Strong convergence theorems of total quasi-ϕ-asymptotically nonexpansive semi-groups in Banach spaces

The purpose of this article is first to introduce the concept of total quasi-ϕ-asymptotically nonexpansive semigroups which contains many kinds of semi-groups as its special cases, and then to modify the Halpern-Mann-type iteration algorithm for this kind of semi-groups to have the strong convergence under a limit condition only in the framework of Banach spaces. The results presented in the paper improve and extend the corresponding recent results announced by many authors.