Strong convergence theorems of the Halpern-Mann's mixed iteration for a totally quasi-φ-asymptotically nonexpansive nonself multi-valued mapping in Banach spaces

In this paper, we introduce a class of totally quasi-phi-asymptotically nonexpansive nonself multi-valued mapping to modify the Halpern-Mann-type iteration algorithm for a totally quasi-phi-asymptotically nonexpansive nonself multi-valued mapping, which has the strong convergence under a limit condition only in the framework of Banach spaces. Our results are applied to study the approximation problem of solution to a system of equilibrium problems. Also, the results presented in the paper improve and extend the corresponding results of Chang et al. (Appl. Math. Comput. 218: 7864-7870, 2012) and others.