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Iterative algorithms for nonexpansive mappings and some of their generalizations. (English) Zbl 1057.47003
Agarwal, Ravi P. (ed.) et al., Nonlinear analysis and applications: To V. Lakshmikantham on his 80th birthday. Vol. 1. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1711-1/hbk). 383-429 (2003).

This article is a small survey of recent and some new results concerning iterative algorithms for approximating fixed points of mappings $T:\phantom{\rule{4pt}{0ex}}K\to K$ ($K$ is a nonempty subset of a Banach space $E$) of different types: nonexpansive ones ($\parallel Tx-Ty\parallel \le \parallel x-y\parallel ,\phantom{\rule{4pt}{0ex}}x,y\in K,$), quasi-nonexpansive ones ($\parallel Tx-T{x}^{*}\parallel \le \parallel x-{x}^{*}\parallel ,\phantom{\rule{4pt}{0ex}}x\in K,{x}^{*}\in \phantom{\rule{4.pt}{0ex}}\text{Fix}\phantom{\rule{0.166667em}{0ex}}T,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{Fix}\phantom{\rule{0.166667em}{0ex}}T\ne \varnothing$, asymptotically nonexpansive mappings ($\parallel {T}^{n}x-{T}^{n}y\parallel \le {k}_{n}\parallel x-y\parallel ,\phantom{\rule{4pt}{0ex}}x,y\in K,\phantom{\rule{4pt}{0ex}}{k}_{n}\to 1$ or ${lim sup}_{n\to \infty }\phantom{\rule{4pt}{0ex}}{sup}_{x,y\in K}\phantom{\rule{0.166667em}{0ex}}$ $\left\{\parallel {T}^{n}x-{T}^{n}y\parallel -\parallel x-y\parallel \right\}\le 0$), asymptotically quasi-nonexpansive ones ($\parallel {T}^{n}x-{T}^{n}{x}^{*}\parallel \le \left(1+{u}_{n}\right)\parallel x-{x}^{*}\parallel ,\phantom{\rule{4pt}{0ex}}x\in K,{x}^{*}\in \phantom{\rule{4.pt}{0ex}}\text{Fix}\phantom{\rule{0.166667em}{0ex}}T,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{Fix}\phantom{\rule{0.166667em}{0ex}}T\ne \varnothing ,\phantom{\rule{4pt}{0ex}}{u}_{n}\to 0$), pseudocontractive ones ($〈Tx-Ty,j\left(x-y\right)〉\le {\parallel x-y\parallel }^{2},\phantom{\rule{4pt}{0ex}}x,y\in K$, $j$ is a selection of the normalized duality mapping in $E$), and asymptotically pseudocontractive mappings ($〈{T}^{n}x-{T}^{n}y,j\left(x-y\right)〉\le {k}_{n}{\parallel x-y\parallel }^{2},\phantom{\rule{4pt}{0ex}}x,y\in K,\phantom{\rule{4pt}{0ex}}{k}_{n}\to 1$).

More than three dozen results about fixed points and the strong and weak convergence of different iterative algorithms are presented here with a sufficiently deep and sophisticated analysis of approaches and proofs of different authors (the bibliography consists of 128 items). The last section is devoted to some remarks concerning the Mann and Ishikawa methods with errors and other Ishikawa-type methods.

##### MSC:
 47-02 Research monographs (operator theory) 47H09 Mappings defined by “shrinking” properties 47H06 Accretive operators, dissipative operators, etc. (nonlinear) 47J05 Equations involving nonlinear operators (general) 47J25 Iterative procedures (nonlinear operator equations) 54H25 Fixed-point and coincidence theorems in topological spaces