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

Abstract: We consider the perturbed Mann's iterative process \begin{equation} x_{n+1}=(1-\theta_n)x_n+\theta_n f(x_n)+r_n, \end{equation} where $f:[0,1]\rightarrow[0,1]$ 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 $t\rightarrow\infty$ 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.