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Zbl 1220.47100
Kim, Tae-Hwa; Xu, Hong-Kun
Convergence of the modified Mann's iteration method for asymptotically strict pseudo-contractions.
(English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 68, No. 9, A, 2828-2836 (2008). ISSN 0362-546X

Summary: Let $C$ be a closed convex subset of a real Hilbert space $H$ and assume that $T$ is an asymptotically $\kappa$-strict pseudo-contraction on $C$ with a fixed point, for some $0\le \kappa<1$. Given an initial guess $x_0\in C$ and a real sequence $\{\alpha_n\}$ in $(0,1)$, the modified Mann algorithm generates a sequence $\{x_n\}$ via the formula: $x_{n+1}=\alpha_nx_n+(1-\alpha_n)T^nx_n$, $n\ge 0$. It is proved that if the control sequence $\{\alpha_n\}$ is chosen so that $\kappa+\delta<\alpha_n-\delta$ for some $\delta\in(0,1)$, then $\{x_n\}$ converges weakly to a fixed point of $T$. We also modify this iteration method by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strongly convergent sequence.
MSC 2000:
*47J25 Methods for solving nonlinear operator equations (general)
47H09 Mappings defined by "shrinking" properties
65J15 Equations with nonlinear operators (numerical methods)

Keywords: asymptotically strict pseudocontraction; modified Mann iteration method; weak convergence; strong convergence; fixed point; projection

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