Quantitative results on Fejér monotone sequences

Abstract

We provide in a unified way quantitative forms of strong convergence results for numerous iterative procedures which satisfy a general type of Fejér monotonicity where the convergence uses the compactness of the underlying set. These quantitative versions are in the form of explicit rates of so-called metastability in the sense of Tao. Our approach covers examples ranging from the proximal point algorithm for maximal monotone operators to various fixed point iterations [Formula: see text] for firmly nonexpansive, asymptotically nonexpansive, strictly pseudo-contractive and other types of mappings. Many of the results hold in a general metric setting with some convexity structure added (so-called [Formula: see text]-hyperbolic spaces). Sometimes uniform convexity is assumed still covering the important class of CAT(0)-spaces due to Gromov.