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

We study a perturbed version of Mann's iterative process (DMP), defined by \[ x_{n+1} = (1 - θ_n)x_n + θ_n f(x_n) + r_n, \] where $f: [0,1] \to [0,1]$ is a continuous function, $\{θ_n\} \subset [0,1]$ is a given sequence, and $\{r_n\} $ represents an error term. We prove that if the sequence $\{θ_n\