On a Notion of Averaged Mappings in \(\operatorname{CAT}(0)\) Spaces

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.