In 1973, Bruck generalized the notion nonexpansive mappings by introducing firmly nonexpansive mappings. Kohsaka and Takahashi introduced nonspreading mappings in 2008 and Takahashi introduced hybrid mappings in 2010. It is worth noting that each nonexpansive mapping is a 1-hybrid mapping and each nonspreading mapping is a 0-hybrid mapping. Thus, the notion of λ-hybrid mappings is a generalization of the notions of firmly nonexpansive mappings and nonspreading mappings. In 2011, Takahashi introduced generalized hybrid mappings and Aoyama and Kohsaka defined α-nonexpansive mappings on Banach spaces. Kocourek, Takahashi and Yao gave the notions of (α, α − 1)- generalized hybrid mappings and (α, β, γ)-super hybrid mappings. In this paper, we discuss (α, β)-generalized hybrid mappings. By using and combining ideas of some recent papers, we generalize the notion of α-nonexpansivity to (α, β)-nonexpansivity and give some results on the subject.
