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Demiclosed principle for asymptotically nonexpansive mappings in CAT(0) spaces. (English) Zbl 1197.54069

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.

Theorem. Let $C$ be a bounded closed convex part of a complete CAT(0) space $\left(X,d\right)$ and $T:C\to C$ be asymptotically nonexpansive, with the sequence $\left({k}_{n}\right)\subset \left[1,\infty \right)$ satisfying ${\sum }_{n=1}^{\infty }\left({k}_{n}-1\right)<\infty$. Then, for each ${x}_{1}\in C$ and $\left({a}_{n}\right)\subset \left(a,b\right)$ (with $0), the iterative process

${x}_{n+1}={a}_{n}{T}^{n}{x}_{n}+\left(1-{a}_{n}\right){x}_{n},\phantom{\rule{1.em}{0ex}}n\ge 1,$

${\Delta }$-converges to a fixed point of $T$.

MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties
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