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

Mean nonexpansive mappings were first introduced in 2007 by Goebel and Japon Pineda and advances have been made by several authors toward understanding their fixed point properties in various contexts. For any given $(\alpha_1, \alpha_2)$-nonexpansive mapping $T$ of a Banach space, many of the positive results have been derived from properties of the mapping $T_\alpha = \alpha_1 T + \alpha_2T^2= (\alpha_1I + \alpha_2T)\circ T$ which is nonexpansive. However, the related mapping $T \circ (\alpha_1I + \alpha_2T)$ has not yet been studied. In this paper, we investigate some fixed point properties of this new mapping and discuss relationships between $(\alpha_1I + \alpha_2T)\circ T$ and $T\circ(\alpha_1I + \alpha_2T)$.