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 T. Tao. Our approach covers examples ranging from the proximal point algorithm for maximal monotone operators to various fixed point iterations (xn) 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 W -hyperbolic spaces). Sometimes uniform convexity is assumed still covering the important class of CAT(0)-spaces due to Gromov.

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.

In this paper we introduce the concept of modulus of regularity as a tool to analyze the speed of convergence, including the finite termination, for classes of Fejér monotone sequences which appear in fixed point theory, monotone operator theory, and convex optimization. This concept allows for a unified approach to several notions such as weak sharp minima, error bounds, metric subregularity, Hölder regularity, etc., as well as to obtain rates of convergence for Picard iterates, the Mann algorithm, the proximal point algorithm and the cyclic algorithm. As a byproduct we obtain a quantitative version of the well-known fact that for a convex lower semi-continuous function the set of minimizers coincides with the set of zeros of its subdifferential and the set of fixed points of its resolvent.MSC: 41A25; 41A52; 41A65; 53C23; 03F10