Generalized Fej\'er monotone sequences and their finitary content

We provide quantitative and abstract strong convergence results for sequences from a compact metric space satisfying a certain form of \emph{generalized Fej\'er monotonicity} where (1) the metric can be replaced by a much more general type of function measuring distances (including, in particular, certain Bregman distances), (2) full Fej\'er monotonicity is relaxed to a partial variant and (3) the distance functions are allowed to vary along the iteration. For such sequences, the paper provides explicit and effective rates of metastability and even rates of convergence, the latter under a regularity assumption that generalizes the notion of metric regularity introduced by Kohlenbach, L\'opez-Acedo and Nicolae, itself an abstract generalization of many regularity notions from the literature. In the second part of the paper, we apply the abstract quantitative results established in the first part to two algorithms: one algorithm for approximating zeros of maximally monotone and maximally $\rho$-comonotone operators in Hilbert spaces (in the sense of Combettes and Pennanen as well as Bauschke, Moursi and Wang) that incorporates inertia terms \emph{every other} term and another algorithm for approximating zeros of monotone operators in Banach spaces (in the sense of Browder) that is only Fej\'er monotone w.r.t.\ a certain Bregman distance.