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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 $$T: \;K \to K$$ ($$K$$ is a nonempty subset of a Banach space $$E$$) of different types: nonexpansive ones ($$\| Tx - Ty\| \leq \| x - y\| , \;x, y \in K,$$), quasi-nonexpansive ones ($$\| Tx - Tx^*\| \leq \| x - x^*\| , \;x \in K, x^* \in \text{ Fix} \, T, \;\text{ Fix} \, T \neq \emptyset$$, asymptotically nonexpansive mappings ($$\| T^nx - T^ny\| \leq k_n\| x - y\| , \;x, y \in K, \;k_n \to 1$$ or $$\limsup_{n \to \infty} \;\sup_{x, y \in K} \,$$ $$\{\| T^nx - T^ny\| - \| x - y\| \} \leq 0$$), asymptotically quasi-nonexpansive ones ($$\| T^nx - T^nx^*\| \leq (1 + u_n)\| x - x^*\| , \;x \in K, x^* \in \text{ Fix} \, T, \;\text{ Fix} \, T \neq \emptyset, \;u_n \to 0$$), pseudocontractive ones ($$\langle Tx - Ty,j(x - y) \rangle \leq \| x - y\| ^2, \;x, y \in K$$, $$j$$ is a selection of the normalized duality mapping in $$E$$), and asymptotically pseudocontractive mappings ($$\langle T^nx - T^ny,j(x - y) \rangle \leq k_n\| x - y\| ^2, \;x, y \in K, \;k_n \to 1$$).
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.
For the entire collection see [Zbl 1030.00016].

##### MSC:
 47-02 Research exposition (monographs, survey articles) pertaining to operator theory 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H06 Nonlinear accretive operators, dissipative operators, etc. 47J05 Equations involving nonlinear operators (general) 47J25 Iterative procedures involving nonlinear operators 54H25 Fixed-point and coincidence theorems (topological aspects)