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Zbl 0747.47041
Xu, Hong-Kun
Existence and convergence for fixed points of mappings of asymptotically nonexpansive type.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 16, No.12, 1139-1146 (1991). ISSN 0362-546X

This article deals with mappings $T:C\to C$ of asymptotically nonexpansive type of a nonempty closed convex subset in a uniformly convex Banach space $X$ or, in other words, with mappings for which $$\varlimsup\sb{n\to\infty}\sup\sb{y\in C}(\Vert T\sp nx-T\sp ny\Vert- \Vert x-y\Vert)\le 0.$$ The main result is a fixed point theorem for such mappings and a theorem on the weak convergence of the Picard approximations to a fixed point.
[P.Zabreiko (Minsk)]
MSC 2000:
*47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47J25 Methods for solving nonlinear operator equations (general)
47H09 Mappings defined by "shrinking" properties

Keywords: mappings of asymptotically nonexpansive type; uniformly convex Banach space; fixed point theorem; weak convergence of the Picard approximations

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