by the considered discrete process and the trajectories $x(t)$ of its corresponding continuous version.