On $\alpha$-Firmly Nonexpansive Operators in $r$-Uniformly Convex Spaces

We introduce the class of $\alpha$-firmly nonexpansive and quasi $\alpha$-firmly nonexpansive operators on $r$-uniformly convex Banach spaces. This extends the existing notion from Hilbert spaces, where $\alpha$-firmly nonexpansive operators coincide with so-called $\alpha$-averaged operators. For our more general setting, we show that $\alpha$-averaged operators form a subset of $\alpha$-firmly nonexpansive operators. We develop some basic calculus rules for (quasi) $\alpha$-firmly nonexpansive operators. In particular, we show that their compositions and convex combinations are again (quasi) $\alpha$-firmly nonexpansive. Moreover, we will see that quasi $\alpha$-firmly nonexpansive operators enjoy the asymptotic regularity property. Then, based on Browder's demiclosedness principle, we prove for $r$-uniformly convex Banach spaces that the weak cluster points of the iterates $x_{n+1}:=Tx_{n}$ belong to the fixed point set $\text{Fix} T$ whenever the operator $T$ is nonexpansive and quasi $\alpha$-firmly. If additionally the space has a Fr\'echet differentiable norm or satisfies Opial's property then these iterates converge weakly to some element in $\text{Fix} T$. Further, the projections $P_{\text{Fix} T}x_n$ converge strongly to this weak limit point. Finally, we give three illustrative examples, where our theory can be applied, namely from infinite dimensional neural networks, semigroup theory, and contractive projections in $L_p$, $p \in (1,\infty) \backslash \{2\}$ spaces on probability measure spaces.