  In this paper we provide a unified treatment of some convex minimization
problems, which allows for a better understanding and, in some cases,
improvement of results in this direction proved recently in spaces of curvature
bounded above. For this purpose, we analyze the asymptotic behavior of
compositions of finitely many firmly nonexpansive mappings in the setting of
$p$-uniformly convex geodesic spaces focusing on asymptotic regularity and
convergence results.
