Quasi-nonexpansive mappings with respect to orbits in Banach spaces and weak fixed point property

In the present work, we introduce quasi-nonexpansive mappings with respect to orbits on the Banach space. Then we show that a Banach space $\mathcal{A}$ has weak normal structure if and only if $\mathcal{A}$ has the weak fixed point property for quasi-nonexpansive mappings with respect to orbits.