Strong convergence for the modified Mann's iteration of $λ$-strict pseudocontraction
Yisheng Song, Hongjun Wang
TL;DR
This paper proves strong convergence of the modified Mann’s iteration of the λ -strict pseudocontraction T, which unify and improve some existing results.
Abstract
In this paper, for an $λ$-strict pseudocontraction $T$, we prove strong convergence of the modified Mann's iteration defined by $$x_{n+1}=β_{n}u+γ_nx_n+(1-β_{n}-γ_n)[α_{n}Tx_n+(1-α_{n})x_n],$$ where $\{α_{n}\}$, $ \{β_{n}\}$ and $\{γ_n\}$ in $(0,1)$ satisfy: (i) $0 \leq α_{n}\leq \fracλ{K^2}$ with $\liminf\limits_{n\to\infty}α_n(λ-K^2α_n)> 0$; (ii) $\lim\limits_{n\to\infty}β_n= 0$ and $\sum\limits_{n=1}^\inftyβ_n=\infty$; (iii) $\limsup\limits_{n\to\infty}γ_n<1$.Our results unify and improve some existing results.