On the convergence of a perturbed one dimensional Mann's process
Ramzi May
TL;DR
This work analyzes the convergence of a perturbed one-dimensional Mann-type process in both discrete and continuous time. Under vanishing step sizes $\{ heta_n\}$ with $\theta_n\to0$, divergent sum $\sum_n\theta_n$, and small perturbations $\{r_n\}$ with $r_n/\theta_n\to0$ and $\sum_n r_n<\infty$, the discrete scheme $x_{n+1}=(1-\theta_n)x_n+\theta_n f(x_n)+r_n$ converges to a fixed point of $f$. A corresponding continuous model $x'(t)+\theta(t)x(t)=\theta(t)f(x(t))+r(t)$ under $\int_0^\infty\theta(t)dt=\infty$, $\int_0^\infty r(t)dt<\infty$, and $r(t)/\theta(t)\to0$ is shown to converge as well. The paper develops omega-limit-based arguments to relate the discrete and continuous trajectories and includes a numerical study of convergence rates under controlled perturbations, plus an annex proving Mann's original convergence. These results establish robustness of fixed-point convergence under small, vanishing errors with practical implications for numerical fixed-point computations.
Abstract
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 $f:[0,1]\rightarrow[0,1]$ is a continuous function, $\{θ_n\}\in [0,1]$ is a given sequence, and $\{r_n\}$ is the error term. We establish that if the sequence $\{θ_n\}$ 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.