Multi-valued totally quasi-ϕ-asymptotically nonexpansive semi-groups and strong convergence theorems in banach spaces

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