On the convergence of a perturbed one dimensional Mann's process
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909588798832640 |
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| author | May, Ramzi |
| author_facet | May, Ramzi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_16154 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the convergence of a perturbed one dimensional Mann's process May, Ramzi General Mathematics 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. |
| title | On the convergence of a perturbed one dimensional Mann's process |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2504.16154 |