SOME NOTES ON $(\alpha, \beta)$-GENERALIZED HYBRID MAPPINGS

Abstract

In 1973, Bruck generalized the notion nonexpansive mappings by introducing firmly nonexpansive mappings. Kohsaka and Takahashi introduced nonspreading mappings in 2010. But, each nonexpansive mapping is a 1-hybrid mapping and each nonspreading mapping is a 0-hybrid mapping. Thus, the notion of $\lambda$-hybrid mappings is a generalization of the notions of firmly nonexpansive mappings and nonspreading mappings. In 2011, Takahashi introduced generalized hybrid mappings and proved some weak convergence theorems for generalized hybrid mappings in Banach spaces. On the other hand, Aoyama and Kohsaka introduced $\alpha$-nonexpansive mappings on Banach spaces in 2011 and proved some fixed point theorems for the mappings. Also, Kocourek, Takahashi and Yao provided the notions of $(\alpha, \alpha-1)$-generalized hybrid mappings and $(\alpha, \beta, \gamma)$-super hybrid mappings in 2011. In this paper, we give some results on $(\alpha, \beta)$-generalized hybrid mapping. Also, by using and combining ideas of some recent papers, we generalize the notion of $\alpha$-nonexpansivity to $(\alpha, \beta)$-nonexpansivity and give some results about the subject.