  The proximal point algorithm is a widely used tool for solving a variety of
convex optimization problems such as finding zeros of maximally monotone
operators, fixed points of nonexpansive mappings, as well as minimizing convex
functions. The algorithm works by applying successively so-called "resolvent"
mappings associated to the original object that one aims to optimize. In this
paper we abstract from the corresponding resolvents employed in these problems
the natural notion of jointly firmly nonexpansive families of mappings. This
leads to a streamlined method of proving weak convergence of this class of
algorithms in the context of complete CAT(0) spaces (and hence also in Hilbert
spaces). In addition, we consider the notion of uniform firm nonexpansivity in
order to similarly provide a unified presentation of a case where the algorithm
converges strongly. Methods which stem from proof mining, an applied subfield
of logic, yield in this situation computable and low-complexity rates of
convergence.
