Generalized Krasnoselskii–Mann-type iterations for nonexpansive mappings in Hilbert spaces

Abstract

The Krasnoselskii---Mann iteration plays an important role in the approximation of fixed points of nonexpansive operators; it is known to be weakly convergent in the infinite dimensional setting. In this present paper, we provide a new inexact Krasnoselskii---Mann iteration and prove weak convergence under certain accuracy criteria on the error resulting from the inexactness. We also show strong convergence for a modified inexact Krasnoselskii---Mann iteration under suitable assumptions. The convergence results generalize existing ones from the literature. Applications are given to the Douglas---Rachford splitting method, the Fermat---Weber location problem as well as the alternating projection method by John von Neumann.