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

The Krasnoselskii-Mann iteration plays an important role in the approximation of fixed points of nonexpansive operators; it is 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.