Geodesic
On a notion of averaged mappings in $\operatorname{CAT}(0)$ spaces
Funkcionalʹnyj analiz i ego priloženiâ, Tome 56 (2022) no. 1, pp. 37-50

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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.
Keywords: averaged mapping, firmly nonexpansive mapping, $\operatorname{CAT}(0)$ space. 
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     author = {A. B\"erd\"ellima},
     title = {On a notion of averaged mappings in $\operatorname{CAT}(0)$ spaces},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {37--50},
     publisher = {mathdoc},
     volume = {56},
     number = {1},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2022_56_1_a2/}
}
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A. Bërdëllima. On a notion of averaged mappings in $\operatorname{CAT}(0)$ spaces. Funkcionalʹnyj analiz i ego priloženiâ, Tome 56 (2022) no. 1, pp. 37-50. http://geodesic.mathdoc.fr/item/FAA_2022_56_1_a2/