In this paper, we
introduce some new classes of the totally quasi-G-asymptotically nonexpansive
mappings and the totally quasi-G-asymptotically nonexpansive semigroups. Then,
with the generalized f-projection operator, we prove some strong convergence
theorems of a new modified Halpern type hybrid iterative algorithm for the
totally quasi-G-asymptotically nonexpansive semigroups in Banach space. The
results presented in this paper extend and improve some corresponding ones by
many others.

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