Strong convergence for the modified Mann's iteration of $\lambda$-strict pseudocontraction

TL;DR

This paper proves strong convergence of a modified Mann's iteration for $\lambda$-strict pseudocontractions, extending and improving previous results in fixed point theory.

Contribution

It establishes new convergence results for a modified Mann's iteration under specific parameter conditions for $\lambda$-strict pseudocontractions.

Findings

Proves strong convergence of the iteration under given conditions.

Unifies and extends existing convergence results.

Provides parameter conditions ensuring convergence.

Abstract

In this paper, for an $\lambda$-strict pseudocontraction $T$, we prove strong convergence of the modified Mann's iteration defined by $$x_{n+1}=\beta_{n}u+\gamma_nx_n+(1-\beta_{n}-\gamma_n)[\alpha_{n}Tx_n+(1-\alpha_{n})x_n],$$ where $\{\alpha_{n}\}$, $ \{\beta_{n}\}$ and $\{\gamma_n\}$ in $(0,1)$ satisfy: (i) $0 \leq \alpha_{n}\leq \frac{\lambda}{K^2}$ with $\liminf\limits_{n\to\infty}\alpha_n(\lambda-K^2\alpha_n)> 0$; (ii) $\lim\limits_{n\to\infty}\beta_n= 0$ and $\sum\limits_{n=1}^\infty\beta_n=\infty$; (iii) $\limsup\limits_{n\to\infty}\gamma_n<1$.Our results unify and improve some existing results.