Abstract

We introduce a notion of averaged mappings in the broader class of \(\operatorname{CAT}(0)\) spaces. We call these mappings \(\alpha\)-firmly nonexpansive and develop basic calculus rules for ones that are quasi-\(\alpha\)-firmly nonexpansive and have a common fixed point. We show that the iterates \(x_n:=Tx_{n-1}\) of a nonexpansive mapping \(T\) converge weakly to an element in \(\operatorname{Fix} T\) whenever \(T\) is quasi-\(\alpha\)-firmly nonexpansive. Moreover, \(P_{\operatorname{Fix} T}x_n\) converge strongly to this weak limit. Our theory is illustrated with two classical examples of cyclic and averaged projections.

中文翻译：

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$$\operatorname{CAT}(0)$$ 空间中的平均映射概念

摘要

我们在更广泛的\(\operatorname{CAT}(0)\)空间类中引入了平均映射的概念。我们称这些映射为\(\alpha\) - 严格非扩张性的，并为准\(\alpha\) - 严格非扩张性且具有共同不动点的映射开发基本的微积分规则。我们证明了当\ ( T \ )为准\(\alpha\) - 完全不膨胀。此外，\(P_{\operatorname{Fix} T}x_n\)强烈收敛到这个弱极限。我们的理论用循环和平均投影的两个经典例子来说明。