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Zbl 1197.54069
Nanjaras, B.; Panyanak, B.
Demiclosed principle for asymptotically nonexpansive mappings in CAT(0) spaces.
(English)
[J] Fixed Point Theory Appl. 2010, Article ID 268780, 14 p. (2010). ISSN 1687-1812/e

A general demiclosed principle is established for asymptotically nonexpansive mappings in CAT(0) spaces. As a consequence, the following Krasnoselskii-Mann fixed point result is established.\par Theorem. Let $C$ be a bounded closed convex part of a complete CAT(0) space $(X,d)$ and $T:C\to C$ be asymptotically nonexpansive, with the sequence $(k_n)\subset [1,\infty)$ satisfying $\sum_{n=1}^\infty (k_n-1)< \infty$. Then, for each $x_1\in C$ and $(a_n)\subset (a,b)$ (with $0<a<b<1$), the iterative process $$x_{n+1}=a_nT^nx_n+(1-a_n)x_n,\quad n\ge 1,$$ $\Delta$-converges to a fixed point of $T$.
[Mihai Turinici (Iaşi)]
MSC 2000:
*54H25 Fixed-point theorems in topological spaces
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: geodesic metric space; CAT(0) property; demiclosed principle; asymptotically nonexpansive mapping; fixed point; Krasnoselskij-Mann iteration.

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