Fixed point results for a new mapping related to mean nonexpansive mappings

Abstract

Mean nonexpansive mappings were first introduced in 2007 by Goebel and Japón Pineda and advances have been made by several authors toward understanding their fixed point properties in various contexts. For any given mean nonexpansive mapping of a Banach space, many of the positive results have been derived from knowing that a certain average of some iterates of the mapping is nonexpansive. However, nothing is known about the properties of a mean nonexpansive mapping which has been averaged with the identity. In this paper we prove some fixed point results for a mean nonexpansive mapping which has been composed with a certain average of itself and the identity and we use this study to draw connections to the original mapping.