Rates of (T-)asymptotic regularity of the generalized Krasnoselskii-Mann-type iteration
Paulo Firmino, Laurentiu Leustean
TL;DR
This work derives uniform rates of ($T$-)asymptotic regularity for the generalized Krasnoselskii-Mann-type iteration $x_{n+1}=\alpha_n x_n+\beta_n T x_n + r_n$ in uniformly convex spaces with nonexpansive $T$ and fixed points. By applying proof mining and the space's modulus of uniform convexity $\eta$, the authors obtain explicit rate functions $\Phi$ (for $\|x_n-Tx_n\|$) and $\Psi$ (for $\|x_{n+1}-x_n\|$), under quantitative parameter hypotheses (C1)–(C3); in Hilbert spaces and under certain $\eta$-decompositions these rates become quadratic. The paper includes two illustrative examples that yield concrete, computable rates, including quadratic growth in the Hilbert-space setting, and discusses how special choices recover known Krasnoselskii-Mann results. Overall, the results provide the first uniform, computable rates for this generalized iteration, enabling quantitative convergence guarantees and informing stopping criteria in nonlinear fixed-point algorithms.
Abstract
In this paper we use proof mining methods to compute rates of ($T$-)asymptotic regularity of the generalized Krasnoselskii-Mann-type iteration associated to a nonexpansive mapping $T:X\to X$ in a uniformly convex normed space $X$. For special choices of the parameter sequences, we obtain quadratic rates.