A note on the Krasnoselski–Mann theorem and its generalizations

Abstract

In this paper, we investigate the convergence behaviour of the Krasnoselski–Mann (KM) iteration and its generalizations. The KM iteration may be written as follows: where N is a nonexpansive operator on a Hilbert space . This scheme aims to find fixed points of the operator N. Many problems from various fields, including the inverse problems area, can be expressed as a fixed point problem of a certain operator N. In earlier articles, the convergence of the KM iteration and its generations have been investigated in the case when the operator N is nonexpansive and has the fixed points. This paper further studies the convergence behaviour of the algorithms discussed. We first extend the convergence result for the KM iteration to the case when the operator N is firmly nonexpansive, in which case the relaxation parameters are allowed to be in the interval [0, 2], instead of [0, 1]. Then, we show that this result remains valid for the generalized KM iterations. Furthermore, we prove that the sequences generated from the KM iteration or the generalized KM iterations are unbounded if the operators related to the iterations have no fixed points.