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.