Strong convergence for the modified Mann's iteration of $\hat{I}»$-strict pseudocontraction

Abstract

In this paper, for an $\hat{I}»$-strict pseudocontraction $T$, we prove strong convergence of the modified Mann's iteration defined by $$x_{n+1}={\hat{I}}_n^{2}u+{\hat{I}}_n^3{x}_n+(1-{\hat{I}}_n^2-{\hat{I}}_n^3)[\hat{I}{\pm }_{n}T{x}_n+(1-\hat{I}{\pm }_n)x_n],$$ where $\{\hat{I}{\pm }_n\}$, $\{{\hat{I}}_n^2\}$ and $\{{\hat{I}}_n^3\}$ in $(0,1)$ satisfy: (i) $0\le \hat{I}{\pm }_n\le \frac{\hat{I}}{»}{K}^2$ with $\underset{n\to \infty }{\mathrm{liminf}}\hat{I}{\pm }_n(\hat{I}»-K^2\hat{I}{\pm }_n)>0$; (ii) $\underset{n\to \infty }{\lim }{\hat{I}}_n^2=0$ and $\sum _{n=1}^{\infty }{\hat{I}}_n^2=\infty$; (iii) $\underset{n\to \infty }{\mathrm{limsup}}{\hat{I}}_n^3<1$.Our results unify and improve some existing results.

8 pages, 2014