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

Ramzi May – GJM, Volume 11, Issue 1 (2026), 77-86.

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• août 4, 2026
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We study a perturbed version of Mann’s iterative process (DMP), defined by

x_{n+1} = (1 - \theta_n)x_n + \theta_n f(x_n) + r_n

where f: [0,1] \to [0,1] is a continuous function, \{\theta_n\} \subset [0,1] is a given sequence, and \{r_n\} represents an error term. We prove that if the sequence \{\theta_n\} converges sufficiently slowly to zero and the error term r_n is suitably small at infinity, then any sequence \{x_n\} \subset [0,1] generated by this process converges to a fixed point of f. In addition, we investigate the asymptotic behavior of the trajectories x(t) as t \to \infty for a continuous-time version of the process (DMP). We emphasize the parallels between the discrete and the continuous dynamics. Furthermore, through numerical experiments, we analyze the influence of the sequence \{\theta_n\} and the error terms on the stability and the convergence rate of the the discrete and the continuous processes. Notably, we observe that the (DMP) algorithm, when affected by stochastic and relatively large error terms, can outperform the bisection method in efficiently identifying the fixed point set of the function f.

Milestones:

Received: April 24, 2026
Accepted: June 04, 2026
Published: July 30, 2026

Authors:

Ramzi May
Department of Mathematics and Statistics, College of Science,
King Faisal University, Al-Ahsa, Kingdom of Saudi Arabia.

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