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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 ($\Vert Tx - Ty\Vert \le \Vert x - y\Vert , \ x, y \in K,$), quasi-nonexpansive ones ($\Vert Tx - Tx^*\Vert \le \Vert x - x^*\Vert , \ x \in K, x^* \in \text{ Fix} \, T, \ \text{ Fix} \, T \ne \emptyset$, asymptotically nonexpansive mappings ($\Vert T^nx - T^ny\Vert \le k_n\Vert x - y\Vert , \ x, y \in K, \ k_n \to 1$ or $\limsup_{n \to \infty} \ \sup_{x, y \in K} \,$ $\{\Vert T^nx - T^ny\Vert - \Vert x - y\Vert \} \le 0$), asymptotically quasi-nonexpansive ones ($\Vert T^nx - T^nx^*\Vert \le (1 + u_n)\Vert x - x^*\Vert , \ x \in K, x^* \in \text{ Fix} \, T, \ \text{ Fix} \, T \ne \emptyset, \ u_n \to 0$), pseudocontractive ones ($\langle Tx - Ty,j(x - y) \rangle \le \Vert x - y\Vert ^2, \ x, y \in K$, $j$ is a selection of the normalized duality mapping in $E$), and asymptotically pseudocontractive mappings ($\langle T^nx - T^ny,j(x - y) \rangle \le k_n\Vert x - y\Vert ^2, \ x, y \in K, \ k_n \to 1$). 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].

47-02Research monographs (operator theory)
47H09Mappings defined by “shrinking” properties
47H06Accretive operators, dissipative operators, etc. (nonlinear)
47J05Equations involving nonlinear operators (general)
47J25Iterative procedures (nonlinear operator equations)
54H25Fixed-point and coincidence theorems in topological spaces