  We extend to $p$-uniformly convex spaces tools from the analysis of fixed
point iterations in linear spaces. This study is restricted to an appropriate
generalization of single-valued, pointwise $\alpha$-averaged mappings. Our main
contribution is establishing a calculus for these mappings in p-uniformly
convex spaces, showing in particular how the property is preserved under
compositions and convex combinations. This is of central importance to
splitting algorithms that are built by such convex combinations and
compositions, and reduces the convergence analysis to simply verifying
$\alpha$-firm nonexpansiveness of the individual components at fixed points of
the splitting algorithms. Our convergence analysis differs from what can be
found in the previous literature in that only $\alpha$-firm nonexpansiveness
with respect to fixed points is required. Indeed we show that, if the fixed
point mapping is pointwise nonexpansive at all cluster points, then these
cluster points are in fact fixed points, and convergence of the sequence
follows. Additionally, we provide a quantitative convergence analysis built on
the notion of gauge metric subregularity, which we show is necessary for
quantifiable convergence estimates. This allows one for the first time to prove
convergence of a tremendous variety of splitting algorithms in spaces with
curvature bounded from above.
